# Quantum Uncertainty Principles for Measurements with Interventions
Supplemental Material

Yunlong Xiao<EMAIL_ADDRESS>Institute of High Performance Computing
(IHPC), Agency for Science Technology and Research (A*STAR), 1 Fusionopolis
Way, #16-16 Connexis, Singapore 138632, Republic of Singapore  Nanyang Quantum
Hub, School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore 637371, Singapore  Yuxiang Yang<EMAIL_ADDRESS>QICI
Quantum Information and Computation Initiative, Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong  Institute for
Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland  Ximing Wang
Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Qing Liu  Nanyang
Quantum Hub, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  Mile Gu
<EMAIL_ADDRESS>Nanyang Quantum Hub, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  Complexity Institute, Nanyang Technological University, Singapore
639673, Singapore

###### Abstract

In this supplemental material, we formulate several uncertainty relations for
measurements with interventions, extending the results presented in the main
text of our work. As a by-product, we have derived causal uncertainty
relations for quantum dynamics, establishing a trade-off between common-cause
and direct-cause in quantum causal inference. Such a fundamental trade-off
have been further utilized to infer the causality associated with
parameterized quantum circuits, which are the essential building blocks for
Noisy Intermediate-Scale Quantum (NISQ) technologies. Detailed analyses and
proofs of the results presented in the main text have also been included. More
precisely, to systemically investigate the most general quantum dynamics with
definite causal orders and the corresponding measuring processes with
interventions, we introduce the framework of _quantum circuit fragments_ and
_interactive measurements_ in Sec. I. The uncertainty principle for multiple
interactive measurements has been demonstrated in Sec. II, which works for any
quantum circuit fragments. We have further developed the causal uncertainty
relation and detailed its application to causal inference in Sec. III.
Finally, numerical experiments for our results have been provided in Sec. IV.
It is worth noting that we may reiterate some of the steps in the main text to
make this supplemental material more explicit and self-contained.

###### Contents

  1. I Quantum Circuit Fragment
    1. I.1 Quantum Channels and Superchannels
    2. I.2 Quantum Circuit Fragments: Multiple Quantum Processes with Definite Causal Order
    3. I.3 Interactive Measurements: How to Measure the Quantum Circuit Fragments?
    4. I.4 Quantum Causal Maps
  2. II Universal Uncertainty Relation for Measurements with Interventions
    1. II.1 Mathematical Toolkit: Majorization Lattice
    2. II.2 Brief Introduction to Uncertainty Relations
    3. II.3 Operational Interpretation of Universal Uncertainty Relation: Quantum Roulette
    4. II.4 Lemma $1$ and Theorem $1$ of the Main Text: Their Proofs, Improvements, and Generalizations
  3. III Causal Uncertainty Relation
    1. III.1 Uncertainty Relation for Common-Cause and Direct-Cause: Eq. $3$ of the Main Text and Its Extension
    2. III.2 Necessary and Sufficient Conditions for Common-Cause and Direct-Cause
    3. III.3 Coherent Mixture of Common-Cause and Direct-Cause
  4. IV Numerical Experiments
    1. IV.1 The Landscape of Joint Uncertainty
    2. IV.2 Advantage in Inferring Causal Structures

##  I Quantum Circuit Fragment

In this section, we introduce our notations and prepare the groundwork for
results presented in the main text of our work. Additionally, the general
quantum dynamics and the corresponding measuring processes with interventions
have been rigorously formulated as _quantum circuit fragments_ and
_interactive measurements_. This preparatory section contains four
subsections: In Subsec. I.1, we give a brief introduction to the concept of
quantum channels, superchannels, and their Choi-Jamiołkowski operators.
Subsequently, in Subsec. I.2, we move on to quantum dynamics with definite
causal order, and formally introduce the concept of quantum circuit fragments.
The most general measuring processes for quantum circuit fragments –
interactive measurements – have been constructed in Subsec. I.3. Finally, a
special type of quantum circuit fragments, known as causal maps in quantum
causal inference, has been discussed in the last subsection, i.e. Subsec. I.4.

###  I.1 Quantum Channels and Superchannels

The fundamental building blocks that makeup quantum technologies and quantum
information processing are quantum channels. Physically, the preparation of
quantum states, the implementation of quantum measurements, the noise arising
from the system-environment interactions, and the operation (including quantum
gates) carried out on quantum devices are all characterized by the concept of
quantum channel. In this subsection, we will give a brief introduction to the
concept of the quantum channel and the corresponding mathematical toolkit. For
detailed consideration of the quantum channels and their mathematical
properties, we refer the reader to Refs. [1, 2, 3].

A linear map $\mathcal{E}$ from system $A$ to system $B$ is called a quantum
channel if it is both completely positive (CP) and trace-preserving (TP). The
property of completely positive implies that by applying quantum channels to
part of a quantum system, the resultant system is still a well-defined quantum
system. Meanwhile, as indicated by the name, the trace-preserving property
guarantees that any channel’s output state still has trace $1$, when the input
is a quantum state.

Instead of dealing with a map (i.e. quantum channel) directly, usually we
prefer to investigate the properties of its matrix representation. To do so,
let us start with the concept of Choi-Jamiołkowski (CJ) operators [4, 5].
Formally, it is defined as

Definition I.1:  Choi-Jamiołkowski (CJ) Operator [4, 5] For a quantum channel
$\mathcal{E}:A\to B$, its Choi-Jamiołkowski operator $J^{\mathcal{E}}_{AB}$ is
defined as $\displaystyle
J^{\mathcal{E}}_{AB}:=\text{id}_{A}\otimes\mathcal{E}_{A^{{}^{\prime}}\to
B}(\outerproduct{I}{I}_{AA^{{}^{\prime}}}),$ (1) where
$\ket{I}_{AA^{{}^{\prime}}}:=\sum_{i}\ket{i}_{A}\ket{i}_{A^{{}^{\prime}}}$ is
the unnormalized maximally entangled state with $A^{{}^{\prime}}$ being a
replica of system $A$, and $\\{\ket{i}\\}$ being an orthonormal basis on $A$.
See Fig. 1 for an illustration of Eq. 1.

Figure 1: (color online) Circuit realization of generating the CJ operator
$J^{\mathcal{E}}_{AB}$ (see Eq. 1). Here $I_{AA^{{}^{\prime}}}$ stands for
$\outerproduct{I}{I}_{AA^{{}^{\prime}}}$, and $\ket{I}_{AA^{{}^{\prime}}}$
represents the unnormalized maximally entangled state
$\sum_{i}\ket{i}_{A}\ket{i}_{A^{{}^{\prime}}}$.

Here the correspondence between a quantum channel $\mathcal{E}$ and its CJ
operator $J^{\mathcal{E}}$ is referred as the Choi-Jamiołkowski isomorphism in
quantum information theory [1, 2, 3]. For any quantum state $\rho$ acting on
system $A$, and a quantum channel $\mathcal{E}:A\to B$, the output state is
fully characterized by the CJ operator $J^{\mathcal{E}}$. More precisely, the
output on system $b$ can be written as

$\displaystyle\mathcal{E}(\rho)=\Tr_{A}[J^{\mathcal{E}}_{AB}\cdot\rho^{\mathbf{T}}_{A}\otimes\mathbb{1}_{B}].$
(2)

Using the language of CJ operators, we know that $\mathcal{E}$ is (i)
completely positive (CP) if and only if its CJ operator satisfies
$J^{\mathcal{E}}\geqslant 0$ and (ii) trace-preserving (TP) if and only if its
CJ operator satisfies $\Tr_{B}[J^{\mathcal{E}}_{AB}]=\mathbb{1}_{A}$. Here $A$
and $B$ represent the input and output systems of $\mathcal{E}$ respectively.
The notation $\Tr_{A}$ stands for the partial trace over system $A$.

The development of quantum telecommunications and networks requires the
investigations of quantum channels with multiple inputs and outputs at
different time ticks, which comply with the causality in the theory of
relativity. More precisely, in the theory of quantum information, such a
causality is restricted by the concept of non-signaling (NS), which is
formally defined as

Definition I.2:  Non-Signaling (NS) Given a bipartite channel
$\mathcal{E}:AC\to BD$, it is called non-signaling from $C\to D$ to $A\to B$
if the following condition is satisfied
$\displaystyle\Tr_{D}\circ\,\mathcal{E}_{AC\to BD}=\mathcal{F}_{A\to
B}\otimes\Tr_{C},$ (3) for some quantum channel $\mathcal{F}$ from $A$ to $B$.
On the other hand, we say $\mathcal{E}$ is non-signaling from $A\to B$ to
$C\to D$, if there exists a quantum channel $\mathcal{G}$ from $C$ to $D$ such
that $\displaystyle\Tr_{B}\circ\,\mathcal{E}_{AC\to BD}=\mathcal{G}_{C\to
D}\otimes\Tr_{A}.$ (4) Finally, the bipartite channel $\mathcal{E}$ is said to
be non-signaling if it meets the condition of both Eq. 3 and Eq. 4.

Noted that in Def. I.1, Eq. 3 means that for any bipartite input state
$\rho_{AC}$ acting on systems $AC$, we have

$\displaystyle\Tr_{D}\circ\,\mathcal{E}_{AC\to
BD}(\rho_{AC})=\Tr_{D}[\mathcal{E}_{AC\to BD}(\rho_{AC})]=\mathcal{F}_{A\to
B}\otimes\Tr_{C}(\rho_{AC})=\mathcal{F}_{A\to B}(\rho_{A}),$ (5)

where $\rho_{A}:=\Tr_{C}[\rho_{AC}]$ is the reduced state of $\rho_{AC}$ on
system $A$. A similar situation applies to Eq. 4, as shown in Fig. 2.

Figure 2: (color online) Pictorial demonstrations of non-signaling (NS)
bipartite quantum channel $\mathcal{E}$ from systems $AC$ to $BD$: (a) NS from
$C\to D$ to $A\to B$ (see Eq. 3); (b) NS from $A\to B$ to $C\to D$ (see Eq.
4).

Physically, Eq. 3 describes a situation where $C\to D$ is a quantum dynamical
process that happened after the process of $A\to B$. Hence, information cannot
be transmitted from the future (i.e. $C\to D$) to the past (i.e. $A\to B$). A
similar statement holds for Eq. 4, where the temporal order for $C\to D$ and
$A\to B$ has been switched.

The non-signaling condition also plays an important role in the composition of
dynamical processes. Let us consider a concrete example. Assume a quantum
channel $\mathcal{E}_{1}:A\to BE$ is followed by another channel
$\mathcal{E}_{2}:CE\to D$, where the two channels are connected by a memory
system $E$, as illustrated in Fig. 3. In this case, the whole quantum dynamics
$\mathcal{E}:=\mathcal{E}_{2}\circ\mathcal{E}_{1}$ is a linear map from $AC$
to $BD$, satisfying the following conditions:

  1. (i)

Completely Positive (CP),

  2. (ii)

Trace-Preserving (TP),

  3. (iii)

Non-Signaling (NS) from $C\to D$ to $A\to B$.

Figure 3: (color online) Sequential composition of quantum channels
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$, i.e.
$\mathcal{E}:=\mathcal{E}_{2}\circ\mathcal{E}_{1}$, where $\mathcal{E}_{1}$ is
a channel from system $A$ to $BE$, and meanwhile $\mathcal{E}_{2}$ is a
channel from systems $CE$ to $D$. System $E$ is the link that connects them
together.

Denote the CJ operators of quantum channels $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ as $J^{1}_{ABE}$ and $J^{2}_{CDE}$ (or simply $J^{1}$ and
$J^{2}$) respectively. It now follows immediately that the CJ operator of
$\mathcal{E}=\mathcal{E}_{2}\circ\mathcal{E}_{1}$ is given by [6]

$\displaystyle J^{\mathcal{E}}=\Tr_{E}[(J^{1})^{\mathbf{T}_{E}}\cdot J^{2}],$
(6)

where $\mathbf{T}_{E}$ stands for the partial transpose over system $E$. Here
the non-signaling condition indicates the temporal order between processes
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. As a by-product, Eq. 6 motivates the
abstract definition of link product $\star$ [7, 6].

Definition I.3:  Link Product [7, 6] Given two operators $M$ and $N$ acting on
systems $XY$ and $YZ$ respectively, we define their link product $M\star N$ as
$\displaystyle M\star N:=\Tr_{Y}[M^{\mathbf{T}_{Y}}\cdot N],$ (7) where the
common space $Y$ is appeared as the link between operators $M$ and $N$, and
has been “swallowed” by the product $\star$. Here $\Tr_{Y}$ and
$\mathbf{T}_{Y}$ represent the partial trace and partial transpose over the
system $Y$ respectively.

Thanks to link product, now Eq. 2 can be simplified as

$\displaystyle\mathcal{E}(\rho)=J^{\mathcal{E}}\star\rho.$ (8)

Equipped with the link product, now we can rewrite the CJ operator of
$\mathcal{E}=\mathcal{E}_{2}\circ\mathcal{E}_{1}$ as

$\displaystyle J^{\mathcal{E}}=J^{1}\star J^{2}.$ (9)

Here $\mathcal{E}=\mathcal{E}_{2}\circ\mathcal{E}_{1}$ is a typical example of
quantum superchannel, where channels $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$
are known as the pre-processing and post-processing of $\mathcal{E}$
respectively. Similar to the case of quantum channels, all above restrictions
on $\mathcal{E}$ can be translated into the language of CJ operators; a map
$\mathcal{E}$ from $AC$ to $BD$ is (i) completely positive (CP) if and only if
$J^{\mathcal{E}}\geqslant 0$, (ii) trace-preserving (TP) if and only if
$\Tr_{BD}[J^{\mathcal{E}}]=\mathds{1}_{AC}$, and (iii) non-signaling (NS) from
$C\to D$ to $A\to B$ if and only if
$\Tr_{D}[J^{\mathcal{E}}]=\Tr_{CD}[J^{\mathcal{E}}]\otimes\mathds{1}_{C}/d_{C}$
with $d_{C}:=\dim C$. Note that here $\Tr_{CD}[J^{\mathcal{E}}]/d_{C}$ forms a
CJ operator for some quantum channel, as it is both completely positive (CP)
and trace-preserving (TP).

In fact, all quantum superchannels satisfy above conditions, namely completely
positive (CP), trace-preserving (TP), and non-signaling (NS) from one process
to another. Furthermore, the inverse statement – a linear map satisfying
completely positive (CP), trace-preserving (TP), and non-signaling (NS) from
one process to another forms a quantum superchannel – is also true [8].
Mathematically, given a bipartite quantum channel $\Phi:AC\to BD$ and non-
signaling (NS) from $C\to D$ to $A\to B$, there exist two quantum channels –
$\Psi^{\text{Pre}}$ (known as pre-processing) and $\Psi^{\text{Post}}$ (known
as post-processing) – such that for any quantum channel $\mathcal{E}$ from
system $B$ to system $C$ and quantum state $\rho$ acting on system $A$, we
have

$\displaystyle\Phi(\mathcal{E})(\rho)=\Psi^{\text{Post}}\circ\mathcal{E}\circ\Psi^{\text{Pre}}(\rho).$
(10)

Here $\Phi$ represents a general manipulation of quantum channels with
definite causal order, which maps a quantum channel of form $B\to C$ to a
resultant channel of form $A\to D$, visualizing in Fig. 4.

Figure 4: (color online) Physical realization of the quantum superchannel
$\Phi=\Psi^{\text{Post}}\circ\Psi^{\text{Pre}}$, whose action on input channel
is described by Eq. 10. Here channel $\Psi^{\text{Pre}}$ stands for the pre-
processing of superchannel $\Phi$, channel $\Psi^{\text{Post}}$ represents the
post-processing of superchannel $\Phi$, and they are connected through the
memory system $E$.

Remark that, on the one hand, a quantum channel can be viewed as a special
kind of superchannel with only one process (either pre- or post-), and no
quantum memory system. On the other hand, quantum superchannels can also be
recognised as bipartite quantum channels with additional conditions, namely NS
from a later process to early one. From this perspective, it is
straightforward to formulate the CJ operator of a superchannel, such as
$\Phi:AC\to BD$.

$\displaystyle
J^{\Phi}_{ABCD}:=\text{id}_{AC}\otimes\Phi_{A^{{}^{\prime}}C^{{}^{\prime}}\to
BD}(\outerproduct{I}{I}_{AA^{{}^{\prime}}}\otimes\outerproduct{I}{I}_{CC^{{}^{\prime}}}),$
(11)

where $\ket{I}_{AA^{{}^{\prime}}}$ and $\ket{I}_{CC^{{}^{\prime}}}$ are
unnormalized maximally entangled states acting on systems $AA^{{}^{\prime}}$
and $CC^{{}^{\prime}}$ respectively. For a quantum channel $\mathcal{E}:B\to
C$, its resultant channel under $\Phi$ has the following CJ operator,

$\displaystyle J^{\Phi(\mathcal{E})}=J^{\Phi}\star J^{\mathcal{E}},$ (12)

where $J^{\mathcal{E}}$ stands for the CJ operator of the channel
$\mathcal{E}$.

###  I.2 Quantum Circuit Fragments: Multiple Quantum Processes with Definite
Causal Order

Figure 5: (color online) Quantum circuit fragments and interactive
measurements: (a) Passive measurement, where the system described by $\rho$ is
left to evolve before feeding into the measuring process $M:=\\{M_{x}\\}_{x}$.
(b) Measuring quantum dynamics at two different time points. To investigate
the system of $\rho$ and its dynamics $\Psi$, we have executed a quantum
instrument $\overline{M}$ at time point $t_{1}$, followed by a quantum
measurements $M$ taken at time point $t_{2}$, where $t_{1}<t_{2}$. The
measurement outcome obtained at time $t_{1}$ and $t_{2}$ are denoted as
$x_{1}$ and $x_{2}$ respectively. In this example, the entire quantum
dynamics, including the preparation of initial state $\rho$ and quantum
evolution $\Psi$, is marked with blue color. Meanwhile, the adaptive measuring
protocol forms a simple example of interactive measurement, which is colored
black. (c) Correlated quantum dynamics and measurement with interventions.
Based on previous measurement outcome $x_{1}$, an intervention
$\mathcal{F}_{x_{1}}$, characterized by a quantum channel, has been applied to
the quantum dynamics. In this case, quantum information has been transmitted
to later time point by using $\mathcal{F}_{x_{1}}$. (d) Quantum circuit
fragment $\Phi$ and interactive measurement
$\mathcal{T}:=\\{\mathcal{T}_{x}\\}_{x}$. The most general picture of quantum
dynamics $\Phi$ is described by quantum circuit fragment (colored blue). Here
$\mathcal{T}_{x}$ (colored black) represents an interactive measurement.
Before obtaining a classical outcome $x$, $\Phi$ and $\mathcal{T}_{x}$ have
interacted multiple times.

Current quantum theories typically concern passive measurements, where a
system is left to evolve freely before observation. A visual illustration of
passive measurement has been provided in Fig. 5a, where the blue box
represents state-preparational channel, emitting independent and identically
distributed (i.i.d.) copies of state $\rho$. On the right side of Fig. 5a, the
black box, waiting to receive an input, stands for the general positive
operator valued measure (POVM) $M:=\\{M_{x}\\}_{x}$. To further investigate
the system of interest (characterized by $\rho$) and its evolution
(illustrated by $\Psi$), the observer might implement measurement
$M:=\\{M_{x}\\}_{x}$ at two different time points $t_{1}$ and $t_{2}$
($t_{1}<t_{2}$) and collect the corresponding outcome $x_{1}$ and $x_{2}$, as
demonstrated in Fig. 5b. For simplicity, here we assume that the measurements
executed in $t_{1}$ and $t_{2}$ are the same. However, generally they do not
have to be the same. In this case, the first measurement occurred at time
$t_{1}$ interacts with the system, influencing the second measurement. Such a
measuring protocol (colored black in Fig. 5b) forms a simple example of
interactive measurement. In addition, systems can have prior correlations. For
example, the quantum dynamics inside blue dashed box of Fig. 5c is what it
looks like. Now based on the measurement outcome $x_{1}$ obtained at $t_{1}$,
we can then apply an intervention, described by a quantum channel
$\mathcal{F}_{x_{1}}$ (depending on the outcome $x_{1}$), and finalize the
measuring process by carrying out a joint measurement. An illustration is
given in the black dashed box of Fig. 5c. Most generally, systems can have
prior correlations over different time points, and measuring processes can
contain multiple interventions, limned by Fig. 5d. The quantum dynamics and
the measurement with interventions are colored blue and black respectively. In
this work, the entire class of quantum dynamics with definite causal order and
measurement with interventions mentioned above are characterized by the
framework of _quantum circuit fragments_ (see Def. I.2 of Subsec. I.2) and
_interactive measurements_ (see Def. I.3 of Subsec. I.3) respectively, and our
results involves the most general picture of them.

In this subsection, we focus on the idea of quantum circuit fragments. Roughly
speaking, quantum channels have been employed to characterize the quantum
dynamics with a single process. Meanwhile, quantum superchannels offer us the
most general way of manipulating quantum channels [8], constituting two
processes that occur at different time ticks. These specific quantum dynamics
give us a glimpse of a more general framework – quantum circuit fragments,
where multiple quantum processes are connected through quantum memories with
definite causal orders. Formally, the ‘Quantum Circuit Fragments’ or briefly
‘Circuit Fragments’ is defined as

Definition I.4:  Quantum Circuit Fragments A bipartite quantum state $\rho$ is
prepared in systems $\mathcal{H}_{1}E_{1}$, and subjected to $a-1$ quantum
channels $\\{\Psi_{2},\ldots,\Psi_{a}\\}$ in the form of
$\displaystyle\Psi^{i}:\,\mathcal{H}_{2i-2}E_{i-1}\to\mathcal{H}_{2i-1}E_{i},\quad
2\leqslant i\leqslant a-1,$ (13)
$\displaystyle\Psi^{a}:\,\mathcal{H}_{2a-2}E_{a-1}\to\mathcal{H}_{2a-1}.$ (14)
Then we call the quantum dynamics
$\displaystyle\Phi:=\Psi^{a}\circ\Psi^{a-1}\circ\cdots\circ\Psi^{2}(\rho)$
(15) a quantum circuit fragment (see Fig. 6a). Denote the set of all quantum
circuit fragments in the form of Eq. 15 as $\mathfrak{F}_{a}$.

Figure 6: (color online) Quantum circuit fragment $\Phi$ (a) and its CJ
operator $J^{\Phi}$ (b). Here the quantum circuit fragment $\Phi$ is
characterized by Eq. 15, including an initial state $\rho$ and $a-1$ multiple
quantum processes $\\{\Psi^{i}\\}_{i=2}^{a}$. The corresponding CJ operator
$J^{\Phi}$ is given by Eq. I.2. In (b), the blue lines indicate the systems
coming from
$\outerproduct{I}{I}_{\otimes_{i=2}^{a}\mathcal{H}_{2i-2}\mathcal{H}_{2i-2}^{{}^{\prime}}}$
(see Eq. 17), and the red lines represent the output systems of quantum
processes $\\{\Psi^{i}\\}_{i=2}^{a}$.

Here $\Phi$ is a quantum channel with $a$ individual processes
$\\{\Psi^{i}\\}_{i=1}^{i=a}$, where $\Psi^{1}:=\rho$ acting on systems
$\mathcal{H}_{1}E_{1}$. Its input systems and output systems are
$\otimes_{i=2}^{a}\mathcal{H}_{2i-2}$ and
$\otimes_{i=1}^{a}\mathcal{H}_{2i-1}$ respectively. More precisely,
$\Phi:\otimes_{i=2}^{a}\mathcal{H}_{2i-2}\to\otimes_{i=1}^{a}\mathcal{H}_{2i-1}$
is a quantum channel satisfying non-signaling (NS) from
$\mathcal{H}_{2i-2}\to\mathcal{H}_{2i-1}$ to
$\mathcal{H}_{2i-4}\to\mathcal{H}_{2i-3}$ for all $2\leqslant i\leqslant a$,
with $\mathcal{H}_{0}=\mathds{C}$. Noted that here all the memory systems
$\\{E_{i}\\}_{i=1}^{a-1}$ of $\Phi$ have been swallowed by the composition of
quantum channels. Similar to the CJ operators of quantum superchannels
discussed in Sec. I.1, the CJ operator of $\Phi$ can be obtained by
considering the following equation

$\displaystyle
J^{\Phi}:=\text{id}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-2}}\otimes\Psi^{a}_{\mathcal{H}_{2a-2}^{{}^{\prime}}E_{a-1}\to\mathcal{H}_{2a-1}}\circ$
$\displaystyle\Psi^{a-1}_{\mathcal{H}_{2a-4}^{{}^{\prime}}E_{a-2}\to\mathcal{H}_{2a-3}E_{a-1}}\circ\cdots$
$\displaystyle\cdots\circ$
$\displaystyle\Psi^{2}_{\mathcal{H}_{2}^{{}^{\prime}}E_{1}\to\mathcal{H}_{3}E_{2}}(\rho_{\mathcal{H}_{1}E_{1}}\otimes\outerproduct{I}{I}_{\otimes_{i=2}^{a}\mathcal{H}_{2i-2}\mathcal{H}_{2i-2}^{{}^{\prime}}}),$
(16)

where the multiple copies of unnormalized maximally entangled state
$\outerproduct{I}{I}_{\otimes_{i=2}^{a}\mathcal{H}_{2i-2}\mathcal{H}_{2i-2}^{{}^{\prime}}}$
is defined as

$\displaystyle\outerproduct{I}{I}_{\otimes_{i=2}^{a}\mathcal{H}_{2i-2}\mathcal{H}_{2i-2}^{{}^{\prime}}}:=\outerproduct{I}{I}_{\mathcal{H}_{2}\mathcal{H}_{2}^{{}^{\prime}}}\otimes\outerproduct{I}{I}_{\mathcal{H}_{4}\mathcal{H}_{4}^{{}^{\prime}}}\otimes\cdots\otimes\outerproduct{I}{I}_{\mathcal{H}_{2a-2}\mathcal{H}_{2a-2}^{{}^{\prime}}}.$
(17)

The physical process of generating $J^{\Phi}$ is demonstrated in Fig. 6b.
Denote the CJ operator of $\Psi^{i}$ (see Def. I.2) as $J^{i}$
($i\in\\{2,3,\ldots,a-1\\}$), then it is straightforward to check that
$J^{\Phi}$ can also be written as a result of link product, namely

$\displaystyle J^{\Phi}=J^{a}\star J^{a-1}\star\cdots\star J^{2}\star\rho.$
(18)

In literature [7, 6], $J^{\Phi}$ is called an $a$-comb, where $a$ indicates
the number of processes contained in quantum circuit fragment $\Phi$.
Analogous to the Choi-Jamiołkowski (CJ) isomorphism between the quantum
channels $\mathcal{E}:A\to B$ and their CJ operators $J^{\mathcal{E}}_{AB}$,
the morphism between the quantum circuit fragment $\Phi$ and the corresponding
$a$-comb $J^{\Phi}$ forms a bijection. This implies that the physical
properties associated with $\Phi$ are completely characterized by its comb
representation $J^{\Phi}$.

In open quantum systems, a special form of quantum circuit fragment – known as
process tensor [9, 10, 11] – has been used to describe the system-environment
unitary dynamics. In this model, it is assumed that the system-environment is
initialized in a state $\rho_{\mathcal{H}_{1}E_{1}}$. Without loss of
generality, we can always assume it is a pure state (namely
$\rho_{\mathcal{H}_{1}E_{1}}=\phi_{\mathcal{H}_{1}E_{1}}$) by enlarging the
environment system $E_{1}$. After the preparation of
$\phi_{\mathcal{H}_{1}E_{1}}$, the system-environment dynamics undergo unitary
interactions $\\{\mathcal{U}_{3:2},\ldots,\mathcal{U}_{2a-1:2a-2}\\}$ at $a-1$
time steps. Now the whole process $\Phi$ is demonstrated by the following map

$\displaystyle\Phi=\Tr_{E_{a}}[\,\mathcal{U}_{2a-1:2a-2}\circ\cdots\circ\mathcal{U}_{3:2}(\phi_{\mathcal{H}_{1}E_{1}})],$
(19)

where, for any input state $\sigma$, the output state under $\mathcal{U}$ is
given by $\mathcal{U}(\sigma)=U\sigma U^{{\dagger}}$. It is easy to verify
that this process tensor can be viewed as a special form of quantum circuit
fragment (see Def. I.2) by setting

$\displaystyle\rho=\phi_{\mathcal{H}_{1}E_{1}}:\,\mathds{C}\to\mathcal{H}_{1}E_{1},$
(20)
$\displaystyle\Psi^{i}=\mathcal{U}_{2i-1:2i-2}:\,\mathcal{H}_{2i-2}E_{i-1}\to\mathcal{H}_{2i-1}E_{i},\quad
2\leqslant i\leqslant a-1,$ (21)
$\displaystyle\Psi^{a}=\mathcal{U}_{2a-1:2a-2}:\,\mathcal{H}_{2a-2}E_{a-1}\to\mathcal{H}_{2a-1}E_{a}.$
(22)

Remark that, here the $a$-th quantum process is a unitary interaction from
systems $\mathcal{H}_{2a-2}E_{a-1}$ to $\mathcal{H}_{2a-1}E_{a}$ followed by
the partial trace over environmental system $E_{a}$.

Besides process tensor, in the context of different theories, different
concepts of quantum dynamics with multiple processes have been introduced,
such as the causal maps in quantum causal inference [12, 13], the pseudo-
density matrices (PDMs) in witnessing temporal correlations [14], the
collisional model of quantum thermometer [15], the quantum neural networks
(QNNs) [16] and more general parameterized quantum circuits (PQCs) [17] in the
noisy intermediate-scale quantum (NISQ) era [18], and so forth. Despite having
different names, there is a common point for all these concepts – they contain
multiple quantum processes with definite causal order. Finally, it is worth
mentioning that all of them can be treated as special cases of our quantum
circuit fragment.

###  I.3 Interactive Measurements: How to Measure the Quantum Circuit
Fragments?

Quantum circuit fragments offer us a general way of demonstrating multiple
quantum processes with definite causal order. To investigate the physical
properties associated with quantum circuit fragments and decoding information
from such quantum circuit fragments, a measuring process is needed. Unlike
state case, where a system is left to evolve freely before observation, now we
can interact with the quantum dynamics and make multiple preceding
interventions before final measurement. Here, the whole process is called
interactive measurement.

Let us consider the quantum circuit fragment $\Phi$ demonstrated in Eq. 15.
The corresponding interactive measurement, denoted as $\mathcal{T}$, contains
$a-1$ rounds of interventions $\\{\Lambda^{i}\\}_{i=1}^{a-1}$ and a joint
measurement $\\{M_{x}\\}_{x}$. Here the general interventions are described by
quantum channels. In particular, the interactive measurement for quantum
circuit fragment $\Phi$ is defined as

Definition I.5:  Interactive Measurements An interactive measurement
$\mathcal{T}$ designed for quantum circuit fragment $\Phi$ (See Eq. 15)
constitutes of $a-1$ rounds of interventions $\\{\Lambda^{i}\\}_{i=1}^{a-1}$
in the form of
$\displaystyle\Lambda^{1}:\,\mathcal{H}_{1}\to\mathcal{H}_{2}R_{1},$ (23)
$\displaystyle\Lambda^{i}:\,\mathcal{H}_{2i-1}R_{i-1}\to\mathcal{H}_{2i}R_{i},\quad
2\leqslant i\leqslant a-1,$ (24) and a joint measurement $\\{M_{x}\\}_{x}$
acting on systems $\mathcal{H}_{2a-1}R_{a-1}$, namely
$M_{x}:\mathcal{H}_{2a-1}R_{a-1}\to\mathds{C}$. Here each intervention
$\Lambda^{i}$ forms a quantum channel, and different interventions are
connected through memory systems $\\{R_{i}\\}_{i=1}^{a-1}$. Mathematically,
the interactive measurement $\mathcal{T}$ is characterized by the following
maps $\\{\mathcal{T}_{x}\\}_{x}$ (see Fig. 7a); that are
$\displaystyle\mathcal{T}_{x}(\cdot):=\Tr[M_{x}\cdot\Lambda^{a-1}\circ\Lambda^{a-2}\circ\cdots\circ\Lambda^{1}(\cdot)].$
(25) Denote the set of all interactive measurements in the form of Eq. 25 as
$\mathfrak{T}_{a}$.

Figure 7: (color online) Interactive measurement $\mathcal{T}_{x}$ (a) and its
CJ operator $J^{\mathcal{T}}_{x}$ (b). Here the interactive measurement
$\mathcal{T}_{x}$ is characterized by Eq. 25, including $a-1$ rounds of
interventions $\\{\Lambda^{i}\\}_{i=1}^{a-1}$ and a joint measurement $M_{x}$.
The corresponding CJ operator $J^{\mathcal{T}}_{x}:=J^{\mathcal{T}_{x}}$ is
given by Eq. I.3. In (b), the blue lines indicate the systems coming from
$\outerproduct{I}{I}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-1}\mathcal{H}_{2i-1}^{{}^{\prime}}}$
(see Eq. 28), and the red lines represent the system after interventions
$\\{\Lambda^{i}\\}_{i=1}^{a-1}$.

Within the framework of interactive measurement, previous interventions can be
used to improve the performance of upcoming interventions. To be more
specific, take $\mathcal{T}$ for instance, the interventions
$\\{\Lambda^{i}\\}_{i=1}^{k-1}$ occurred before the implementation of
$\Lambda^{k}$ strengthen our capability of gaining information from subsequent
quantum interventions $\\{\Lambda^{i}\\}_{i=k}^{a-1}$. Given quantum circuit
fragment $\Phi$, the probability of obtaining outcome $x$ by measuring
interactive measurement $\mathcal{T}$ is given by

$\displaystyle
p_{x}(\Phi,\mathcal{T}):=\Tr[M_{x}\cdot\Psi^{a}\circ\Lambda^{a-1}\circ\Psi^{a-1}\circ\Lambda^{a-2}\circ\cdots\circ\Psi^{2}\circ\Lambda^{1}(\rho)].$
(26)

Since the dependencies of probability distribution $p_{x}(\Phi,\mathcal{T})$
is usually clear from the context, we simply re-express it as $p_{x}$.

To characterize interactive measurements and simplify related calculations,
the CJ operator of interactive measurements is a good choice. Again, let us
take $\mathcal{T}$ (see Eq. 25) for an illustration. Here the input and output
systems of $\mathcal{T}:=\\{\mathcal{T}_{x}\\}_{x}$ (see Eq. 25 for the
definition of $\mathcal{T}_{x}$) are $\otimes_{i=1}^{a}\mathcal{H}_{2i-1}$ and
$\otimes_{i=1}^{a}\mathcal{H}_{2i-2}$, and the trivial system $\mathds{C}$ has
been ignored. Thus, the CJ operator of each $\mathcal{T}_{x}$ becomes

$\displaystyle J^{\mathcal{T}}_{x}:=J^{\mathcal{T}_{x}}$ $\displaystyle=$
$\displaystyle\text{id}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-1}}\otimes\Tr[M_{x}\cdot\Lambda^{a-1}_{\mathcal{H}_{2a-3}^{{}^{\prime}}R_{a-2}\to\mathcal{H}_{2a-2}R_{a-1}}\circ\Lambda^{a-2}_{\mathcal{H}_{2a-5}^{{}^{\prime}}R_{a-3}\to\mathcal{H}_{2a-4}R_{a-2}}\circ\cdots\circ\Lambda^{1}_{\mathcal{H}_{1}^{{}^{\prime}}\to\mathcal{H}_{2}R_{1}}(\outerproduct{I}{I}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-1}\mathcal{H}_{2i-1}^{{}^{\prime}}})].$
(27)

A circuit illustration of $J^{\mathcal{T}}_{x}$ is depicted in Fig. 7b. Here
the measurement $M_{x}$ is acting on systems
$\mathcal{H}_{2a-1}^{{}^{\prime}}R_{a-1}$, and the operator
$\outerproduct{I}{I}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-1}\mathcal{H}_{2i-1}^{{}^{\prime}}}$
is defined as

$\displaystyle\outerproduct{I}{I}_{\otimes_{i=1}^{a}\mathcal{H}_{2i-1}\mathcal{H}_{2i-1}^{{}^{\prime}}}:=\outerproduct{I}{I}_{\mathcal{H}_{1}\mathcal{H}_{1}^{{}^{\prime}}}\otimes\outerproduct{I}{I}_{\mathcal{H}_{3}\mathcal{H}_{3}^{{}^{\prime}}}\otimes\cdots\otimes\outerproduct{I}{I}_{\mathcal{H}_{2a-1}\mathcal{H}_{2a-1}^{{}^{\prime}}}.$
(28)

Let us further denote the CJ operator of quantum circuit fragment $\Phi$ as
$J^{\Phi}$, then probability $p_{x}$ considered in Eq. 26 can be re-expressed
as

$\displaystyle p_{x}(\Phi,\mathcal{T})=J^{\Phi}\star J^{\mathcal{T}}_{x},$
(29)

where $J^{\mathcal{T}}_{x}$ is defined in Eq. I.3.

Another new concept that will help us demonstrate the trade-off between
incompatible interactive measurements is _eigencircuit_ , which is formally
defined as

Definition I.6:  Eigencircuit Given an interactive measurement
$\mathcal{T}:=\\{\mathcal{T}_{x}\\}_{x}$, we call quantum circuit fragment
$\Phi$ an eigencircuit of $\mathcal{T}$ if $\displaystyle J^{\Phi}\star
J_{x}=1,$ (30) for some $x$. Here $J^{\Phi}$ and $J_{x}$ represent the CJ
operators of $\Phi$ and $\mathcal{T}_{x}$ respectively.

After building the general framework of interactive measurements (see Def.
I.3), let us now investigate some special forms of measurements for quantum
dynamics and show that they are all special cases of our interactive
measurements. First of all, consider the channel measurement, which was first
introduced in Ref. [19] and named as process positive-operator-valued measure
(PPOVM). Such a measuring process can be viewed as an interactive measurement
by trivializing the input system of the first intervention, which is followed
by a joint measurement directly. Second, we move on to investigating the
framework of testers introduced in Ref. [6]. Mathematically, the testers are
linear maps that take quantum causal networks as their inputs and output
probability distributions. Physically, the testers generalize the concept of
PPOVMs. In particular, a tester is consisted of the preparation of an initial
state, finite rounds of interventions, and a joint measurement at the end of
dynamical process. Analogous to relation between PPOVMs and interactive
measurements, all testers can be regarded as interactive measurements with
trivialized input system of the first intervention. More precisely, for
general quantum testers, they can be written in the form of Eq. 25 with
restricted $\Lambda^{1}$, where $\mathcal{H}_{1}=\mathds{C}$, i.e.
$\Lambda^{1}=\rho_{\mathcal{H}_{2}R_{1}}$ holds for some bipartite state
acting on systems $\mathcal{H}_{2}R_{1}$. To summarize, both PPOVMs and
testers are all special cases of interactive measurements introduced in this
work.

Different from previous works of investigating quantum dynamics, in this work
we are not interested in the statistical properties associated with quantum
circuit fragments and interactive measurements. Instead, we are caring about
the physical properties that can be learned from quantum circuit fragments by
implementing interactive measurements, such as their causal structures, non-
Markovianity, and so on. More details will be unfolded in subsequent sections,
especially the investigation of causal inference in Sec. III.

Figure 8: (color online) Circuit demonstration of process positive-operator-
valued measure (PPOVM) (a) and tester (b). Both of them are obtained by
trivializing the input system, i.e. $\mathcal{H}_{1}$, of our interactive
measurement model $\mathcal{T}_{x}$ (see Eq. 28). In particular, for
interactive measurement $\mathcal{T}_{x}\in\mathfrak{T}_{a}$ depicted in Fig.
7a, taking $\Lambda^{1}$ as
$\rho_{\mathcal{H}_{2}R_{1}}\otimes\Tr_{\mathcal{H}_{1}}$ leads to testers.
When $a=2$, we cover the framework of PPOVMs.

###  I.4 Quantum Causal Maps

In this subsection, we turn our attention to the dynamics of quantum causal
map, which was first introduced in the Ref. [12], and further discuss how to
obtain a causal map from process tensor in open quantum system (see Refs. [9,
10, 11] for more details about related topics). We start by writing down the
system-environment dynamics described by process tensor $\Phi$ of the
following form (see Eq. 31):

$\displaystyle\Phi=\Tr_{E_{a}}[\,\mathcal{U}_{2a-1:2a-2}\circ\cdots\circ\mathcal{U}_{3:2}(\phi_{\mathcal{H}_{1}E_{1}})],$
(31)

where each $\mathcal{U}_{2i-1,2i-2}$ is a unitary map from system-environment
dynamics $\mathcal{H}_{2i-2}E_{i-1}$ to $\mathcal{H}_{2i-1}E_{i}$ with
$2\leqslant i\leqslant a$. Assume each system $\mathcal{H}_{k}$ happens at
time point $t_{k}$ ($1\leqslant k\leqslant 2a-1$), and we can only interact
with the system-environment dynamics in two time periods – $(t_{2i-1},t_{2i})$
and $(t_{2j-1},t_{2j})$ ($1\leqslant i<j\leqslant a-1$). Based on the feedback
from these two time periods, our goal is to identify its causal structure.

In this model, non-intervention, i.e. letting the system-environment dynamics
evolve automatically, is equivalent to apply
$\text{id}_{\mathcal{H}_{2k-1}\to\mathcal{H}_{2k}}$ to system
$\mathcal{H}_{2k-1}$, where $1\leqslant k\leqslant a-1$ and $k\neq i,j$. To
gain information from the quantum circuit fragment $\Phi$, we need to end our
interactions with a measurement. Therefore, a POVM has been applied to the
system $\mathcal{H}_{2j-1}$, and after the time point $t_{2j}$ all systems
will be discarded, including $\mathcal{H}_{2j}$. Denote the state
$\phi_{\mathcal{H}_{2i-1}E_{i}}^{{}^{\prime}}$ as

$\displaystyle\phi_{\mathcal{H}_{2i-1}E_{i}}^{{}^{\prime}}:=\mathcal{U}_{2i-1:2i-2}\circ\text{id}_{\mathcal{H}_{2i-3}\to\mathcal{H}_{2i-2}}\circ\cdots\circ\mathcal{U}_{3:2}\circ\text{id}_{\mathcal{H}_{1}\to\mathcal{H}_{2}}(\phi_{\mathcal{H}_{1}E_{1}}),$
(32)

and define the following unitary map
$\mathcal{U}^{{}^{\prime}}_{\mathcal{H}_{2i}E_{i}\to\mathcal{H}_{2j-1}E_{j}}$,

$\displaystyle\mathcal{U}^{{}^{\prime}}_{\mathcal{H}_{2i}E_{i}\to\mathcal{H}_{2j-1}E_{j}}:=\mathcal{U}_{2j-1:2j-2}\circ\text{id}_{\mathcal{H}_{2j-3}\to\mathcal{H}_{2j-2}}\circ\cdots\circ\text{id}_{\mathcal{H}_{2i+1}\to\mathcal{H}_{2i+2}}\circ\mathcal{U}_{2i+1:2i}.$
(33)

Now, besides the time periods $(t_{2i-1},t_{2i})$ and $(t_{2j-1},t_{2j})$, the
whole quantum circuit fragment turns out to be $\Phi^{{}^{\prime}}$, which is
demonstrated by the following map

$\displaystyle\Phi^{{}^{\prime}}_{\mathcal{H}_{2i}\to\mathcal{H}_{2i-1}\mathcal{H}_{2j-1}}=\Tr_{E_{j}}[\,\mathcal{U}^{{}^{\prime}}_{\mathcal{H}_{2i}E_{i}\to\mathcal{H}_{2j-1}E_{j}}(\phi_{\mathcal{H}_{2i-1}E_{i}}^{{}^{\prime}})].$
(34)

Figure 9: (color online) Investigating the causality of $\Phi$ (see Eq. 35)
between time periods $(t_{2i-1},t_{2i})$ and $(t_{2j-1},t_{2j})$. For this
purpose, id channels have been applied to systems $\mathcal{H}_{1}$ and
$\mathcal{H}_{7}$, followed by $\Tr_{E_{5}}$ and $\Tr_{\mathcal{H}_{9}}$. The
final quantum circuit fragment $\Phi^{{}^{\prime}}$ belongs to the set of all
causal maps, i.e. $\mathfrak{F}_{2}$, and has been characterized by Eq. 36.

Take a system-environment dynamics with four rounds of unitary evaluations for
instance (see Fig. 9); that is

$\displaystyle\Phi=\Tr_{E_{5}}[\,\mathcal{U}_{9:8}\circ\mathcal{U}_{7:6}\circ\mathcal{U}_{5:4}\circ\mathcal{U}_{3:2}(\phi_{\mathcal{H}_{1}E_{1}})]\in\mathfrak{F}_{5}.$
(35)

To infer the causal models between time periods $(t_{3},t_{4})$ and
$(t_{5},t_{6})$, we apply noiseless quantum channels to systems
$\mathcal{H}_{1}$ and $\mathcal{H}_{7}$, i.e. described by
$\text{id}_{\mathcal{H}_{1}\to\mathcal{H}_{2}}$ and
$\text{id}_{\mathcal{H}_{7}\to\mathcal{H}_{8}}$ respectively. Then trace out
all the systems happened after time point $t_{6}$, namely implementing
$\Tr_{E_{5}}$ and $\Tr_{\mathcal{H}_{9}}$. Thus, the resultant quantum
dynamics turns out to be

$\displaystyle\Phi^{{}^{\prime}}_{\mathcal{H}_{4}\to\mathcal{H}_{3}\mathcal{H}_{5}}=\Tr_{E_{3}}[\,\mathcal{U}_{5:4}(\phi_{\mathcal{H}_{3}E_{2}}^{{}^{\prime}})],$
(36)

where
$\phi_{\mathcal{H}_{3}E_{2}}^{{}^{\prime}}:=\mathcal{U}_{3:2}(\text{id}_{\mathcal{H}_{1}\to\mathcal{H}_{2}}(\phi_{\mathcal{H}_{1}E_{1}}))$.

Let us return to the model of general system-environment dynamics investigated
in Eq. 34. To simplify our notations, we relabel the systems as

$\displaystyle A$ $\displaystyle:=\mathcal{H}_{2i-1},$ (37) $\displaystyle B$
$\displaystyle:=\mathcal{H}_{2i},$ (38) $\displaystyle C$
$\displaystyle:=\mathcal{H}_{2j-1},$ (39) $\displaystyle E$
$\displaystyle:=E_{i},$ (40) $\displaystyle F$ $\displaystyle:=E_{j}.$ (41)

Equipped with new labels, the quantum circuit fragment $\Phi^{{}^{\prime}}$
(see Eq. 34) can be re-expressed as

$\displaystyle\Phi^{{}^{\prime}}_{B\to AC}=\Tr_{F}[\,\mathcal{U}_{BE\to
CF}^{{}^{\prime}}(\phi^{{}^{\prime}}_{AE})].$ (42)

Above example can be seen as a prototype of causal map in open quantum system.
Generally, we call all the quantum circuit fragments $\Phi_{B\to AC}$ with
$t_{A}\leqslant t_{B}\leqslant t_{C}$ quantum causal maps (see Fig. 2b of our
main text), which form the set of $\mathfrak{F}_{2}$ (see Def. I.2). Such a
quantum dynamics can also be viewed as a special form of superchannel with
trivialized input system of pre-processing. In particular, given a causal map
$\Phi_{B\to AC}\in\mathfrak{F}_{2}$, it can be divided into two quantum
processes: pre-processing $\Psi^{\text{Pre}}_{\mathds{C}\to AE}$ and post-
processing $\Psi^{\text{Post}}_{BE\to C}$, namely

$\displaystyle\Phi_{B\to AC}=\Psi^{\text{Post}}_{BE\to
C}\circ\Psi^{\text{Pre}}_{\mathds{C}\to AE}.$ (43)

Here the subscript $\mathds{C}\to AE$ indicates that
$\Psi^{\text{Pre}}_{\mathds{C}\to AE}$ is a state preparational channel. For
Eq. 42, the map $\Phi^{{}^{\prime}}_{B\to AC}$ can also be written as the
composition of pre-processing $\phi^{{}^{\prime}}_{AE}$ and post-processing
$\Tr_{F}\circ\,\mathcal{U}_{BE\to CF}^{{}^{\prime}}$. Denote the CJ operators
of pre-processing and post-processing as $J^{\text{Pre}}$ and
$J^{\text{Post}}$ respectively, then the CJ operator of causal map $\Phi_{B\to
AC}$ can be obtained from their link product; that is

$\displaystyle J^{\Phi}_{ABC}=J^{\text{Pre}}_{AE}\star J^{\text{Post}}_{BCE}.$
(44)

Here the CJ operator $J^{\Phi}$ of causal map $\Phi_{B\to AC}$, acting on
systems $ABC$, satisfies the conditions of CPTP and NS from post-processing to
pre-processing. It now follows immediately that

  1. (i)

CP: $J^{\Phi}\geqslant 0$,

  2. (ii)

TP: $\Tr_{AC}[J^{\Phi}]=\mathds{1}_{B}$,

  3. (iii)

NS from post-processing to pre-processing:
$\Tr_{C}[J^{\Phi}]=\Tr_{BC}[J^{\Phi}]\otimes\mathds{1}_{B}/d_{B}$.

Actually, these conditions can be further simplified as $J^{\Phi}\geqslant 0$
and $\Tr_{C}[J^{\Phi}]=\rho_{A}\otimes\mathds{1}_{B}$ for some quantum state
$\rho_{A}$.

From an experimental viewpoint, the normalized CJ operator (also called CJ
state in this work) $\Phi_{B\to AC}/d_{B}$ can be obtained directly by feeding
maximally entangled state to the dynamics, which is given by

$\displaystyle\text{id}_{B}\otimes\Phi_{B^{{}^{\prime}}\to
AC}(\phi^{+}_{BB^{{}^{\prime}}}),$ (45)

where
$\phi^{+}_{BB^{{}^{\prime}}}:=\outerproduct{I}{I}_{BB^{{}^{\prime}}}/d_{B}$ be
the maximally entangled state with $B^{{}^{\prime}}$ being a replica of system
$B$. The circuit realization is demonstrated in Fig. 10.

Figure 10: (color online) Circuit realization of the CJ state $J^{\Phi_{B\to
AC}}/d_{B}$, where
$\phi^{+}_{BB^{{}^{\prime}}}:=\outerproduct{I}{I}_{BB^{{}^{\prime}}}/d_{B}$
stands for the maximally entangled state on systems $BB^{{}^{\prime}}$.

The interactive measurement is essential in inferring the causal structures
associated with $\Phi_{B\to AC}$. Such a measuring process $\mathcal{T}$,
which is an element of $\mathfrak{T}_{2}$ (see Def. I.3), consists of a
quantum channel $\Lambda$ from system $A$ to $BR$, a joint POVM
$M:=\\{M_{x}\\}_{x}$ acting on systems $CR$, and an ancillary system $R$
connecting them together (see Fig. 1a of our main text), leading to a set of
linear maps

$\displaystyle\mathcal{T}_{AC\rightarrow
B\mathds{C}}(\cdot):=\Big{\\{}\Tr_{CR}[M_{x}\cdot\Lambda_{A\rightarrow
BR}(\cdot)]\Big{\\}}_{x},$ (46)

where each $\mathcal{T}_{x}(\cdot):=\Tr_{CR}[M_{x}\cdot\Lambda_{A\rightarrow
BR}(\cdot)]$ forms a CP and trace-non-increasing (TNI) map from $AC$ to
$B\mathds{C}$. Since the CJ operator of the positive operator-valued measure
(POVM) $M_{x}$ is given by $M_{x}^{\mathbf{T}_{CR}}$, the CJ operator of
$\mathcal{T}_{x}$, denoted by $J_{x}$, can be obtained by using link product
between $J^{\Lambda}$ and $M_{x}^{\mathbf{T}_{CR}}$. Writing everything out
explicitly, we have

$\displaystyle J_{x}=M_{x}^{\mathbf{T}_{CR}}\star J^{\Lambda}.$ (47)

To realize the CJ state $J_{x}/d_{A}d_{C}$ experimentally, we employ the
quantum circuit shown in Fig. 11. For any incompatible interactive
measurements $\mathcal{T}_{1}=\\{J^{\mathcal{T}_{1}}_{x}\\}_{x=1}^{m}$ and
$\mathcal{T}_{2}=\\{J^{\mathcal{T}_{2}}_{y}\\}_{y=1}^{n}$ with

$\displaystyle J^{\mathcal{T}_{1}}_{x}$
$\displaystyle:=M_{x}^{\mathbf{T}_{CR}}\star J^{\Lambda_{A\rightarrow BR}},$
(48) $\displaystyle J^{\mathcal{T}_{2}}_{y}$
$\displaystyle:=N_{y}^{\mathbf{T}_{CR}}\star J^{\Upsilon_{A\rightarrow BR}},$
(49)

the probability of obtain classical outcomes $x$ and $y$ from causal map
$\Phi_{B\to AC}=\Psi^{\text{Post}}_{BE\to
C}\circ\Psi^{\text{Pre}}_{\mathbb{C}\to AE}$ is given by

$\displaystyle p_{x}$ $\displaystyle=\Tr[M_{x}\cdot\Psi^{\text{Post}}_{BE\to
C}\circ\Lambda_{A\rightarrow BR}(\Psi^{\text{Pre}}_{\mathbb{C}\to AE})],$ (50)
$\displaystyle q_{y}$ $\displaystyle=\Tr[N_{y}\cdot\Psi^{\text{Post}}_{BE\to
C}\circ\Upsilon_{A\rightarrow BR}(\Psi^{\text{Pre}}_{\mathbb{C}\to AE})].$
(51)

Using the language of CJ operators and link product, Eqs. 50 and 51 can also
be simplified as

$\displaystyle p_{x}$ $\displaystyle=J^{\Phi}\star J^{\mathcal{T}_{1}}_{x},$
(52) $\displaystyle q_{y}$ $\displaystyle=J^{\Phi}\star
J^{\mathcal{T}_{2}}_{y},$ (53)

where $J^{\Phi}$ stands for the CJ operator of causal map $\Phi_{B\to AC}$.
Collect $p_{x}$ and $q_{y}$ into two probability vector $\mathbf{p}$ and
$\mathbf{q}$, namely $\mathbf{p}:=\\{p_{x}\\}_{x=1}^{m}$ and
$\mathbf{q}:=\\{q_{y}\\}_{y=1}^{n}$. In upcoming sections, we will show how to
analyze the causal structure of $\Phi$ by using the uncertainty trade-off
between $\mathbf{p}$ and $\mathbf{q}$, and exhibit the corresponding
advantages.

Figure 11: (color online) Circuit realization of the CJ state
$J_{x}/d_{A}d_{C}$, where
$\phi^{+}_{AA^{{}^{\prime}}}:=\outerproduct{I}{I}_{AA^{{}^{\prime}}}/d_{A}$
and
$\phi^{+}_{CC^{{}^{\prime}}}:=\outerproduct{I}{I}_{CC^{{}^{\prime}}}/d_{C}$
stand for the maximally entangled states on systems $AA^{{}^{\prime}}$ and
$CC^{{}^{\prime}}$ respectively.

##  II Universal Uncertainty Relation for Measurements with Interventions

In this section, starting from the algebraic structure of majorization
lattice, we investigate the connections between universal uncertainty relation
and quantum roulette, and provide the optimal bound of universal uncertainty
relation for all quantum circuit fragments, leading to the entropic
uncertainty relation for measurements with interventions (viz., Theorem $1$ of
the main text). Specifically, in Subsec. II.1, we briefly review the concept
of majorization lattice and discuss its completeness. The standard steps of
finding least upper bound (LUB) for any set and a useful tool known as
flatness process, which was originally introduced to investigate the
supermodularity and subadditivity of entropies [20], are also supplied. In
Subsec. II.2, we offer a short review for the historical developments of
uncertainty principle, and introduce the requirement for uncertainty measures,
which gives rise to the majorization uncertainty relation (known as universal
uncertainty relation in literature). In Subsec. II.3, we formulate an
universal uncertainty relation for arbitrary quantum circuit fragments,
showing its direct operational meaning in quantum roulette. With different
uncertainty measures and different quantum dynamics, such a form leads to an
infinite number of uncertainty relations. The generality and optimality of our
results have also been analyzed. Lemma $1$ and Theorem $1$ of the main text
are proved in the last subsection, i.e. Subsec. II.4. Additionally, we have
also introduced the entropic uncertainty relation with multiple interactive
measurements, extending the results presented in the main text of our work.

###  II.1 Mathematical Toolkit: Majorization Lattice

In this subsection, we turn our attention to the mathematical concept of
_lattice_ , which is crucial in constructing the optimal bound of universal
uncertainty relation (see Subsec. II.2 and Refs. [21, 22, 23] for more
details). Let us start with the definition of Lattice:

Definition II.1:  Lattice A quadruple $(S,\sqsubset,\wedge,\vee)$ is called
lattice if $\ \sqsubset$ leads to a partial oder on the set $S$ such that for
all $\mathbf{p}$, $\mathbf{q}\in S$ there exists a unique greatest lower bound
(GLB) $\mathbf{p}\wedge\mathbf{q}$ and a unique least upper bound (LUB)
$\mathbf{p}\vee\mathbf{q}$ satisfying
$\displaystyle\mathbf{x}\sqsubset\mathbf{p},\,\mathbf{x}\sqsubset\mathbf{q}$
$\displaystyle\Rightarrow\mathbf{x}\sqsubset\mathbf{p}\wedge\mathbf{q},$
$\displaystyle\mathbf{p}\sqsubset\mathbf{y},\,\mathbf{q}\sqsubset\mathbf{y}$
$\displaystyle\Rightarrow\mathbf{p}\vee\mathbf{q}\sqsubset\mathbf{y}.$ (54)
for any $\mathbf{x}$, $\mathbf{y}\in S$.

A special class of lattices are those which have GLB and LUB for all their
subsets, namely Complete Lattice:

Definition II.2:  Complete Lattice A lattice $(S,\sqsubset,\wedge,\vee)$ is
called complete, if for any nonempty subset $R\subset S$, it has a LUB,
denoted by $\vee R$ and a GLB, denoted by $\wedge R$. More precisely, if there
exist $\mathbf{x},\mathbf{y}\in S$ such that $\mathbf{x}\sqsubset
R\sqsubset\mathbf{y}$, i.e.
$\displaystyle\mathbf{x}\sqsubset\mathbf{p}\sqsubset\mathbf{y},\quad\forall\,\mathbf{p}\in
R,$ (55) then we immediately have $\displaystyle\mathbf{x}\sqsubset\wedge
R\quad\text{and}\quad\vee R\sqsubset\mathbf{y}.$ (56)

We now move on to introducing the concept of majorization [24], which plays
important roles in quantum resource theory and uncertainty relations. In
particular, majorization was originally defined by Hardy, Littlewood, and
Pólya

Definition II.3:  Majorization For vectors $\mathbf{x}=(x_{k})_{k}$,
$\mathbf{y}=(y_{k})_{k}\in\mathds{R}^{d}$, we say $\mathbf{x}$ is majorized by
$\mathbf{y}$, denoted as $\displaystyle\mathbf{x}\prec\mathbf{y},$ (57) if the
following inequality
$\displaystyle\sum_{k=1}^{i}x_{k}^{\downarrow}\leqslant\sum_{k=1}^{i}y_{k}^{\downarrow}$
(58) holds for all $1\leqslant i\leqslant d-1$, and
$\displaystyle\sum_{k=1}^{d}x_{k}^{\downarrow}=\sum_{k=1}^{d}y_{k}^{\downarrow}.$
(59) Here the down-arrow notation ↓ means that the components of corresponding
vector are arranged in non-increasing order.

Let us consider a concrete example of majorization. For any $3$-dimensional
probability vector $\mathbf{p}$, we see that

$\displaystyle(1/3,1/3,1/3)\prec\mathbf{p}\prec(1,0,0).$ (60)

To build a lattice structure for probability simplex, we hope that
majorization can induce a partial order for all probability vectors. However,
this is not the case. Majorization only forms a preorder. It is
straightforward to check that majorization satisfies the properties of
reflexivity ($\mathbf{x}\prec\mathbf{x}$ holds for any $\mathbf{x}$) and
transitivity ($\mathbf{x}\prec\mathbf{y}$ and $\mathbf{y}\prec\mathbf{z}$
imply $\mathbf{x}\prec\mathbf{z}$). However, the property of antisymmetry is
absent. Take vectors $(1,0)$ and $(0,1)$ for instance, it is clear that
$(1,0)\prec(0,1)$ and $(0,1)\prec(1,0)$, but $(1,0)\neq(0,1)$. To remedy this
problem, we should work on the ordered set of
$\mathds{P}_{n}^{d,\,\downarrow}:=\\{\mathbf{x}\in\mathds{R}^{d}\,\|\,x_{k}\geqslant
x_{k+1}\geqslant 0,\,\forall 1\leqslant k\leqslant d-1,\,\sum_{k}x_{k}=n\\}$
instead of the probability simplex. Noted that when $n=1$, the set
$\mathds{P}_{1}^{d,\,\downarrow}$ stands for the ordered probability simplex.

As proven in Ref. [25], the quadruple
$(\mathds{P}_{n}^{d,\,\downarrow},\prec,\wedge,\vee)$ indeed forms a complete
lattice under majorization, called majorization lattice. The properties of
majorization lattice lead to a standard approach in finding the optimal bounds
for any subset $S$ of $\mathds{P}_{n}^{d,\,\downarrow}$ [25]. Formally, given
a subset $S\subset\mathds{P}_{n}^{d,\,\downarrow}$, there are two steps in
constructing its LUB $\vee S$. The first step is to find the quantities
$\mathbf{b}_{S,k}$, which is defined as

$\displaystyle\mathbf{b}_{S,k}:=\left(\max_{\mathbf{x}\in
S}\sum_{i=1}^{k}x_{i}\right)-\sum_{i=1}^{k-1}\mathbf{b}_{S,i},$ (61)

for $1\leqslant k\leqslant d$. Remark that the resultant vector
$\mathbf{b}_{S}:=(\mathbf{b}_{S,k})_{k}$ might not always belongs to the set
$\mathds{P}_{n}^{d,\,\downarrow}$. To gain some intuitions, recall the example
constructed in Ref. [20]. Take $S=\\{\mathbf{x},\mathbf{y}\\}$ with

$\displaystyle\mathbf{x}$ $\displaystyle=(0.6,0.15,0.15,0.1),$ (62)
$\displaystyle\mathbf{y}$ $\displaystyle=(0.5,0.25,0.2,0.05).$ (63)

In this case, the vector

$\displaystyle\mathbf{b}_{S}=(0.6,0.15,0.2,0.05),$ (64)

obtained from (61) does not belong to the set
$\mathds{P}_{1}^{d,\,\downarrow}$, since

$\displaystyle\mathbf{b}_{S,2}=0.15<\mathbf{b}_{S,3}=0.2.$ (65)

Actually, even if we rearrange the vector $\mathbf{b}_{S}$ into non-increasing
order

$\displaystyle\mathbf{b}_{S}^{\downarrow}=(0.6,0.2,0.15,0.05),$ (66)

the re-ordered vector $\mathbf{b}_{S}^{\downarrow}$ is not the optimal upper
bound, i.e. $\mathbf{b}_{S}^{\downarrow}\neq\vee S$.

In order to achieve the optimal bound $\vee S$ of the subset $S$, an
additional process $\mathcal{F}$ on $\mathbf{b}_{S}$ (not
$\mathbf{b}_{S}^{\downarrow}$) is needed, named flatness process [20]:

Definition II.4:  Flatness Process Let $\mathbf{x}\in\mathds{R}^{d}_{+}$ be a
non-negative $d$-dimensional vector, and $j$ be the smallest integer in
$\left\\{2,\ldots,d\right\\}$ such that $x_{j}>x_{j-1}$, and $i$ be the
greatest integer in $\left\\{1,\ldots,j-1\right\\}$ such that
$x_{i-1}\geqslant(\sum_{k=i}^{j}x_{k})/(j-i+1):=a$. Define
$\displaystyle\mathcal{T}(\mathbf{x}):=\left(x_{1}^{\prime},\ldots,x_{n}^{\prime}\right)\,\quad\text{with}\quad\,x_{k}^{\prime}=\begin{cases}a&\text{for}\quad
k=i,\ldots,j\\\ x_{k}&\text{otherwise.}\\\ \end{cases}$ (67) and
$\mathcal{F}(\mathbf{x}):=\mathcal{T}^{d-1}(\mathbf{x})=\mathcal{T}(\mathcal{T}^{d-2}(\mathbf{x}))$,
i.e. applying $\mathcal{T}$ on the vector $\mathbf{x}$ successively $d-1$
times. We call $\mathcal{F}$ the flatness process of vector $\mathbf{x}$. If
the vector $\mathbf{x}$ is already arranged in non-increasing order, or
equivalently there does not exist an integer $j$ in
$\left\\{2,\ldots,d\right\\}$ such that $x_{j}>x_{j-1}$, then we simply have
$\mathcal{T}(\mathbf{x})=\mathbf{x}$.

Based on the definition of flatness process, we will show how to obtain the
LUB for $\mathbf{x}$ and $\mathbf{y}$ considered in Eqs. 62 and 63. In this
case, $\mathbf{b}_{S}=(0.6,0.15,0.2,0.05)$ is not arranged in non-increasing
order. Let us define $x_{k}$ ($k=1,2,3,4$) as

$\displaystyle x_{1}$ $\displaystyle=0.6,$ (68) $\displaystyle x_{2}$
$\displaystyle=0.15,$ (69) $\displaystyle x_{3}$ $\displaystyle=0.2,$ (70)
$\displaystyle x_{4}$ $\displaystyle=0.05.$ (71)

Then the smallest integer in $\left\\{2,3,4\right\\}$ such that
$x_{j}>x_{j-1}$ is $3$ as $x_{3}=0.2>x_{2}=0.15$. Note that now the quantity
$(\sum_{k=2}^{3}x_{k})/2$ is given by

$\displaystyle\frac{x_{2}+x_{3}}{2}=0.175\leqslant x_{1}=0.6.$ (72)

Thus, we can write $\mathcal{T}(\mathbf{b}_{S})=(0.6,0.175,0.175,0.05)$. As
$\mathcal{T}(\mathbf{b}_{S})$ is already in non-increasing order, we have
$\mathcal{F}(\mathbf{b}_{S})=\mathcal{T}(\mathbf{b}_{S})$, which is exactly
the optimal bound for $\mathbf{x}=(0.6,0.15,0.15,0.1)$ and
$\mathbf{y}=(0.5,0.25,0.2,0.05)$, namely

$\displaystyle\vee
S=\mathbf{x}\vee\mathbf{y}=\mathcal{F}(\mathbf{b}_{S})=(0.6,0.175,0.175,0.05),$
(73)

with

$\displaystyle(0.6,0.15,0.15,0.1)$
$\displaystyle\prec(0.6,0.175,0.175,0.05)\prec(0.6,0.15,0.2,0.05),$ (74)
$\displaystyle(0.5,0.25,0.2,0.05)$
$\displaystyle\prec(0.6,0.175,0.175,0.05)\prec(0.6,0.15,0.2,0.05),$ (75)

and for any probability vector $\mathbf{z}$ satisfying
$\mathbf{x}\prec\mathbf{z}$ and $\mathbf{y}\prec\mathbf{z}$, it follows that

$\displaystyle(0.6,0.175,0.175,0.05)\prec\mathbf{z}.$ (76)

It is worth mentioning that the flatness process $\mathcal{F}$ introduced in
Ref. [20] is exactly the second step of formulating the optimal bound for $S$.
More precisely, $\vee S=\mathcal{F}(\mathbf{b}_{S})$ holds in general [25].

To summarize, the standard approach in finding the optimal bounds for a subset
$S$ of $\mathds{P}_{n}^{d,\,\downarrow}$ contains two steps:

  1. Step $1$.

Formulating $\mathbf{b}_{S}$ defined in Eq. (61),

  2. Step $2$.

Applying the flatness process defined in Eq. (67) to obtain
$\mathcal{F}(\mathbf{b}_{S})$,

leading to the following lemma.

Lemma II.5:  Least Upper Bound (LUB) For any subset
$S\subset\mathds{P}_{n}^{d,\,\downarrow}$, its least upper bound (LUB) $\vee
S$ is given by $\displaystyle\vee S=\mathcal{F}(\mathbf{b}_{S}),$ (77) where
the vector elements of $\mathbf{b}_{S}$ are defined in Eq. 61, and
$\mathcal{F}$ represents the flatness process introduced in Def. II.1.

Finally, we note that majorization is naturally connected with entropies. More
precisely, all _Schur-concave_ functions, including Rényi entropies, are
order-reversing functions for majorization. The relations between
majorization, doubly stochastic matrix (square matrix of non-negative real
numbers, whose rows and columns sums to $1$), and Schur-concave functions are
detailed in the following lemma.

Lemma II.6:  Majorization Equivalents [24] For two vectors $\mathbf{x}$ and
$\mathbf{y}$, the following conditions are equivalent: (i) $\mathbf{x}$ is
majorized by $\mathbf{y}$, i.e. $\mathbf{x}\prec\mathbf{y}$; (ii) $x=y\cdot D$
for some doubly stochastic matrix $D$; (iii) $f(\mathbf{x})\geqslant
f(\mathbf{y})$ holds for any Schur-concave function $f$.

Let us check some simple applications of Lem. II.1. First, Shannon entropy is
invariant under permutations of its components since
$\mathbf{p}\prec\mathbf{p}\cdot D$ and $\mathbf{p}\cdot D\prec\mathbf{p}$ hold
for any probability vector $\mathbf{p}$ and permutation matrix $D$. Second,
for any $d$-dimensional probability vector $\mathbf{p}$, we always have $\log
d\geqslant H(\mathbf{p})$ as $(1/d,\ldots,1/d)\prec\mathbf{p}$, where $H$
stands for the Shannon entropy.

###  II.2 Brief Introduction to Uncertainty Relations

If an observer is well-equipped, then estimating the position and momentum of
a moving football simultaneously is not a difficult task. However, if the item
of observation has been replaced by a particle, then it is impossible to
predict the outcomes for both position and momentum with arbitrary precision,
no matter how well the observer is equipped. Quantum mechanics constrains what
we can learn about the observables of particle. Such a restriction is known as
uncertainty principle, which was first introduced by Heisenberg in Ref. [26].
Quantitatively, the fundamental trade-off between position and momentum is
characterized by the following inequality [27, 28]

$\displaystyle\Delta\mathbf{x}\cdot\Delta\mathbf{p}\geqslant\frac{\hbar}{2},$
(78)

where $\mathbf{x}$ and $\mathbf{p}$ stand for the position and momentum
operators, and $\Delta$ is the standard deviation. For bounded operators $M$
and $N$, Robertson gave a more general form in terms of the commutator $[M,N]$
[29]

$\displaystyle\Delta M\cdot\Delta
N\geqslant\frac{1}{2}|\bra{\psi}[M,N]\ket{\psi}|.$ (79)

From an information-theoretic perspective, it is more natural to quantify the
uncertainty associated with quantum measurements in terms of entropy rather
than the statistical tools, such as the standard deviation. Using this
insight, the entropic version of position-momentum uncertainty relation can be
formulated as

$\displaystyle h(\mathbf{x})+h(\mathbf{p})\geqslant\log(\pi e\hbar)$ (80)

Remark that the entropic uncertainty relation of Eq. 80 is obtained by
Białynicki-Birula and Mycielski in Ref. [30]. Here $e$ is the Euler’s number,
and $h$ stands for the differential entropy: consider a random variable $X$
with density $f(x)$, then its differential entropy is defined as

$\displaystyle h(X):=-\int_{\infty}^{\infty}f(x)\log f(x)\,d\,f(x).$ (81)

The first entropic uncertainty relation for general bounded operators was
introduced by Deutsch in Ref. [31], which had later been improved by Maassen
and Uffink. In particular, given projective measurements
$M:=\\{\ket{u_{i}}\\}$ and $N:=\\{\ket{v_{j}}\\}$, the Maassen-Uffink entropic
uncertainty relation reads [32]

$\displaystyle H(M)+H(N)\geqslant-\log c,$ (82)

where $H(M)$ represents the Shannon entropy of probability distribution
obtained by implementing measurement $M$, and $c$ is the maximal overlap
between measurements $M$ and $N$. More precisely, here the quantity $c$ is
defined by

$\displaystyle c:=\max_{i,j}|\innerproduct{u_{i}}{v_{j}}|^{2},$ (83)

which depends only on the measurements, or equivalently state-independent, and
hence captures their inherent incompatibility. Technically, the quantity $c$
is derived from _Riesz theorem_ in functional analysis.

In the ensuing decades, quantum uncertainty associated with measurements has
been quantified by a variety of entropies. Various forms of entropic
uncertainty relations have also been proposed [33]. However, none of these
forms answer the fundamental question – which uncertainty measure is the most
adequate to use? Shannon entropy, Collision entropy, Min-entropy, or even
Tsallis entropy? To fully understand the uncertainty induced by measurements
and address this question, the concept of ‘reasonable measure’ for uncertainty
has been introduced in Ref. [21]: any reasonable measure $f$ of the quantum
uncertainty should be a function only of the probability vector associated
with measurement and satisfies monotonicity under random relabeling
$\mathfrak{R}$. For example, given a probability vector $\mathbf{p}$, function
$f$ is an uncertainty measure if the following condition is satisfied

$\displaystyle f(\mathfrak{R}(\mathbf{p}))\geqslant f(\mathbf{p}).$ (84)

In other words, the procedure of random relabeling – known as forgetting
outcome labels in classical world – turns the results to be more uncertain.
Technically speaking, the process of random relabeling $\mathfrak{R}$ is
characterized by the convex hull of permutations [24], which implies that

$\displaystyle\mathfrak{R}(\mathbf{p})=\mathbf{p}\cdot D_{\mathfrak{R}},$ (85)

holds for some doubly stochastic matrix $D_{\mathfrak{R}}$. As a direct
consequence of Lem. II.1, we know that for two probability vectors
$\mathbf{p}$ and $\mathbf{q}$, $\mathbf{p}$ is more uncertain than
$\mathbf{q}$ if and only if $\mathbf{p}=\mathbf{q}\cdot D$ for some doubly
stochastic matrix $D$, or equivalently $\mathbf{p}\prec\mathbf{q}$. Therefore,
all Schur-concave functions (including Rényi entropies) are reasonable
uncertainty measures. Historically, majorization $\prec$ is initially
introduced as a tool to extend known inequalities and unify all inequalities
based on convex functions. Here, in the context of uncertainty relation,
majorization will generate an infinite family of uncertainty relations. Thus,
the uncertainty relation in the forms of majorization is also known as
_universal uncertainty relation_. A typical example is the direct-sum
majorization uncertainty relation [23]. Consider the probability vectors
$\mathbf{p}:=(p_{i})_{i=1}^{m}$ and $\mathbf{q}:=(q_{j})_{j=1}^{n}$ obtained
by measuring a quantum state $\rho$ with respect to a pair of incompatible
measurements $M$ and $N$, their direct-sum $\oplus$ is defined as the union of
vectors. Formally,

$\displaystyle\mathbf{p}\oplus\mathbf{q}:=(p_{1},\ldots,p_{m},q_{1},\ldots,q_{n}).$
(86)

For example, given probability vectors $\mathbf{p}=(1,0)$ and
$\mathbf{q}=(1/2,1/2)$, their direct-sum $\mathbf{p}\oplus\mathbf{q}$ is
simply $(1,0,1/2,1/2)$. The goal is find the optimal upper bound
$\mathbf{w}_{M,N}$ such that

$\displaystyle\mathbf{p}\oplus\mathbf{q}\prec\mathbf{w}_{M,N},$ (87)

holds for all quantum state $\rho$, namely $\mathbf{w}_{M,N}$ is a vector
independent of the state $\rho$. For any Schur-concave function $f$,
uncertainty relation 87 leads to

$\displaystyle f(\mathbf{p}\oplus\mathbf{q})\geqslant f(\mathbf{w}_{M,N}),$
(88)

which includes all entropic uncertainty relations as special cases. Take
Shannon entropy for instance, we immediately obtain the entropic uncertainty
relation in the form of $H(\mathbf{p}\oplus\mathbf{q})\geqslant
H(\mathbf{w}_{M,N})$. Before the end of this subsection, we make two remarks.
First, direct-sum majorization uncertainty relation is not the only form of
universal uncertainty relations. There also exists a direct-product
majorization uncertainty relation, which was formulated by Friedland,
Gheorghiu, Gour in Ref. [21] and Puchała, Rudnicki, Życzkowski in Ref. [22].
Specifically, they are trying to bound the uncertainty associated with
$\mathbf{p}\otimes\mathbf{q}:=(p_{i}\cdot q_{j})_{i,j}$. In this work, we
mainly focus on the direct-sum form since it shows a provable advantage over
the direct-product form (see Ref. [23] for more details). Second, all the
uncertainty relations considered in this work belong to the category of
preparational uncertainty relation. The noise-disturbance uncertainty relation
of interactive measurements is beyond the scope of this work, and merits
future investigation.

###  II.3 Operational Interpretation of Universal Uncertainty Relation:
Quantum Roulette

Universal uncertainty relations capture the essential trade-off between
incompatible measurements in terms of their probability vectors, leading to a
family of uncertainty relations. Although it can outperform previous well-
known results, such as the Maassen-Uffink entropic uncertainty relation (see
Eq. 82) to some extent, its operational interpretation remains unknown. In
this subsection, we will connect the direct-sum majorization uncertainty
relation with a guessing game which we refer to as the quantum roulette.

In particular, quantum roulette is a game played between two parties – Alice
and Bob. In this game, Alice is an agent that is capable of probing the
quantum circuit fragment $\Phi$ (see Def. I.2) supplied by Bob with two
possible interactive measurements,
$\mathcal{T}_{1}:=\\{\mathcal{T}_{1,x_{1}}\\}_{x_{1}=1}^{m_{1}}$ and
$\mathcal{T}_{2}:=\\{\mathcal{T}_{2,x_{2}}\\}_{x_{2}=2}^{m_{2}}$ (see Def.
I.3). Generally, these interactive measurements do not need to have the same
number of outcomes, i.e. $m_{1}\neq m_{2}$. In each round of the game, Alice
and Bob begin with a ‘roulette table’, whose layout consists of all tuples
$(b,x_{b})$, where $b\in\\{1,2\\}$ denotes Alice’s choice of interactive
measurements, and $x_{b}\in\\{1,\ldots,m_{b}\\}$ represents the corresponding
measurement outcome by implementing $\mathcal{T}_{b}$. Bob starts the game
with $k$ chips, which he can use to place bets on $k$ of the possible tuples
and supplies Alice with any $\Phi\in\mathfrak{F}_{a}$ of his choosing. Alice
will then select some $b$ at random and probe $\Phi$ with interactive
measurement $\mathcal{T}_{b}\in\mathfrak{T}_{a}$. She finally announces both
$b$ and the resulting measurement outcome $x_{b}$. Bob wins if one of his
chips is on $(b,x_{b})$. We denote Bob’s maximum winning probability as
$p_{\text{win},k}$, and our goal is to characterize this maximum winning
probability for any quantum circuit fragment $\mathfrak{F}_{a}$.

To gain some intuition, let us first consider the simplest case in which Bob
can only provide quantum causal map $\Phi_{B\to AC}\in\mathfrak{F}_{2}$ (see
Subsec. I.4 for more details). Meanwhile, assume the interactive measurements
chosen by Alice are $\mathcal{T}_{1}$ of Eq. 48 and $\mathcal{T}_{2}$ of Eq.
49, where $x_{1}=x\in\\{1,\ldots,m\\}$ and $x_{2}=y\in\\{1,\ldots,n\\}$. In
this case, the outcomes $x$ of $\mathcal{T}_{1}$ and $y$ of $\mathcal{T}_{2}$
happen with the following probabilities (see Eqs. 52 and 53)

$\displaystyle\frac{1}{2}p_{x}$ $\displaystyle=\frac{1}{2}J^{\Phi}\star
J^{\mathcal{T}_{1}}_{x},$ (89) $\displaystyle\frac{1}{2}q_{y}$
$\displaystyle=\frac{1}{2}J^{\Phi}\star J^{\mathcal{T}_{2}}_{y}.$ (90)

Here the coefficient $1/2$ comes from the fact that interactive measurements
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ have the same probability of
occurring. Now Bob’s guessing strategy can be described by two sets
$\mathcal{S}_{1}\subset\\{1,\ldots,m\\}$ and
$\mathcal{S}_{2}\subset\\{1,\ldots,n\\}$ satisfying

$\displaystyle|\mathcal{S}_{1}|+|\mathcal{S}_{2}|=k,$ (92)

where the set $\mathcal{S}_{b}$ ($b\in\\{1,2\\}$) demonstrates the collection
of his guesses in the form of $(b,\cdot)$. For example, when Bob has $2$
chips, $\mathcal{S}_{1}=\\{2,3\\}$ means that he will put his chips on $(1,2)$
and $(1,3)$. When Bob has $3$ chips, with $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ are given by the sets $\\{2,3\\}$ and $\\{2\\}$
respectively, then he will place the chips on $(1,2)$, $(1,3)$ and $(2,2)$.
Given $k$ chips and a quantum causal map $\Phi$, Bob’s maximum winning
probability is characterized by

$\displaystyle p_{\text{win},k}(\Phi)$
$\displaystyle=\max_{|\mathcal{S}_{1}|+|\mathcal{S}_{2}|=k}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{1}\subset\\{1,\ldots,m\\}\\\
y\in\mathcal{S}_{2}\subset\\{1,\ldots,n\\}\end{subarray}}\left(\frac{1}{2}p_{x}+\frac{1}{2}q_{y}\right)$
(93)
$\displaystyle=\max_{|\mathcal{S}_{1}|+|\mathcal{S}_{2}|=k}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{1}\subset\\{1,\ldots,m\\}\\\
y\in\mathcal{S}_{2}\subset\\{1,\ldots,n\\}\end{subarray}}\frac{1}{2}\left(J^{\Phi}\star(J^{\mathcal{T}_{1}}_{x}+J^{\mathcal{T}_{2}}_{y})\right).$
(94)

The second equation comes from Eqs. 52 and 53. Note that the largest
probability of $k$ possible tuples is given by

$\displaystyle\sum_{i=1}^{k}(\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q})_{i}^{\downarrow},$
(95)

where the superscript $\downarrow$ indicates that the corresponding vector is
arranged in non-increasing order, and the subscript $i$ means it is the $i$-th
element of the vector. Take $(0.1,0.1,0.6,0.2)$ for instance, we have

$\displaystyle(0.1,0.1,0.6,0.2)_{1}^{\downarrow}$ $\displaystyle=0.6,$ (96)
$\displaystyle(0.1,0.1,0.6,0.2)_{2}^{\downarrow}$ $\displaystyle=0.2,$ (97)
$\displaystyle(0.1,0.1,0.6,0.2)_{3}^{\downarrow}$ $\displaystyle=0.1,$ (98)
$\displaystyle(0.1,0.1,0.6,0.2)_{4}^{\downarrow}$ $\displaystyle=0.1.$ (99)

It now follows immediately that

$\displaystyle\sum_{i=1}^{k}(\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q})_{i}^{\downarrow}=p_{\text{win},k}(\Phi)=\max_{|\mathcal{S}_{1}|+|\mathcal{S}_{2}|=k}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{1}\subset\\{1,\ldots,m\\}\\\
y\in\mathcal{S}_{2}\subset\\{1,\ldots,n\\}\end{subarray}}\frac{1}{2}\left(J^{\Phi}\star(J^{\mathcal{T}_{1}}_{x}+J^{\mathcal{T}_{2}}_{y})\right).$
(100)

There are two trivial case of quantum roulette: (i) $k=0$, which means Bob
does not have any chips on hand. Thus, his winning probability is simply zero.
In this case, we can write $p_{\text{win},0}(\Phi)=0$; (i) $k=m+n$. In this
case, Bob has enough chips to select all tuples, namely
$p_{\text{win},m+n}(\Phi)=1$. Our goal here is to find the fundamental
limitations of winning the game for all causal maps, which is defined as

$\displaystyle p_{\text{win},k}$
$\displaystyle:=\max_{\Phi\in\mathfrak{F}_{2}}p_{\text{win},k}(\Phi)$ (101)
$\displaystyle=\max_{|\mathcal{S}_{1}|+|\mathcal{S}_{2}|=k}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{1}\subset\\{1,\ldots,m\\}\\\
y\in\mathcal{S}_{2}\subset\\{1,\ldots,n\\}\end{subarray}}\frac{1}{2}\left(\max_{\Phi\in\mathfrak{F}_{2}}J^{\Phi}\star(J^{\mathcal{T}_{1}}_{x}+J^{\mathcal{T}_{2}}_{y})\right).$
(102)

Thus, for any k ($1\leqslant k\leqslant m+n$) and quantum causal map supplied
by Bob, the probability vectors $\mathbf{p}$ and $\mathbf{q}$ obtained from
interactive measurements $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ satisfy

$\displaystyle\sum_{i=1}^{k}(\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q})_{i}^{\downarrow}\leqslant
p_{\text{win},k},\quad 1\leqslant k\leqslant m+n,$ (103)

leading to

$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\left(p_{\text{win},1},(p_{\text{win},2}-p_{\text{win},1}),\ldots,(p_{\text{win},m+n}-p_{\text{win},m+n-1})\right).$
(104)

Above majorization inequality connects the direct-sum majorization uncertainty
relation with Bob’s maximum winning probability in quantum roulette, showing
that the largest summation of $k$ elements of
$\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}$ is completely determined by
Bob’s maximum winning probability with $k$ chips in quantum roulette. Denote
the increment of Bob’s maximum winning probability with $k$ rather than $k-1$
chips as $w_{k}$, i.e.

$\displaystyle w_{k}:=p_{\text{win},k}-p_{\text{win},k-1},\quad 1\leqslant
k\leqslant m+n,$ (105)

with $p_{\text{win},0}:=0$, and set
$\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}:=(w_{k})_{k=1}^{m+n}$, we see
that

$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}.$
(106)

To further simplify the representation, we take the union of CJ operators
$J^{\mathcal{T}_{1}}_{x}$ and $J^{\mathcal{T}_{2}}_{y}$, and denote it by
$\\{J_{z}\\}_{z}$, where

$J_{z}=\left\\{\begin{array}[]{rcl}J^{\mathcal{T}_{1}}_{z}&&1\leqslant
z\leqslant m,\\\ J^{\mathcal{T}_{2}}_{z-m}&&m+1\leqslant z\leqslant
m+n.\end{array}\right.$ (107)

We adopt the convention $J_{\mathcal{S}}$ for

$\displaystyle J_{\mathcal{S}}:=\sum_{z\in\mathcal{S}}J_{z},$ (108)

where $\mathcal{S}\subset\\{1,\ldots,m+n\\}$, e.g. for
$\mathcal{S}=\\{2,5\\}$, we immediately have $J_{\\{2,5\\}}=J_{2}+J_{5}$. To
more precise, if $m=n=3$, then we then see that

$\displaystyle
J_{\\{2,5\\}}=J_{2}+J_{5}=J^{\mathcal{T}_{1}}_{2}+J^{\mathcal{T}_{2}}_{2}.$
(109)

Based on CJ operators, now the universal uncertainty relation in the form of
direct-sum can be formulated as

Theorem II.7:  Universal Uncertainty Relation for $\mathfrak{F}_{2}$ For any
quantum causal map $\Phi\in\mathfrak{F}_{2}$, the probability vectors
$\mathbf{p}$ and $\mathbf{q}$ obtained by measuring $\Phi$ with respect to
interactive measurements $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}\in\mathfrak{T}_{2}$ satisfy the following trade-off
$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}=(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{m+n},$
(110) where $p_{\text{win},0}:=0$, and for $1\leqslant k\leqslant m+n$, the
quantity $p_{\text{win},k}$ is characterized by the following optimization
problem. $\displaystyle
p_{\text{win},k}=\frac{1}{2}\max_{|\mathcal{S}|=k}\max\quad$
$\displaystyle\Tr[J_{\mathcal{S}}\cdot J]$ $\displaystyle{\rm s.t.}\quad$
$\displaystyle\,J\geqslant
0,\quad\Tr[\rho_{A}]=1,\quad\Tr_{C}[J]=\rho_{A}\otimes\mathds{1}_{B}.$ (111)
In particular, here the optimal solution of $p_{\text{win},k}$ forms a
semidefinite program (SDP), and hence can be computed efficiently in
polynomial time (up to desired accuracy) by using the ellipsoid method or an
interior point method [34, 35].

###### Proof.

From Eq. 102, we can write

$\displaystyle p_{\text{win},k}$
$\displaystyle=\frac{1}{2}\max_{|\mathcal{S}|=k}\max_{\Phi\in\mathfrak{F}_{2}}J^{\Phi}\star
J_{\mathcal{S}}$ (112)
$\displaystyle=\frac{1}{2}\max_{|\mathcal{S}|=k}\max_{\Phi\in\mathfrak{F}_{2}}\Tr[J^{\Phi}\cdot
J_{\mathcal{S}}],$ (113)

where in the second equation, namely Eq. 113, we have used the fact that the
set of all CJ operators associated with causal map is algebraically closed
under the transpose over all of its systems. Remark that, given a CJ operator
$J^{\Phi}_{ABC}$ for some causal map $\Phi_{B\to AC}$, the operator
$(J^{\Phi}_{ABC})^{\mathbf{T}}$ is still a CJ operator for some causal map.
However, this may not be the case for partial transpose, such as
$(J^{\Phi}_{ABC})^{\mathbf{T}_{B}}$. As discussed in Subsec. I.4, a quantum
dynamics $\Phi_{B\to AC}$ is a causal map if and only if its CJ operator
$J^{\Phi}$ satisfies the requirements of (i) CP: $J^{\Phi}\geqslant 0$, (ii)
TP: $\Tr_{AC}[J^{\Phi}]=\mathds{1}_{B}$ and (iii) NS from $B\to C$ to $A$:
$\Tr_{C}[J^{\Phi}]=\Tr_{BC}[J^{\Phi}]\otimes\mathds{1}_{B}/d_{B}$. It hence
follows that

$\displaystyle p_{\text{win},k}=\frac{1}{2}\max_{|\mathcal{S}|=k}\max\quad$
$\displaystyle\Tr[J_{\mathcal{S}}\cdot J]$ $\displaystyle{\rm s.t.}\quad$
$\displaystyle\,J\geqslant 0,\quad\Tr_{AC}[J]=\mathds{1}_{B},$
$\displaystyle\Tr_{C}[J]=\Tr_{BC}[J]\otimes\mathds{1}_{B}/d_{B},$ (114)

which is equivalent to Eq. 111, as required. ∎

In general, for different quantum causal map $\Phi$, we will have different
probability vectors $\mathbf{p}$ and $\mathbf{q}$ with respect to interactive
measurements $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$. To indicate their
dependencies, we re-express them as $\mathbf{p}(\Phi)$ and $\mathbf{q}(\Phi)$,
then the set
$\\{\frac{1}{2}\mathbf{p}(\Phi)\oplus\frac{1}{2}\mathbf{q}(\Phi)\\}_{\Phi}$
forms a subset of the probability simplex. Constructing the probability vector
$\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}$ is exactly the first step of
finding the optimal bounds for
$\\{\frac{1}{2}\mathbf{p}(\Phi)\oplus\frac{1}{2}\mathbf{q}(\Phi)\\}_{\Phi}$
(see Eq. 61 of Subsec. II.1), and the next step is flatness process
$\mathcal{F}$ (see Def. II.1), which implies that

$\displaystyle\vee\left\\{\frac{1}{2}\mathbf{p}(\Phi)\oplus\frac{1}{2}\mathbf{q}(\Phi)\right\\}_{\Phi}^{\downarrow}=\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}).$
(115)

Writing above result formally, we have

Theorem II.8:  Optimal Universal Uncertainty Relation for $\mathfrak{F}_{2}$
For any quantum causal map $\Phi\in\mathfrak{F}_{2}$, the probability vectors
$\mathbf{p}$ and $\mathbf{q}$ obtained by measuring $\Phi$ with respect to
interactive measurements $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}\in\mathfrak{T}_{2}$ satisfy the following trade-off
$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}),$
(116) where $\mathcal{F}$ is the flatness process defined in Def. II.1. The
vector $\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}$ is defined as
$\displaystyle\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}:=(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{m+n},$
(117) with $p_{\text{win},0}:=0$ and $p_{\text{win},k}$ is solved by SDPs in
the form of Eq. 110 for $1\leqslant k\leqslant m+n$. Here the bound
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ is optimal, due to
the structure of majorization lattice, namely for any probability vector
$\mathbf{x}$ satisfying
$\displaystyle\left\\{\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\right\\}_{\Phi}\prec\mathbf{x},$
(118) then we also have
$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})\prec\mathbf{x}.$
(119) Physically, the causal map independence of $\mathbf{x}$ leads to an
universal uncertainty relation
$\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathbf{x}$, and
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ constructed here
is the optimal one. Mathematically,
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ is the GLB (see
Def. II.1) for all such $\mathbf{x}$. In particular, taken any Schur-concave
function $f$, we have $\displaystyle
f(\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q})\geqslant
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}))\geqslant
f(\mathbf{x}).$ (120)

Here we would like to remark that permutations will induce an equivalence
relation, denoted as $\sim$, on the probability simplex. Given two vectors
$\mathbf{x}$ and $\mathbf{y}$, we say that they are equivalent, written as

$\displaystyle\mathbf{x}\sim\mathbf{y},$ (121)

if there exists a permutation matrix $D$ such that

$\displaystyle\mathbf{x}=\mathbf{y}\cdot D.$ (122)

To illustrate $\sim$, let us consider $(0.2,0.1,0.7)$ and $(0.7,0.2,0.1)$.
Clearly, there exists a permutation matrix satisfying Eq. 122, then we have

$\displaystyle(0.2,0.1,0.7)\sim(0.7,0.2,0.1).$ (123)

The equivalence class of $\mathbf{x}$ under $\sim$, denoted
$[\mathbf{x}^{\downarrow}]$, is defined as

$\displaystyle[\mathbf{x}^{\downarrow}]:=\\{\mathbf{y}\,|\,\mathbf{y}\sim\mathbf{x}\\}.$
(124)

Thus, as a corollary of Thm. II.3, we obtain the following result about
universal uncertainty relation

Corollary II.9:  Optimal Universal Uncertainty Relation for $\mathfrak{F}_{2}$
under Permutations Given a quantum causal map $\Phi\in\mathfrak{F}_{2}$, let
us denote the probability vectors obtained by measuring $\Phi$ with respect to
interactive measurements $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}\in\mathfrak{T}_{2}$ as $\mathbf{p}$ and $\mathbf{q}$. Define
the vector $\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}$ as
$(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{m+n}$ with $p_{\text{win},0}:=0$
and $p_{\text{win},k}$ in the form of Eq. 110 for $1\leqslant k\leqslant m+n$.
Then for any $\mathbf{p}^{{}^{\prime}}\in[\mathbf{p}^{\downarrow}]$,
$\mathbf{q}^{{}^{\prime}}\in[\mathbf{q}^{\downarrow}]$, and
$\bf{w}^{{}^{\prime}}\in[\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})]$,
we have the following universal uncertainty relation
$\displaystyle\frac{1}{2}\mathbf{p}^{{}^{\prime}}\oplus\frac{1}{2}\mathbf{q}^{{}^{\prime}}\prec{\bf
w}^{{}^{\prime}}.$ (125) In particular, given any permutation matrices
$D_{1}$, $D_{2}$, and $D_{3}$, we also have
$\displaystyle\frac{1}{2}(\mathbf{p}^{{}^{\prime}}\cdot
D_{1})\oplus\frac{1}{2}(\mathbf{q}^{{}^{\prime}}\cdot D_{2})\prec{\bf
w}^{{}^{\prime}}\cdot D_{3}.$ (126) Moreover, if there exists a causal map
independent probability vector $\mathbf{x}$, such that
$\displaystyle\left\\{\frac{1}{2}\mathbf{p}^{{}^{\prime}}\oplus\frac{1}{2}\mathbf{q}^{{}^{\prime}}\right\\}_{\Phi}\prec\mathbf{x},$
(127) then it is straightforward to see that $\displaystyle{\bf
w}^{{}^{\prime}}\cdot D\prec\mathbf{x},$ (128) holds for any permutation
matrix $D$.

We now move on to investigating the most general form of quantum roulette,
where the player Bob can engineer any quantum circuit fragment
$\Phi\in\mathfrak{F}_{a}$ with finite integer $a$. On top of that, the agent
Alice will select an interactive measurement $\mathcal{T}_{b}$ with
$b\in\\{1,2,\ldots,c\\}$ at random and probe $\Phi$ with her choice of
interactive measurement $\mathcal{T}_{b}$. Without loss of generality, let us
assume the quantum circuit fragment (see Def. I.2) provided by Bob is in the
form of

$\displaystyle\Phi=\Psi^{a}\circ\Psi^{a-1}\circ\cdots\circ\Psi^{2}(\rho)\in\mathfrak{F}_{a}.$
(129)

Meanwhile, the interactive measurement
$\mathcal{T}_{b}:=\\{\mathcal{T}_{b,x_{b}}\\}_{x_{b}=1}^{m_{b}}\in\mathfrak{T}_{a}$
($b=1,2,\ldots,c$) (see Def. I.3) is constituted of

$\displaystyle\mathcal{T}_{b,x_{b}}(\cdot):=\Tr[M_{b,x_{b}}\cdot\Lambda^{a-1}_{b}\circ\Lambda^{a-2}_{b}\circ\cdots\circ\Lambda^{1}_{b}(\cdot)].$
(130)

Here $\\{M_{b,x_{b}}\\}_{x_{b}=1}^{m_{b}}$ forms a POVM for each
$b\in\\{1,2,\ldots,c\\}$. In this case, the layout of roulette table consists
of $\sum_{b=1}^{c}m_{b}$ tuples $(b,x_{b})$, where $b\in\\{1,2,\ldots,c\\}$
and $x_{b}\in\\{1,2,\ldots,m_{b}\\}$. In particular, for Bob’s quantum circuit
fragment $\Phi$, the tuple $(b,x_{b})$ happens with probability
$p_{x_{b}}(\Phi,\mathcal{T}_{b})/c$, where the quantity
$p_{x_{b}}(\Phi,\mathcal{T}_{b})$ is given by

$\displaystyle
p_{x_{b}}(\Phi,\mathcal{T}_{b}):=\Tr[M_{b,x_{b}}\cdot\Psi^{a}\circ\Lambda^{a-1}_{b}\circ\Psi^{a-1}\circ\Lambda^{a-2}_{b}\circ\cdots\circ\Psi^{2}\circ\Lambda^{1}_{b}(\rho)].$
(131)

Similar to the case of quantum causal maps, when starts with $k$ chips, Bob’s
guessing strategy is characterized by $c$ sets
$\mathcal{S}_{b}\subset\\{1,\ldots,m_{b}\\}$ such that

$\displaystyle\sum_{b=1}^{c}|\mathcal{S}_{b}|=k,$ (132)

and his maximum winning probability is given by

$\displaystyle p_{\text{win},k}(\Phi)$
$\displaystyle=\max_{\sum_{b=1}^{c}|\mathcal{S}_{b}|=k}\sum_{x_{b}\in\mathcal{S}_{b}\subset\\{1,\ldots,m_{b}\\}}\frac{1}{c}\cdot
p_{x_{b}}(\Phi,\mathcal{T}_{b})$ (133)
$\displaystyle=\max_{\sum_{b=1}^{c}|\mathcal{S}_{b}|=k}\sum_{x_{b}\in\mathcal{S}_{b}\subset\\{1,\ldots,m_{b}\\}}\frac{1}{c}\left(J^{\Phi}\star
J^{\mathcal{T}_{b}}_{x_{b}}\right),$ (134)

where the operator $J^{\Phi}$ and $J^{\mathcal{T}_{b}}_{x_{b}}$ stand for the
CJ operators of quantum circuit fragment $\Phi$ and the component
$\mathcal{T}_{b,x_{b}}$ of interactive measurement $\mathcal{T}_{b}$
respectively. To simplify our representation, we take the union of CJ
operators $J^{\mathcal{T}_{b}}_{x_{b}}$, and denote it by $\\{J_{z}\\}_{z}$,
which is formally defined as

$\displaystyle
J_{z}:=J^{\mathcal{T}_{b}}_{z-\sum_{j=1}^{b-1}m_{j}},\quad\text{for}\,\,\,\sum_{j=1}^{b-1}m_{j}+1\leqslant
z\leqslant\sum_{j=1}^{b}m_{j},$ (135)

and set $J_{\mathcal{S}}$ as

$\displaystyle J_{\mathcal{S}}:=\sum_{z\in\mathcal{S}}J_{z},$ (136)

where $\mathcal{S}$ is a subset of $\\{1,\ldots,\sum_{b=1}^{c}m_{b}\\}$.
Before going further, let us consider an example of $J_{\mathcal{S}}$. Assume
$m_{1}=m_{2}=m_{3}=3$, then $J_{\\{2,5,8\\}}$ is an abbreviation of the
following summation,

$\displaystyle
J_{\\{2,5,8\\}}=J_{2}+J_{5}+J_{8}=J^{\mathcal{T}_{1}}_{2}+J^{\mathcal{T}_{2}}_{2}+J^{\mathcal{T}_{3}}_{2}.$
(137)

Thus, given quantum circuit fragment $\Phi$, Bob’s maximum winning
probability, i.e. Eq. 134, can be simplified to the following form in terms of
the CJ operators and link product,

$\displaystyle
p_{\text{win},k}(\Phi)=\frac{1}{c}\max_{|\mathcal{S}|=k}J^{\Phi}\star
J_{\mathcal{S}}.$ (138)

Denote the probability vector obtained by measuring $\Phi$ with respect to
interactive measurements $\mathcal{T}_{b}$ as $\mathbf{p}_{b}$, then the
largest summation of $k$ elements of $\oplus_{b=1}^{c}\mathbf{p}_{b}/c$ is
exactly $p_{\text{win},k}(\Phi)$. Thus, written out explicitly, we have

$\displaystyle\sum_{i=1}^{k}(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})_{i}^{\downarrow}=p_{\text{win},k}(\Phi)=\frac{1}{c}\max_{|\mathcal{S}|=k}J^{\Phi}\star
J_{\mathcal{S}},$ (139)

where the superscript $\downarrow$ indicates that the corresponding vector is
arranged in non-increasing order, and the subscript $i$ means it is the $i$-th
element of the vector. By maximizing over all quantum circuit fragment
$\Phi\in\mathfrak{F}_{a}$, we have the following fundamental limitation
$p_{\text{win},k}$ for Bob’s winning probability, namely

$\displaystyle
p_{\text{win},k}:=\max_{\Phi\in\mathfrak{F}_{a}}p_{\text{win},k}(\Phi).$ (140)

It hence follows that for any quantum circuit fragment
$\Phi\in\mathfrak{F}_{a}$ prepared by Bob, his winning probability with $k$
chips is upper-bounded by $p_{\text{win},k}$,

$\displaystyle\sum_{i=1}^{k}(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})_{i}^{\downarrow}=p_{\text{win},k}(\Phi)\leqslant
p_{\text{win},k},$ (141)

for $1\leqslant k\leqslant\sum_{b=1}^{c}m_{b}$. It is worth mentioning that
the maximum winning probability $p_{\text{win},k}$ for Bob is achievable.
Then, by the definition of majorization (see Def. II.1), we can unify
$\sum_{b=1}^{c}m_{b}$ inequalities in the form of Eq. 141 into a single
majorization inequality; that is

$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}},$
(142)

where the probability vector
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ only relies on the set
of interactive measurements, i.e.
$\\{\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c}\\}$, and
independent of the quantum circuit fragment $\Phi$ chosen by Bob. Formally, we
define $\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ as

$\displaystyle\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}:=(w_{k})_{k=1}^{\sum_{b=1}^{c}m_{b}}=(w_{1},\ldots,w_{\sum_{b=1}^{c}m_{b}}),$
(143)

with each component $w_{k}$ is defined as

$\displaystyle w_{k}:=p_{\text{win},k}-p_{\text{win},k-1},$ (144)

for $1\leqslant k\leqslant\sum_{b=1}^{c}m_{b}$, and $p_{\text{win},0}:=0$.
Using the language of CJ operators, the maximum winning probability
$p_{\text{win},k}$ can be solved by using SDPs [7, 6],

$\displaystyle
p_{\text{win},k}=\frac{1}{c}\max_{|\mathcal{S}|=k}\max_{\Phi\in\mathfrak{F}_{a}}J^{\Phi}\star
J_{\mathcal{S}}=\frac{1}{c}\max_{|\mathcal{S}|=k}\max\quad$
$\displaystyle\Tr[J(a)\cdot J_{\mathcal{S}}]$ $\displaystyle{\rm s.t.}\quad$
$\displaystyle\,J(a)\geqslant 0,\quad\Tr_{\mathcal{H}_{1}}[J(1)]=1,$
$\displaystyle\Tr_{\mathcal{H}_{2i-1}}[J(i)]=\Tr_{\mathcal{H}_{2i-1}\mathcal{H}_{2i-2}}[J(i)]\otimes\frac{\mathds{1}_{\mathcal{H}_{2i-2}}}{d_{\mathcal{H}_{2i-2}}},\quad\text{for}\,\,\,2\leqslant
i\leqslant a,$ (145)

where $d_{\mathcal{H}_{2i-2}}:=\dim\mathcal{H}_{2i-2}$. From a causal
perspective, the last restriction of Eq. II.3 indicates that the quantum
circuit fragment $\Phi$ is NS from $\Psi^{i}$ to $\Psi^{i-1}$ for $2\leqslant
i\leqslant a$ with $\Psi^{1}:=\rho$ (see Eq. 129). Now the universal
uncertainty relation for multiple interactive measurements can be formulated
as

Theorem II.10:  Multiple Universal Uncertainty Relation for $\mathfrak{F}_{a}$
For any quantum circuit fragment $\Phi\in\mathfrak{F}_{a}$, the probability
vector $\mathbf{p}_{b}$ obtained by measuring $\Phi$ with respect to
interactive measurements $\mathcal{T}_{b}\in\mathfrak{T}_{a}$
($b\in\\{1,2,\ldots,c\\}$) satisfies the following trade-off
$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}=(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{\sum_{b=1}^{c}m_{b}},$
(146) where $p_{\text{win},0}:=0$, and for $1\leqslant
k\leqslant\sum_{b=1}^{c}m_{b}$, the quantity $p_{\text{win},k}$ is
characterized by Eq. II.3. Similar to the case of causal maps (see Eq. 111),
the optimal solution of $p_{\text{win},k}$ forms a semidefinite program (SDP),
and hence can be computed efficiently in polynomial time (up to desired
accuracy) by using the ellipsoid method or an interior point method [34, 35].

As discussed in Subsec. II.1, constructing
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ is the first step of
formulating the optimal bound of
$\left\\{\oplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\right\\}_{\Phi}$, and the
next step is the flatness process $\mathcal{F}$, which implies that

Theorem II.11:  Optimal Multiple Universal Uncertainty Relation for
$\mathfrak{F}_{a}$ For any quantum circuit fragment $\Phi\in\mathfrak{F}_{a}$,
the probability vector $\mathbf{p}_{b}$ obtained by measuring $\Phi$ with
respect to interactive measurements $\mathcal{T}_{b}\in\mathfrak{T}_{a}$
($b\in\\{1,2,\ldots,c\\}$) satisfies the following trade-off
$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}),$
(147) where $\mathcal{F}$ is the flatness process defined in Def. II.1. The
vector $\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ is defined as
$\displaystyle\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}:=(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{\sum_{b=1}^{c}m_{b}},$
(148) with $p_{\text{win},0}:=0$ and $p_{\text{win},k}$ is solved by SDPs in
the form of Eq. II.3 for $1\leqslant k\leqslant\sum_{b=1}^{c}m_{b}$. Here the
bound $\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$ is
optimal, due to the structure of majorization lattice, namely for any
probability vector $\mathbf{x}$ satisfying
$\displaystyle\left\\{\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\right\\}_{\Phi}\prec\mathbf{x},$
(149) then we also have
$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})\prec\mathbf{x}.$
(150) Physically, the quantum circuit fragment independence of $\mathbf{x}$
leads to a universal uncertainty relation
$\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{x}$, and
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$ constructed
here is the optimal one. Mathematically,
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$ is the GLB
(see Def. II.1) for all such $\mathbf{x}$. In particular, taken any Schur-
concave function $f$, we have $\displaystyle
f(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})\geqslant
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))\geqslant
f(\mathbf{x}).$ (151)

From the equivalence relation $\sim$ based on permutations, we know that
permutations will not change the order of majorization. As an illustration,
let us consider two equivalence class $[\mathbf{x}]$ and $[\mathbf{y}]$. If
$\mathbf{x}\prec\mathbf{y}$, then it is straightforward to check that
$\mathbf{x}\cdot D_{1}\prec\mathbf{y}\cdot D_{2}$ holds for any permutation
matrices $D_{1}$ and $D_{2}$. Using this insight, we obtain the following
corollary.

Corollary II.12:  Optimal Universal Uncertainty Relation for
$\mathfrak{F}_{a}$ under Permutations Given a quantum circuit fragment
$\Phi\in\mathfrak{F}_{a}$, let us denote the probability vectors obtained by
measuring $\Phi$ with respect to interactive measurement
$\mathcal{T}_{b}\in\mathfrak{T}_{a}$ ($b\in\\{1,2,\ldots,c\\}$) as
$\mathbf{p}_{b}$. Define the vector
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ as
$(p_{\text{win},k}-p_{\text{win},k-1})_{k=1}^{\sum_{b=1}^{c}m_{b}}$ with
$p_{\text{win},0}:=0$ and $p_{\text{win},k}$ in the form of Eq. II.3 for
$1\leqslant k\leqslant\sum_{b=1}^{c}m_{b}$. Then for any
$\mathbf{p}^{{}^{\prime}}_{b}\in[\mathbf{p}^{\downarrow}_{b}]$, and
$\bf{w}^{{}^{\prime}}\in[\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})]$,
we have the following universal uncertainty relation
$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}{{}^{\prime}}\prec{\bf
w}^{{}^{\prime}}.$ (152) In particular, given any permutation matrices
$D_{1},D_{2},\ldots,D_{c+1}$, we also have
$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}(\mathbf{p}_{b}{{}^{\prime}}\cdot
D_{b})\prec{\bf w}^{{}^{\prime}}\cdot D_{c+1}.$ (153) Moreover, if there
exists a quantum circuit fragment independent probability vector $\mathbf{x}$,
such that
$\displaystyle\left\\{\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}{{}^{\prime}}\right\\}_{\Phi}\prec\mathbf{x},$
(154) then it is straightforward to see that $\displaystyle{\bf
w}^{{}^{\prime}}\cdot D\prec\mathbf{x},$ (155) holds for any permutation
matrix $D$.

So far we’ve been talking about the formulation of universal uncertainty
relation and its intrinsic connection with quantum roulette, but the
advantages of using this form are just as important. One of the greatest
advantages of formulating Eq. 147 is its universality. Here we addressed the
question of how universal our framework is, and highlight the generality of
our uncertainty relations.

Uncertainty principle in quantum mechanics has revolutionized our view of the
physical world. Roughly speaking, it captures the uncertainty trade-off
between observables associated with static system – quantum states. However,
systems in quantum mechanics evolve. Hence, quantum dynamics with definite
causal order, known as quantum circuit fragment in this work, emerge as a
central ingredient in state-of-the-art quantum technologies. Our framework
extends uncertainty principle to incorporate quantum dynamics, allowing for
any quantum circuit fragment $\Phi\in\mathfrak{F}_{a}$. In particular, our
framework, including theorems (Thms. II.3 and II.3) and corollary (Cor. II.3),
does not focus on a particular choice of quantum circuit fragment. By
employing our theory, one can build uncertainty relations for any quantum
dynamics with definite causal order, e.g., based upon the universal
uncertainty relation for quantum causal maps, i.e. $\Phi\in\mathfrak{F}_{2}$
(see Thms. II.3 and II.3), we can derived the causal uncertainty relation,
which characterizes the fundamental trade-off between causal structures, such
as direct-cause and common-cause in causal inference. Besides causality, it
would also be interesting to apply our framework to other physical properties,
such as completely positive divisibility, non-Markovianity, and so forth, of
quantum dynamics, and identify the corresponding trade-off. This could be a
good issue for future works.

Finally, we ask if the uncertainty relation
$f(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})\geqslant
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))$ induced
by some Schur-concave function $f$ is optimal? Unfortunately, we cannot
guarantee the optimality of
$f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))$ with
specific uncertainty measure $f$. There might exist a quantum circuit fragment
independent vector $\mathbf{x}$ such that

$\displaystyle f(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})\geqslant
f(\mathbf{x})\geqslant
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})).$ (156)

Remark that Eq. 156 only holds for a specific $f$, and is not equivalent to
$\oplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{x}\prec\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$.
In this case, there must exist another uncertainty measure $g$, leading to the
following uncertainty relation

$\displaystyle g(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})\geqslant
g(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))\geqslant
g(\mathbf{x}).$ (157)

This in turn implies that, as a lower bound of uncertainty relation, the
quantity $g(\mathbf{x})$ is weaker than
$g(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))$. From
above discussion, it is clear that there does not exist another circuit
fragment independent vector $\mathbf{x}$ that always outperform our bound
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$. Actually,
if there exist a quantum circuit fragment independent vector $\mathbf{x}$ such
that Eq. 156 holds for any uncertainty measure $f$, i.e. Schur-concave
function, then Lem. II.1 immediately implies that

$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{x}\prec\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}),$
(158)

holds for any $\Phi\in\mathfrak{F}_{a}$. Note that vector
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$ is the LUB
of set $\left\\{\oplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\right\\}_{\Phi}$, we
then find that the reverse is also true, i.e.
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})\prec\mathbf{x}$.
Therefore, they are equivalent under permutations, namely

$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})\sim\mathbf{x},$
(159)

or there exists a permutation matrix $D$ such that

$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})=\mathbf{x}\cdot
D,$ (160)

which implies

$\displaystyle
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))=f(\mathbf{x}),$
(161)

holds for any Schur-concave function $f$. Put simply, the completeness of
majorization lattice guarantee the optimality of Eq. 147, and there does exist
another circuit fragment independent vector that outperforms
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$.

###  II.4 Lemma $1$ and Theorem $1$ of the Main Text: Their Proofs,
Improvements, and Generalizations

In this subsection, we prove the results introduced in the main text of our
work, including Lemma $1$ and Theorem $1$. Let us begin with Lemma $1$ of the
main text, namely

Corollary II.13:  Lemma $1$ of the Main Text Collect the probability
distributions obtained by implementing interactive measurements
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ on some dynamical process $\Phi$ as
$\mathbf{p}$ and $\mathbf{q}$, then there exists a probability vector
$\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2})$ such that
$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2}).$
(162) The vector-type bound
$\displaystyle\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2}):=\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}},$
(163) is independent of the dynamical process $\Phi$ being measured, and hence
captures the essential incompatibility between $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$.

###### Proof.

By restricting Thm. II.3 to the case of two interactive measurements, i.e.
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, we immediately obtain Eq. 164, where
the upper bound under majorization is completely characterized by the winning
probability of quantum roulette. ∎

From the algebraic structure of majorization lattice (see Subsec. II.1), we
know that the upper bound $\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}$
provided by Cor. II.4 (or originally by Thm. II.3) may not be optimal. Thanks
to Thm. II.3, the optimal bound for $\mathbf{p}/2\oplus\mathbf{q}/2$ under
majorization can be obtained by further applying the flatness process
$\mathcal{F}$ (see Def. II.1); that is

Corollary II.14:  Improved Lemma $1$ of the Main Text Collect the probability
distributions obtained by implementing interactive measurements
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ on some dynamical process $\Phi$ as
$\mathbf{p}$ and $\mathbf{q}$, then there exists a probability vector
$\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2})$ such that
$\displaystyle\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\prec\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2}).$
(164) The vector-type bound
$\displaystyle\mathbf{v}(\mathcal{T}_{1},\mathcal{T}_{2}):=\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}),$
(165) is independent of the dynamical process $\Phi$ being measured, and hence
captures the essential incompatibility between $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$. Here $\mathcal{F}$ is the flatness process defined in Def.
II.1. Remark that, the bound
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ is optimal: for
any probability vector $\mathbf{x}$ satisfying
$\displaystyle\left\\{\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q}\right\\}_{\Phi}\prec\mathbf{x},$
(166) then we also have
$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})\prec\mathbf{x}.$
(167) Physically, the quantum circuit fragment independence of $\mathbf{x}$
leads to a universal uncertainty relation
$\mathbf{p}/2\oplus\mathbf{q}/\prec\mathbf{x}$, and
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ constructed here
is the optimal one. Mathematically,
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})$ is the GLB (see
Def. II.1) for all such $\mathbf{x}$. In particular, taken any Schur-concave
function $f$, we have $\displaystyle
f(\frac{1}{2}\mathbf{p}\oplus\frac{1}{2}\mathbf{q})\geqslant
f(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}))\geqslant
f(\mathbf{x}).$ (168)

We now move on to demonstrating a generalized entropic uncertainty relation
with multiple interactive measurements, which includes the Theorem $1$ in the
main text as a special case. Moreover, compare with the bound
$C(\mathcal{T}_{1},\mathcal{T}_{2}):=2H(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})-2$
offered in the main text, an improved entropic uncertainty relation has also
been provided. We start by writing down a direct consequence of Thm. II.3:

Theorem II.15:  Entropic Uncertainty Relation for Measurements with
Interventions Given $c$ interactive measurements
$\mathcal{T}_{b}\in\mathfrak{T}_{a}$ ($b\in\\{1,2,\ldots,c\\}$) acting on some
quantum circuit fragment $\Phi\in\mathfrak{F}_{a}$. The entropy of their
measurement outcomes, when summed, satisfies
$\displaystyle\sum_{b=1}^{c}H(\mathcal{T}_{b})_{\Phi}\geqslant
C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c}):=cH(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-c\log
c,$ (169) where $H(\mathcal{T}_{b})_{\Phi}:=H(\mathbf{p}_{b})$, with
$\mathbf{p}_{b}$ denotes the probability vectors obtained by measuring $\Phi$
with respect to interactive measurement $\mathcal{T}_{b}$. The bound
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ – measuring
incompatibility between interactive measurements $\mathcal{T}_{b}$ – is non-
negative and independent of $\Phi$.
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ can be explicitly
computed and is strictly non-zero whenever these interactive measurements have
no common eigencircuit.

###### Proof.

From Eq. 146 of Thm. II.3, we see that

$\displaystyle\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b}\prec\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}.$
(170)

Then, by applying Shannon entropy $H$, it now follows immediately that

$\displaystyle H(\bigoplus_{b=1}^{c}\frac{1}{c}\mathbf{p}_{b})$
$\displaystyle=-\sum_{b,x_{b}}\frac{p_{x_{b}}(\Phi,\mathcal{T}_{b})}{c}\log\frac{p_{x_{b}}(\Phi,\mathcal{T}_{b})}{c}$
(171)
$\displaystyle=-\frac{1}{c}\sum_{b,x_{b}}p_{x_{b}}(\Phi,\mathcal{T}_{b})\log
p_{x_{b}}(\Phi,\mathcal{T}_{b})+\frac{\sum_{b,x_{b}}p_{x_{b}}(\Phi,\mathcal{T}_{b})}{c}\log
c$ (172)
$\displaystyle=\frac{1}{c}\sum_{b=1}^{c}H(\mathcal{T}_{b})_{\Phi}+\log c$
(173) $\displaystyle\geqslant
H(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}),$ (174)

where the second equation comes from the fact that, for each
$b\in\\{1,2,\ldots,c\\}$, we have
$\sum_{x_{b}}p_{x_{b}}(\Phi,\mathcal{T}_{b})=1$, and thus
$\sum_{b}\sum_{x_{b}}p_{x_{b}}(\Phi,\mathcal{T}_{b})=c$. The third equation
follows from the definition, i.e.

$\displaystyle
H(\mathcal{T}_{b})_{\Phi}:=H(\mathbf{p}_{b})=-\sum_{x_{b}}p_{x_{b}}(\Phi,\mathcal{T}_{b})\log
p_{x_{b}}(\Phi,\mathcal{T}_{b}).$ (175)

Note that throughout this supplemental material, all logarithms are base $2$.
Now we have

$\displaystyle\sum_{b=1}^{c}H(\mathcal{T}_{b})_{\Phi}\geqslant
c\left(H(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-\log
c\right)=cH(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-c\log c.$
(176)

Bu further define

$\displaystyle
C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c}):=cH(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-c\log
c,$ (177)

we obtain the entropic uncertainty relation of Eq. 169, as required. Remark
that, as a probability vector with $\sum_{b=1}^{c}m_{b}$ components, our bound
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ is bounded by

$\displaystyle(\underbrace{\frac{1}{\sum_{b=1}^{c}m_{b}},\ldots,\frac{1}{\sum_{b=1}^{c}m_{b}}}_{\sum_{b=1}^{c}m_{b}})\prec\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}\prec(\underbrace{\underbrace{\frac{1}{c},\ldots,\frac{1}{c}}_{c},\underbrace{0,\ldots,0}_{\sum_{b=1}^{c}m_{b}-c}}_{\sum_{b=1}^{c}m_{b}}),$
(178)

where the captions in Eq. 178 indicate the number of elements inside the
vector. Hence

$\displaystyle
H(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})\geqslant\log c,$ (179)

and the bound
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})=cH(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-c\log
c$ is non-negative. The majorization upper bound of Eq. 178 is achieved, i.e.
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}=(1/c,\ldots,1/c,0,\ldots,0)$,
if and only if $(\mathbf{p}_{b})^{\downarrow}_{1}=1$ for all
$b\in\\{1,2,\ldots,c\\}$, which is equivalent to say that the bound
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ is zero whenever
these interactive measurements
$\\{\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c}\\}$ have a common
eigencircuit $\Phi\in\mathfrak{F}_{a}$ (see Def. I.3). According to the
definition of $\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$, we know
that $C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ is
independent of the quantum circuit fragment $\Phi$, and hence quantifies the
inherent incompatibility between interactive measurements. Eq. II.3 further
ensures that $\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ (and thus
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$) can be explicitly
computed. ∎

Thm. II.4 forms a fundamental limitation of incompatible interactive
measurements in terms of Shannon entropy, covering all quantum dynamics with
definite causal order (called quantum circtuit fragment in this work, see Def.
I.2 for more details). In fact, Eq. 169 works for quantum circuit fragment
$\Phi\in\mathfrak{F}_{a}$ with arbitrary $a$ and arbitrary number of
interactive measurements, and hence generates an infinite number of entropic
uncertainty relations. As a special case, we consider the situation where
$c=2$, i.e. a pair of interactive measurements, and obtain the following
corollary straightforwardly, which is exactly Theorem $1$ of the main text.

Corollary II.16:  Theorem $1$ of the Main Text Given two interactive
measurements $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ acting on some quantum
circuit fragment $\Phi\in\mathfrak{F}_{a}$. The entropy of their measurement
outcomes, when summed, satisfies $\displaystyle
H(\mathcal{T}_{1})_{\Phi}+H(\mathcal{T}_{2})_{\Phi}\geqslant
C(\mathcal{T}_{1},\mathcal{T}_{2}),$ (180) where
$H(\mathcal{T}_{b})_{\Phi}:=H(\mathbf{p}_{b})$, with $\mathbf{p}_{b}$ denotes
the probability vectors obtained by measuring $\Phi$ with respect to
interactive measurement $\mathcal{T}_{b}$ ($b=1,2$). The bound $\displaystyle
C(\mathcal{T}_{1},\mathcal{T}_{2}):=2H(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})-2,$
(181) which measures incompatibility between $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, is non-negative and independent of $\Phi$.
$C(\mathcal{T}_{1},\mathcal{T}_{2})$ can be explicitly computed and is
strictly non-zero whenever these interactive measurements have no common
eigencircuit.

Here we have two remarks: First, when the quantum circuit fragment $\Phi$
degenerates to the case of a quantum channel, we can simply adopt the
framework introduced in [25] to construct
$C(\mathcal{T}_{1},\mathcal{T}_{2})$, which includes a Maassen-Uffink type
uncertainty relation. However, in such a case, the intervention is absent.
Second, till now we have a variety of different forms of entropic uncertainty
relations [33], but none of them can surpass all the others. Among them, the
form derived from universal uncertainty relation outperforms other forms of
entropic uncertainty relations in a large number of instances. For readers who
are unfamiliar with entropic uncertainty relation and majorization, recent
works, and overviews exist (e.g., Refs. [21, 23, 33]).

If the vector $\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ is not
arranged in non-increasing order, then the flatness process $\mathcal{F}$ will
further improve the result [20]. Recall the example of
$S=\\{\mathbf{x}=(0.6,0.15,0.15,0.1),\mathbf{y}=(0.5,0.25,0.2,0.05)\\}$
considered in Subsec. II.1, where

$\displaystyle\mathcal{F}(\mathbf{b}_{S})=(0.6,0.175,0.175,0.05)\prec\mathbf{b}_{S}=(0.6,0.15,0.2,0.05).$
(182)

Using this insight, we can employ Eq. 147 instead of Eq. 146 to formulate
entropic uncertainty relation. Thus, written out explicitly, we have

Theorem II.17:  Improved Entropic Uncertainty Relation for Measurements with
Interventions Given $c$ interactive measurements
$\mathcal{T}_{b}\in\mathfrak{T}_{a}$ ($b\in\\{1,2,\ldots,c\\}$) acting on some
quantum circuit fragment $\Phi\in\mathfrak{F}_{a}$. The entropy of their
measurement outcomes, when summed, satisfies
$\displaystyle\sum_{b=1}^{c}H(\mathcal{T}_{b})_{\Phi}\geqslant
C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c}):=cH(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))-c\log
c,$ (183) where $H(\mathcal{T}_{b})_{\Phi}:=H(\mathbf{p}_{b})$, with
$\mathbf{p}_{b}$ denotes the probability vectors obtained by measuring $\Phi$
with respect to interactive measurement $\mathcal{T}_{b}$, and $\mathcal{F}$
stands for the flatness process defined in Def. II.1. The bound
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ – measuring
incompatibility between interactive measurements $\mathcal{T}_{b}$ – is non-
negative and independent of $\Phi$.
$C(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{c})$ can be explicitly
computed and is strictly non-zero whenever these interactive measurements have
no common eigencircuit.

Thm. II.4 follows directly by replacing
$\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}$ with
$\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})$ in the
proof of Thm. II.4. Thanks to the property of flatness process $\mathcal{F}$,
now we have

$\displaystyle\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})\prec\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}},$
(184)

so that

$\displaystyle
H(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))\geqslant
H(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}).$ (185)

Written in full, that is

$\displaystyle\sum_{b=1}^{c}H(\mathcal{T}_{b})_{\Phi}\geqslant
cH(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}}))-c\log
c\geqslant cH(\mathbf{w}_{\mathcal{T}_{1},\ldots,\mathcal{T}_{c}})-c\log c.$
(186)

In particular, given two interactive measurements $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, we would like to show that

Corollary II.18:  Improved Theorem $1$ of the Main Text Given two interactive
measurements $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ acting on some quantum
circuit fragment $\Phi\in\mathfrak{F}_{a}$. The entropy of their measurement
outcomes, when summed, satisfies $\displaystyle
H(\mathcal{T}_{1})_{\Phi}+H(\mathcal{T}_{2})_{\Phi}\geqslant
C(\mathcal{T}_{1},\mathcal{T}_{2}),$ (187) where
$H(\mathcal{T}_{b})_{\Phi}:=H(\mathbf{p}_{b})$, with $\mathbf{p}_{b}$ denotes
the probability vectors obtained by measuring $\Phi$ with respect to
interactive measurement $\mathcal{T}_{b}$ ($b=1,2$). The bound $\displaystyle
C(\mathcal{T}_{1},\mathcal{T}_{2}):=2H(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}))-2\log
2,$ (188) which measures incompatibility between $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, is non-negative and independent of $\Phi$.
$C(\mathcal{T}_{1},\mathcal{T}_{2})$ can be explicitly computed and is
strictly non-zero whenever these interactive measurements have no common
eigencircuit.

Similar to Eq. 186, we have the following relation between Cor. II.4 and Cor.
II.4 (i.e. Theorem $1$ of the main text),

$\displaystyle H(\mathcal{T}_{1})_{\Phi}+H(\mathcal{T}_{2})_{\Phi}\geqslant
2H(\mathcal{F}(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}}))-2\log 2\geqslant
2H(\mathbf{w}_{\mathcal{T}_{1},\mathcal{T}_{2}})-2\log 2.$ (189)

To summarize, in this subsection, Thm. II.4 introduces the entropic
uncertainty relation for arbitrary quantum circuit fragments with multiple
interactive measurements, which includes Cor. II.4 (namely, Theorem $1$ of the
main text) as a special case. By applying the flatness process, Thm. II.4 and
Cor. II.4 further improve the results presented in Thm. II.4 and Cor. II.4
respectively.

##  III Causal Uncertainty Relation

In this section, we extend the uncertainty principle from studying observables
for quantum states to inferring causal structures of quantum dynamics,
providing a lower bound for the fundamental trade-off between maximal common-
cause indicator and the maximal direct-cause indicator. Such a trade-off is
called causal uncertainty relation in this work. Specifically, the causal
uncertainty relations in terms of both Shannon entropy and majorization are
provided. We further infer the causality associated with system-environment
unitary dynamics by utilizing entropic causal uncertainty relation. In Subsec.
III.1, we introduce the concepts of maximal common-cause indicator and maximal
direct-cause indicator, and show that it is possible to formulate a universal
uncertainty relation for these two families of interactive measurements. As a
special case, we obtain the entropic uncertainty relation for maximal common-
cause indicators and maximal direct-cause indicators, characterizing the
incompatibility between common-cause and direct-cause in quantum causal
inference. We then discuss the necessary and sufficient conditions for system-
environment unitary dynamics to be purely common-cause and purely direct-
cause, and further detail the application of our causal uncertainty relation
to inferring causality in Subsec. III.2. Finally, in Subsec. III.3, we
demonstrate that the parameterized quantum circuit considered in our main text
contains a coherent superpositions of common-cause and direct-cause as a
special case.

###  III.1 Uncertainty Relation for Common-Cause and Direct-Cause: Eq. $3$ of
the Main Text and Its Extension

Physicists have studied the fundamental limitations of observable pairs for
quantum states – like the position and momentum of a particle, the phase and
excitation number of a harmonic oscillator, the orthogonal components of spin
angular momentum, and etc. Even with the complete description of the quantum
state, it is impossible to predict the outcomes of these observable pairs.
Such a restriction leads to a key principle in quantum mechanics – known as
Heisenberg’s uncertainty principle. However, less is known about the intrinsic
constraints on underlying physical properties of quantum dynamics. In this
subsection, we give a complete characterization of the uncertainty trade-off
between common-cause and direct-cause, establishing uncertainty relations for
the maximal common-cause indicator and the maximal direct-cause indicator.
Unlike previous studies with static object (e.g. quantum states), here we
focus on the dynamical process of quantum causal maps, i.e. $\mathfrak{F}_{2}$
(see Subsec. I.4).

There are so many types of interactive measurements that can be used to get
information about the dynamical properties of quantum causal maps. For the
purpose of causal inference, which of them are likely to be most useful to us?
For these useful interactive measurements, how to characterize their
incompatibility and formulate the corresponding trade-off between them? To
address these questions and explain our uncertainty relation for common-cause
and direct-cause, let us first recall two families of interactive measurements
introduced in the main text – the set of all maximal common-cause indicators
$\mathcal{M}_{\text{CC}}\subset\mathfrak{T}_{2}$ and the set of all maximal
direct-cause indicators $\mathcal{M}_{\text{DC}}\subset\mathfrak{T}_{2}$ (see
Fig. $4$ of the main text for an illustration). For consistency purposes, we
use the same notation as in Subsec. I.4, in particular that a causal map
$\Phi\in\mathfrak{F}_{2}$ is a linear map from system $B$ to systems $A$ and
$C$. Without loss of generality, we assume $d_{A}=d_{B}=d_{C}=d$ in this work.
Formally, the set $\mathcal{M}_{\text{CC}}$ is defined as

Definition III.1:  Maximal Common-Cause Indicator An interactive measurement
$\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}):=\\{\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})\\}_{i}\in\mathfrak{T}_{2}$
is called maximal common-cause indicator, if it has the form of
$\displaystyle\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})(\cdot)=\Tr[\Phi_{i}\cdot\mathcal{U}_{2}(\cdot)\otimes\frac{\mathbb{1}_{B}}{d}\otimes\mathcal{U}_{1}(\cdot)],$
(190) where $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are some local unitary
channels acting on systems $A$ and $C$ respectively, namely
$\mathcal{U}_{b}(\cdot)=U_{b}(\cdot)U_{b}^{{\dagger}}$ ($b=1,2$), with
$\ket{\Phi_{1}}:=\ket{\phi^{+}}=\sum_{k=0}^{d-1}\ket{kk}/\sqrt{d}$ being the
maximally entangled state. Measurements are done with respect to a maximally
entangling basis $\\{\Phi_{i}\\}_{i}$ with $d^{2}$ possible outcomes. Denote
the collection of all maximal common-cause indicators as
$\mathcal{M}_{\text{CC}}$.

Remark that, in the main text of our work, we simply denote the elements of
$\mathcal{M}_{\text{CC}}$ as $\mathcal{T}_{1}$. But, to identify the
dependence and causation, we keep all the subscripts and arguments of
interactive measurements in this Supplemental Material. On the other hand, the
maximal direct-cause indicator is defined as

Definition III.2:  Maximal Direct-Cause Indicator An interactive measurement
$\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4}):=\\{\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})\\}_{j}\in\mathfrak{T}_{2}$
is called maximal direct-cause indicator, if it has the form of
$\displaystyle\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})(\cdot)=\Tr[\Phi_{j}\cdot\mathcal{U}_{4}(\cdot)\otimes\mathcal{U}_{3}(\Phi_{1})\otimes\Tr_{A}(\cdot)],$
(191) where $\mathcal{U}_{3}$ and $\mathcal{U}_{4}$ are some local unitary
channels acting on systems $B$ and $C$ respectively, namely
$\mathcal{U}_{b}(\cdot)=U_{b}(\cdot)U_{b}^{{\dagger}}$ ($b=1,2$).
$\ket{\Phi_{1}}:=\ket{\phi^{+}}=\sum_{k=0}^{d-1}\ket{kk}/\sqrt{d}$ is the
maximally entangled state acting on systems $BR$. Measurements are done with
respect to a maximally entangling basis $\\{\Phi_{j}\\}_{j}$ with $d^{2}$
possible outcomes. Denote the collection of all maximal direct-cause
indicators as $\mathcal{M}_{\text{DC}}$.

The circuit illustrations of Def. III.1 and Def. III.1 are given by Figs. 4a
and 4b of the main text. The principal objective of this subsection is to
characterize the fundamental trade-off between
$\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2})\in\mathcal{M}_{\text{CC}}$
and
$\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4})\in\mathcal{M}_{\text{DC}}$,
and formulate their uncertainties in terms of Shannon entropy. To simplify our
discussion and gain some intuitions, here we mainly focus on the qubit case,
where $d_{A}=d_{B}=d_{C}=2$. The general qudit case can be obtained by a
straightforward extension of the procedure.

Before stating our uncertainty relation for common-cause and direct-cause, let
us first introduce some notations related with maximal common-cause indicator
and maximal direct-cause indicator. Denote the probability vectors obtained by
measuring causal map $\Phi$ with respect to interactive measurements
$\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}):=\\{\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})\\}_{i}\in\mathfrak{T}_{2}$
and
$\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4}):=\\{\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})\\}_{j}\in\mathfrak{T}_{2}$
as
$\mathbf{p}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}=(p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi})_{i}$
and
$\mathbf{q}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}=(q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi})_{j}$,
where each $p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}$ and
$q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}$ are given by

$\displaystyle p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}$
$\displaystyle=\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})(\Phi)=J^{\Phi}\star
J^{\text{CC}}_{i}(\mathcal{U}_{1},\mathcal{U}_{2}),$ (192) $\displaystyle
q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}$
$\displaystyle=\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})(\Phi)=J^{\Phi}\star
J^{\text{DC}}_{j}(\mathcal{U}_{3},\mathcal{U}_{4}).$ (193)

Here $J^{\text{CC}}_{i}(\mathcal{U}_{1},\mathcal{U}_{2})$ and
$J^{\text{DC}}_{j}(\mathcal{U}_{3},\mathcal{U}_{4})$ are CJ operators of the
measuring processes
$\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})$ and
$\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})$ respectively. The
probability $p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}$ is a functional of
unitary channels $\mathcal{U}_{1}$, $\mathcal{U}_{2}$, and the causal map
$\Phi$. Similarly, the probability
$q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}$ is a functional of unitary
channels $\mathcal{U}_{3}$, $\mathcal{U}_{4}$, and the causal map $\Phi$. In
the case of qubit systems, the CJ operators of
$\mathcal{T}_{\text{CC},i}(\mathcal{U}_{1},\mathcal{U}_{2})$ and
$\mathcal{T}_{\text{DC},j}(\mathcal{U}_{3},\mathcal{U}_{4})$ are written as

$\displaystyle J^{\text{CC}}_{i}(\mathcal{U}_{1},\mathcal{U}_{2})$
$\displaystyle=\frac{\mathds{1}_{B}}{2}\otimes(U_{1}^{\dagger}\otimes
U_{2}^{\dagger}\outerproduct{\Phi_{i}}{\Phi_{i}}U_{1}\otimes
U_{2})^{\mathbf{T}},\quad i\in\\{1,2,3,4\\}.$ (194) $\displaystyle
J^{\text{DC}}_{j}(\mathcal{U}_{3},\mathcal{U}_{4})$
$\displaystyle=\mathds{1}_{A}\otimes(U_{3}\outerproduct{\Phi_{1}}{\Phi_{1}}_{BR}U_{3}^{\dagger})\star(U_{4}^{\dagger}\outerproduct{\Phi_{j}}{\Phi_{j}}_{CR}U_{4})^{\mathbf{T}},\quad
j\in\\{1,2,3,4\\}.$ (195)

The coefficient of $1/2$ in Eq. 194 comes from the fact that $d_{B}=2$,
ensuring $\mathbb{1}_{B}/2$ is a maximally mixed state on system $B$. In Eq.
195, $\mathbb{1}_{A}$ is the CJ operator of $\Tr_{A}$ (partial trace over
system $A$), i.e. $J^{\Tr_{A}}=\mathbb{1}_{A}$. For example, given a bipartite
quantum state $\rho_{AB}$, it reduced state on $B$ system can be rewritten as
$\rho_{B}:=\Tr_{A}[\rho_{AB}]=\rho_{AB}\star\mathbb{1}_{A}$.

Now the quantum uncertainty associated with the maximal common-cause indicator
$\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2})$ and the maximal
direct-cause indicator
$\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4})$ can be quantified
through the Shannon entropy of probability vectors
$\mathbf{p}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}$ and
$\mathbf{q}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}$. Mathematically, they are
given by

$\displaystyle
H(\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}))_{\Phi}$
$\displaystyle:=H(\mathbf{p}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi})=-\sum_{i}p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}\log
p_{i}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi},$ (196) $\displaystyle
H(\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4}))_{\Phi}$
$\displaystyle:=H(\mathbf{q}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi})=-\sum_{j}q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}\log
q_{j}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}.$ (197)

Particularly, we are interested in the minimization of their joint uncertainty
over all causal maps $\Phi\in\mathfrak{F}_{2}$, maximal common-cause
indicators
$\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2})\in\mathcal{M}_{\text{CC}}$,
and maximal direct-cause indicators
$\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4})\in\mathcal{M}_{\text{DC}}$,
which is

$\displaystyle\mathcal{B}:$
$\displaystyle=\min_{\begin{subarray}{c}\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2})\in\mathcal{M}_{\text{CC}}\\\
\mathcal{T}_{\text{DC}}(\mathcal{U}_{3},\mathcal{U}_{4})\in\mathcal{M}_{\text{DC}}\end{subarray}}\min_{\Phi}\Big{\\{}H(\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}))_{\Phi}+H(\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}))_{\Phi}\Big{\\}}$
(198)
$\displaystyle=\min_{\mathcal{U}_{1},\mathcal{U}_{2},\mathcal{U}_{3},\mathcal{U}_{4}}\min_{\Phi}\Big{\\{}H(\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}))_{\Phi}+H(\mathcal{T}_{\text{CC}}(\mathcal{U}_{1},\mathcal{U}_{2}))_{\Phi}\Big{\\}}.$
(199)

We note that the characterization of $\mathcal{B}$ identifies the uncertainty
trade-off between two families of interactive measurements, leading to a
quantitative connection between different causal structures – common-cause and
direct-cause – in quantum theories.

Roughly speaking, the universal uncertainty relation investigated in Subsec.
II.3 is the key to solving $\mathcal{B}$. In the first stage, we collect all
the measurement data into a set $\mathcal{Q}$, and study its algebraic
properties. Formally, the set $\mathcal{Q}$ is defined as

$\displaystyle\mathcal{Q}:=\\{\mathbf{p}(\mathcal{U}_{1},\mathcal{U}_{2})_{\Phi}\oplus\mathbf{q}(\mathcal{U}_{3},\mathcal{U}_{4})_{\Phi}\\}_{\mathcal{U}_{1},\ldots,\mathcal{U}_{4},\Phi},$