# Certifying Temporal Correlations

Harshank Shrotriya  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, Singapore 117543  Leong-Chuan Kwek  Centre for
Quantum Technologies, National University of Singapore, 3 Science Drive 2,
Singapore 117543  MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit,
UMI 3654, Singapore  National Institute of Education, Nanyang Technological
University, 1 Nanyang Walk, Singapore 637616  School of Electrical and
Electronic Engineering Block S2.1, 50 Nanyang Avenue, Singapore 639798  Kishor
Bharti  Joint Center for Quantum Information and Computer Science and Joint
Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742,
USA  Institute of High Performance Computing, Agency for Science, Technology
and Research (A*STAR), 1 Fusionopolis Way, Connexis, Singapore 138632,
Singapore

###### Abstract

Self-testing has been established as a major approach for quantum device
certification based on experimental statistics with minimal assumptions.
However, despite more than 20 years of research effort, most of the self-
testing protocols are restricted to spatial scenarios (Bell scenarios),
without many temporal generalizations known. Under the scenario of sequential
measurements performed on a single quantum system, we build on previous works
using semi-definite programming (SDP) techniques to bound sequential
measurement inequalities. For such SDPs, we show that the optimizer matrix is
unique and, moreover, this uniqueness is robust to small deviations from the
quantum bound. Furthermore, we consider a generalized scenario in the presence
of quantum channels and draw analogies to the structure of Bell and sequential
inequalities using the pseudo-density matrix formalism. These analogies allow
us to show a practical use of maximal violations of sequential inequalities in
the form of certification of quantum channels up to isometries.

Introduction – Newton, in his Principia Mathematica, defined time to be
absolute and uniform throughout the universe. This notion was altered in 1905
by Einstein in his Special Theory of Relativity when he showed that inertial
reference frames have their own relative clocks, thus placing time and space
on equal footing. More recently, such analogies in space-time structure have
also been explored in quantum time crystals [1] where the atoms in a crystal
are arranged periodically in both space and time. From the perspective of
quantum correlations, similar ideas have been explored with respect to spatial
and temporal scenarios [2, 3, 4, 5]. These examples also illuminate a symmetry
in the role played by space and time in our understanding of the universe. The
goal of this work is to explore the similar structure for the temporal
analogue of self-testing, which emerges from temporal quantum correlations.

Since its inception in [6], self-testing [7] has become a prominent technique
to certify quantum states in the device independent scenario [8, 9, 10, 11,
12], with extensions in literature to semi-device independent [13, 14, 15, 16,
17] and single party contextuality based scenarios [18, 19, 20]. Broadly
speaking, self testing is a technique to infer the hidden quantum state and
measurements that lead to extremum quantum correlations in a black-box
scenario. In the prominent scenario of spatially separated parts of an
entangled state, self testing is made possible due to properties of Bell
inequalities[21] i.e. quantum realizations that achieve maximal violations of
Bell inequalities are unique up to local isometries. Applications of self
testing in such scenarios have been explored for device-independent (DI)
randomness generation[22, 23, 24, 25, 26, 27], DI quantum cryptography via
quantum key distribution [28, 29, 30, 31] and delegated quantum computing [32,
33, 34] among others. All these applications inherently share the idea of
quantum state certification without explicitly using self-testing protocols
making them essential for the progress of quantum technologies. Also, self
testing has recently garnered interest as a theoretical tool with applications
in quantum complexity theory [35].

However, most of the applications of self testing are limited to scenarios
with entanglement shared between spatially separated observers without any
temporal analogue. A scenario involving sequential measurements on a single
quantum system has already been explored from the perspective of self-testing
in [18] where non-contextuality inequalities [36] supplant the role of Bell
inequalities. However, sequential measurements for this purpose need to form a
context or, in other words, they need to commute. This constraint on the
sequential measurements is lifted in [37] by considering sequential
correlation inequalities, such as the Leggett-Garg inequality [38], where the
measurements performed in the sequence do not necessarily commute and the
classical bound is applicable for all theories that obey macro-realism. In
this unconstrained scenario, a semi-definite programming (SDP) based method
can be used to find quantum bounds of sequential correlation-based
inequalities. An extension of sequential measurements to Bell scenarios is
considered in [39] where an infinite hierarchy of SDPs is shown to bound the
set of sequential correlations. The sequential measurement scenario also
allows for the interpretation of such correlations as temporal correlations in
analogy with spatial correlations since measurements are done on the same
state at different times. Temporal correlations, as obtained in the sequential
measurement scenarios, have yet to find applications in quantum technologies
save in few cases such as [40] where authors show that genuine temporal
correlations can be used to certify the minimum dimension of the underlying
quantum state. In [41] authors consider the prepare-transform-measure scenario
to certify a single qubit preparation (state) and intermediate measurements up
to global unitary degree of freedom based on sequential correlation based
witnesses. More recently, in [42], a sequential measurement scenario has been
considered where maximal violations of temporal inequalities were used to
certify arbitrary outcome measurements under certain assumptions regarding the
underlying state.
A radical approach is adopted in [43] where the authors show how time
evolution of a qubit can be viewed as temporal teleportation using Pseudo-
density matrix (PDM) formalism [44] in analogy to the spatial teleportation
protocol using entangled states [45]. In [5], authors study the geometry of
two sequential Pauli measurements on a qubit in a more general scenario where
the effect of an arbitrary quantum channel between the two sequential
measurements is considered.

In this work, we study correlations arising from sequential measurements on a
single party system and its generalizations to scenarios with quantum
channels. Since such correlations cannot be used to certify the underlying
state or measurements due to the single system isometries involved, we
identify the correlation matrix itself as the candidate for certification.
Under the SDP-based formulation, we use maximal violations of sequential
inequalities to infer certain properties of uniqueness of the optimizer
matrix. Furthermore, we consider the generalized scenario where quantum
channels act between the sequence of measurements and use the PDM formalism to
put spatial (Bell) and temporal (sequential) inequality violations on the same
footing. The PDM based approach helps to identify local isometries in the
temporal setting and also helps to formulate statements about the underlying
channels that lead to maximal violations of sequential inequalities.

##  I Bounding temporal correlations via SDP

Consider the scenario of sequential measurements performed on a single qudit
state. In order to compute quantum bounds on the inequalities constructed from
such correlations between sequential measurements of binary valued $\left(\pm
1\right)$ observables $\\{A_{i}\\}$ a semi-definite programming (SDP) based
strategy was proposed in [37]. Optimization is performed for temporal
inequalities constructed from correlations between sequential measurements on
a single quantum state $\rho$. For an inequality
$C=\sum_{ij}\lambda_{ij}X_{ij}$ where $\lambda_{ij}$ are the coefficients
corresponding to the term $X_{ij}=\langle A_{i}A_{j}\rangle_{seq}$ with

$\langle A_{i}A_{j}\rangle_{seq}=\frac{1}{2}\left[Tr\left(\rho
A_{i}A_{j}\right)+Tr\left(\rho A_{j}A_{i}\right)\right]$ (1)

the semi-definite program is given by;

$\displaystyle\mathrm{maximize:}\,$
$\displaystyle\sum_{ij}\lambda_{ij}X_{ij},$ (2)
$\displaystyle\mathrm{subject\,to}:\,$ $\displaystyle X=X^{T}\succeq
0\;\mathrm{and\,}X_{ii}=1\;\forall\,i$ (3)

The constraint $X\succeq 0$ follows from the fact that $X$ is the real part of
$Y=\mathrm{Tr}\left[\rho(A_{i}A_{j})\right]$ and $v^{T}Yv\geq 0$ for any real
vector $v$. Consider a special case of such an inequality, called an N-cycle
inequality (4), given by

$S_{N}\equiv\sum_{i=1}^{N-1}\langle A_{i}A_{i+1}\rangle_{seq}-\langle
A_{N}A_{1}\rangle_{seq}\leq N-2$ (4)

with $N\geq 3$ and the classical bound $N-2$ obeyed by all macro-realistic
theories. Invoking the strong duality for the corresponding SDP gives the
quantum bound $S_{N}\leq N\cos\left(\frac{\pi}{N}\right)$.

######  Remark 1.

Reinterpreting the correlation term $X_{ij}$ as an inner product of unit
vectors $\\{x_{i}\\}$, we can write $X_{ij}=\left(x_{i},x_{j}\right)$ thus
obtaining $\\{x_{i}\\}$ as the columns of the matrix $\sqrt{X}$. As a
consequence of this reinterpretation, for every PSD $X$ one can find a set
$\\{x_{i}\\}$ and a set of binary ($\pm 1$) outcome observables $\\{A_{i}\\}$
such that

$\langle
A_{i}A_{j}\rangle_{seq}=Tr\left[\frac{1}{2}\rho\left(A_{i}A_{j}+A_{j}A_{i}\right)\right]=\left(x_{i},x_{j}\right),$
(5)

for all quantum states $\rho$. Thus, for every quantum state $\rho$ there
exist observables $\\{A_{i}\\}$ such that $X_{ij}^{opt}=\langle
A_{i}A_{j}\rangle_{seq}=Tr\left[\frac{1}{2}\rho\left(A_{i}A_{j}+A_{j}A_{i}\right)\right]$
which maximally violate the inequality (4). Further, in the 2-qubit case it
can be seen that taking $A_{i}=\vec{a_{i}}.\vec{\sigma}$ gives $\langle
A_{i}A_{j}\rangle_{seq}=\vec{a_{i}}.\vec{a_{j}}$ implying that the
correlations do not depend on the underlying quantum state.

In section II, we show the uniqueness of the optimizer matrix $X^{opt}$ for
the N-cycle inequality (4) for all $N\geq 3$. One interesting aspect of this
approach is the fact that in the temporal scenario a single SDP suffices to
find a tight upper bound in the general case which is in contrast with the
infinite (NPA) hierarchy of SDPs required in the general case to tightly bound
Bell inequalities in the spatial correlation scenario as shown in [46, 47].
This significant increase in complexity can be attributed to the fact that the
observables acting on the spatially separated halves of the entangled state
are required to commute but no such restriction is placed on observables
acting on a single quantum system at distinct time points in the sequential
measurement scenario.

##  II Certification of temporal correlations using optimizer matrix

For the purposes of this work, we define robust certification of temporal
correlations as follows
$(\epsilon,r)$ robust certification of temporal correlations: Given 2 outcome
observables $\\{A_{i}\\}_{i}$ and physical set of sequential correlations
$\\{\langle A_{i},A_{i+1}\rangle_{seq}\\}_{i}$, they give an $(\epsilon,r)$
certificate of reference temporal correlations
$\\{\langle\tilde{A}_{i},\tilde{A}_{j}\rangle\\}_{ij}$ if the matrices $X$ and
$\tilde{X}$ defined via $X_{i,j}=\langle A_{i}A_{j}\rangle$ (similarly for
$\tilde{X}$) are close in Frobenius norm distance s.t.

$|X-\tilde{X}|\leq\mathcal{O}(\epsilon^{r}).$

######  Theorem 1.

The optimizer matrix $X^{opt}$ that maximizes the objective function
$C=\sum_{ij}\lambda_{ij}X_{ij}$ in (3) for the $N$-cycle inequality defined in
(4) is unique for all $N\geq 3$.

###### Proof.

The proof has been deferred to Appendix section C.1. ∎

We mentioned earlier that the matrix $X$ that is being optimized over is PSD
and gives us the set of unit vectors $\\{x_{i}\\}$ as the columns of
$\sqrt{X}$; however, these $\\{x_{i}\\}$ vectors are not directly relatable to
any state or measurements. Noting that $X$ is the real part of PSD matrix $Y$,
where $Y_{ij}=\mathrm{Tr}(\rho A_{i}A_{j})$, allows us to introduce set of
unit vectors $\\{y_{i}\\}$ as columns of $\sqrt{Y}$ matrix. In the special
case when $\rho=|\psi\rangle\langle\psi|$ is a pure state, we can write
$y_{i}=A_{i}|\psi\rangle$. For $X^{opt}$, we thus have $\displaystyle
X^{opt}_{ij}=\mathrm{Tr}\left[\frac{1}{2}\tilde{\rho}\left(\tilde{A_{i}}\tilde{A_{j}}+\tilde{A_{j}}\tilde{A_{i}}\right)\right]=\mathrm{Re\left\\{\mathrm{Tr}\left[\tilde{\rho}\tilde{A_{i}}\tilde{A_{j}}\right]\right\\}}=\mathrm{Re}\left\\{\langle\tilde{\psi}|\tilde{A_{i}}\tilde{A_{j}}|\tilde{\psi}\rangle\right\\}=\mathrm{Re}\left\\{\left(\tilde{y_{i}},\tilde{y_{j}}\right)\right\\}$
where in the third step we have assumed that $\tilde{\rho}$ is a pure state,
since $X^{opt}$ can be obtained using any quantum state and suitable 2-outcome
measurements. However, note that while defining the set
$\\{\tilde{y_{i}}\\}_{i}$ we have a global isometry such that there could be
another set of unit vectors $\\{y^{\prime}_{i}\\}_{i}$ satisfying
$\mathrm{Re}\left\\{\left(\tilde{y_{i}},\tilde{y_{j}}\right)\right\\}=\mathrm{Re}\left\\{\left(Uy^{\prime}_{i},Uy^{\prime}_{j}\right)\right\\}$
with $UU^{\dagger}=I$. This leads to the following,

######  Corollary 1.

For the optimizer matrix $X^{opt}$ which maximizes an N-cycle inequality of
the kind (4), for every set $\\{y_{i}\\}_{i}$ which satisfies
$X^{opt}_{ij}=\mathrm{Re}\left\\{\left(y_{i},y_{j}\right)\right\\}$, we have a
global isometry $U$ and a reference set $\\{\tilde{y_{i}}\\}_{i}$ such that

$\mathrm{Re}\left\\{\left(\tilde{y_{i}},\tilde{y_{j}}\right)\right\\}=\mathrm{Re}\left\\{\left(Uy_{i},Uy_{j}\right)\right\\}$
(6)

Furthermore this implies the existence of reference measurements
$\tilde{\\{A_{i}\\}_{i}}$ and pure state $\tilde{|\psi\rangle}$ such that
$\displaystyle\tilde{y_{i}}=\tilde{A_{i}}|\tilde{\psi}\rangle$.

Note that in Bell inequality based self testing scenarios uniqueness up to
local isometries of $A_{i}|\psi\rangle$ implies self test of the measurement
$A_{i}$; however, in the temporal scenario due to the single system global
isometry involved we can only claim that uniqueness of $X^{opt}$ fixes
$\mathrm{Re}\\{(y_{i},y_{j})\\}$.

######  Theorem 2.

Robustness: Consider the SDP (3), given that matrix $X^{real}$ achieves a
near-optimal value for the objective function $C=\sum_{ij}\lambda_{ij}X_{ij}$
such that
$|\sum_{ij}\lambda_{ij}X^{opt}_{ij}-\sum_{ij}\lambda_{ij}X^{real}_{ij}|\leq\epsilon$
and $X^{opt}$ is unique, then we can upper bound the Frobenius-norm distance
between $X^{opt}$ and $X^{real}$ as

$|X^{opt}-X^{real}|\leq\mathcal{O}(\epsilon)$ (7)

.

###### Proof.

The proof has been deferred to Appendix section C.1. ∎

The multiplicative constant going with $\mathcal{O}(\epsilon)$ certification
guarantee in Theorem 2 can be approximated by numerical SWAP based methods.
Such SWAP based techniques along with SDP based methods were used to estimate
constants in CHSH and CGLMP inequality based robust self tests in [48]. Also,
analytical robust self testing bounds for certain Bell inequalities have been
obtained in [49, 50]. The sequential measurement scenario considered here is
closer in spirit to the scenario considered in [20] where numerical SWAP based
techniques were adapted for robust self testing of a single quantum system.

######  Remark 2.

It follows from Theorem 2 that near maximal violation of (4) by the set
$\\{\langle A_{i},A_{i+1}\rangle_{seq}\\}_{i}$ such that

$\left|\left(\sum_{i=1}^{N-1}\langle A_{i}A_{i+1}\rangle_{seq}-\langle
A_{N}A_{1}\rangle_{seq}\right)-\beta_{q}\right|\leq\epsilon$

(where $\beta_{q}$ is the quantum bound for (4)) is an $(\epsilon,1)$
certificate for the reference set
$\\{\langle\tilde{A}_{i},\tilde{A}_{j}\rangle_{seq}\\}_{ij}$ with
$\langle\tilde{A}_{i},\tilde{A}_{j}\rangle_{seq}=X^{opt}_{i,j}$.

##  III Generalized scenario with quantum channels

In this section we generalize the sequential measurement scenario considered
in the previous part by introducing a quantum channel between the measurements
of Alice and Bob as shown in Fig. 1. In the case of arbitrary channels
$\mathcal{E}_{A|B}$ acting in between the time points, the sequential
correlation is

$\langle
A_{m}A_{n}\rangle_{seq}=\frac{1}{2}\mathrm{Tr}\Big{[}(A_{m}\rho_{A}+\rho_{A}A_{m})\mathcal{E}_{A|B}^{\dagger}(A_{n})\Big{]},$

as a result of which the matrix $X_{ij}=\langle A_{i}A_{j}\rangle_{seq}$ is
not generally PSD in this scenario. Hence, one cannot use the SDP method
described earlier in Sec. I to bound sequential measurement inequalities.

Figure 1: In the generalized scenario, Alice obtains black box containing
$\rho_{A}$ and makes measurement $A_{m}\in\\{M_{i}\\}$ at time $t_{A}$, then
the black box containing post measured state $\rho_{A}^{\prime}$ is sent via
channel $\mathcal{E}_{A|B}$ to Bob who performs measurement
$A_{n}\in\\{M_{i}\\}$ at time $t_{B}$.

The sequential measurement scenario can be viewed from the perspective of
Pseudo Density Matrix (PDM) formalism (see Appendix section A for a review) by
considering the measurement events in a sequence and writing the corresponding
PDM. The PDM formalism in general is more powerful since it can also take into
account time evolution (via quantum channels) of the quantum state between the
two sequential measurement events. Consider a 2-dimensional quantum state
$\rho_{A}$ with sequential measurements being performed at time points $t_{A}$
and $t_{B}$, additionally the channel $\mathcal{E}_{A|B}$ acts on $\rho_{A}$
mapping operators from state space $\mathcal{H}_{A}$ at $t_{A}$ to state space
$\mathcal{H}_{B}$ at $t_{B}$. Then it can be shown [5] that the PDM $R_{AB}$
for this scenario is given by

$\displaystyle R_{AB}$
$\displaystyle=(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})\Big{\\{}\rho_{A}\otimes\frac{I}{2},\;\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}\Big{\\}}$
(8)

where $\mathcal{I}_{A}$ is the identity superoperator.

######  Lemma 1.

In this generalized scenario

$\mathrm{Tr}\left[(A_{m}\otimes A_{n})R_{AB}\right]=\langle
A_{m}A_{n}\rangle_{seq}$

###### Proof.

The proof has been deferred to Appendix section C.2. ∎

The tensor structure introduced in Lemma 1 allows for certain local isometries
to act in the temporal sense as

$\displaystyle\begin{aligned} \mathrm{Tr}\left[(A_{m}\otimes
A_{n})R_{AB}\right]&=\mathrm{Tr}\left[(A_{m}\otimes
A_{n})(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})\Big{\\{}\rho_{A}\otimes\frac{I}{2},\;\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}\Big{\\}}\right]\\\
&=\mathrm{Tr}\left[(VA_{m}V^{\dagger}\otimes
UA_{n}U^{\dagger})(\mathcal{I}_{A}\otimes\tilde{\mathcal{E}}_{A|B})\Big{\\{}V\rho_{A}V^{\dagger}\otimes\frac{I}{2},\;\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}\Big{\\}}\right]\\\
\end{aligned}$ (9)

where $V$ is restricted to the set of Pauli matrices i.e. $V\in\\{X,Y,Z\\}$
acting locally at time point $t_{A}$, $U$ could be any unitary operator acting
locally at $t_{B}$ and $\tilde{\mathcal{E}}_{A|B}$ is the channel with
transformed Kraus operators
$\\{K_{i}\\}\longrightarrow\\{UK_{i}V^{\dagger}\\}$ as those of
$\mathcal{E}_{A|B}$ (see Appendix section D for details). Thus, the
transformation of the channel $\mathcal{E}_{A|B}$’s Kraus operators along with
corresponding rotation of observables at times $t_{A}$ and $t_{B}$ does not
change the correlation value. This local isometry in time is reminiscent of
the local isometries present in the scenario with spatially separated
entangled states and observables.

Here, we will use the PDM formalism to explore the maximal violation of
sequential inequalities in the generalized scenario. Going back to $N$-cycle
sequential inequality (4), using Lemma 1 we can write:

$\mathrm{Tr}\Bigg{[}\Bigg{(}\sum_{i=0}^{N-2}A_{i}\otimes
A_{i+1}-A_{N-1}\otimes A_{0}\Bigg{)}R_{AB}\Bigg{]}\leq N-2$ (10)

We note that (10) possesses a structure similar to the Bell inequalities under
entanglement-based scenarios where the tensor product appears due to spatially
separated Hilbert spaces.

Remark – Considering the spatial analogue of (10) where the measurement events
are performed on spatially separated qubits with $N=4$ shows that the PDMs
maximally violating the inequality are the maximally entangled Bell states
(which are valid density matrices).

Case: $N=3$ – In this case the N-cycle inequality is the Leggett-Garg
inequality [38]

$S_{3}\equiv\langle A_{1}A_{2}\rangle_{seq}+\langle
A_{2}A_{3}\rangle_{seq}-\langle A_{3}A_{1}\rangle_{seq},$ (11)

we will consider maximal violation of this inequality in the presence of
channels. Throughout the draft we highlighted the various degrees of freedom
involved in obtaining correlations from sequential measurements on a quantum
system. We saw that in the simple case without channels being present,
sequential correlations depend only on the angle between the observables and
not on the particular direction of the observable and not even on the
underlying quantum state. However, in the generalised scenario with channels
other isometries also come into play thus allowing us to choose non ideal
angles to obtain maximal violation of the 3-cycle sequential inequality (11).
Following this discussion, we do not base our self testing statement on
certification of the quantum state or of the measurements but rather on the
quantum channel acting in between the sequence of measurements. A protocol for
channel certification has been previously described in [51] where the physical
channel was compared to a target reference channel via the action of the
channel on the maximally entangled state thus requiring the physical input
state to be an entangled one. On the contrary, our channel certification
scheme does not require entanglement but utilizes sequential correlations
instead.

######  Theorem 3.

Temporal certification statement: In a sequential measurement setting with
channel $\mathcal{E}_{A|B}$ acting between the sequence of 2-dim. measurements
$\\{A_{i}\\}$ on 2-dim. state $\rho$, maximal violation of the 3-cycle
sequential inequality (11) implies that $\mathcal{E}_{A|B}$ is a 1-qubit Pauli
channel with Kraus rank 1.

###### Proof.

The proof has been deferred to Appendix section C.3. ∎

##  IV Discussion

Motivated by the SDP based formulations of bounding temporal correlations, we
show uniqueness of $X^{opt}$ as well as robustness of this uniqueness
property. Further, we show that uniqueness of $X^{opt}$ allows for existence
of sets of unit vectors $\\{y_{i}\\}_{i}$ which are unique up to a global
isometry. Next, we considered the generalized scenario with quantum channels
and connected sequential correlations in this scenario with PDMs. This
connection allows us to establish analogies between the structure of spatial
and temporal inequalities along with local isometries involved. Further, we
show maximal violation of sequential inequality in the generalised scenario
implies unitarity of the channel. This result could prove to be useful in
developing quantum process tomography protocols [52, 53] which make the
assumption of working with unitary channels.

We made use of the $N=3$ case for our channel certification result; this
result could be extended to non-cyclic sequential inequalities. We believe
that our work is experimentally testable. Also, robustness of the channel
certification result remains to be explored which we leave for future work.

###### Acknowledgements.

The authors thank Otfried Gühne, Valerio Scarani and Adan Cabello for helpful
discussions. We are grateful to the National Research Foundation and the
Ministry of Education, Singapore for financial support. K.B. acknowledges
funding by AFOSR, DoE QSA, NSF QLCI (award No. OMA-2120757), DoE ASCR
Accelerated Research in Quantum Computing program (award No. DE-SC0020312),
NSF PFCQC program, the DoE ASCR Quantum Testbed Pathfinder program (award No.
DE-SC0019040), U.S. Department of Energy Award No. DE-SC0019449, ARO MURI,
AFOSR MURI, and DARPA SAVaNT ADVENT.

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##  Appendix A Pseudo Density Matrices

The density matrix of a system contains information of all possible Pauli
observables acting on the system thus the density matrix has information of
all correlations between spatially separated subsystems. The Pseudo-Density
Matrix (PDM) formalism was introduced in [44] by extending this analogy to
account for causal correlations between observables acting on the same
subsystem at different time points. The PDM of an $n$-measurement event is
defined as

$R=\frac{1}{2^{n}}\sum_{i_{1}=0}^{3}\cdots\sum_{i_{n}=0}^{3}\langle\\{\sigma_{i_{j}}\\}_{j=1}^{n}\rangle\bigotimes_{j=1}^{n}\sigma_{i_{j}}$

where $\sigma_{0}=\mathrm{I},$ $\sigma_{1}=\mathrm{X}$,
$\sigma_{2}=\mathrm{Y}$, and $\sigma_{3}=\mathrm{Z}.$ The sub-indices $j$ go
over all measurement events which could be done in a sequence on a single
qubit, on spatially separated qubits or a combination of both thus accounting
for both spatial and causal correlations arising due to the $n$ measurement
events. The factor $\langle\\{\sigma_{i_{j}}\\}_{j=1}^{n}\rangle$ denotes the
correlation term corresponding to the size-$n$ measurement events. The
properties retained by the PDM $R$ are hermiticity and unit trace. If all the
measurement events are performed on distinct qubits then $R$ is positive
semidefinite (PSD) and thus a valid density matrix, however; in the presence
of causal or temporal correlations $R$ is not necessarily PSD with possible
negative eigenvalues. As a measure of causality, authors in [44] introduced
$f_{tr}(R)$=$\left\|R\right\|_{tr}-1$. Let us expound on PDMs using the
following example,
Example: Consider a qubit system on which 2-measurement events $E_{1}$ and
$E_{2}$ are made. Next, on calculating
$\langle\\{\sigma_{i_{1}},\sigma_{i_{2}}\\}\rangle$ associated with the 2
events, we observe that only the following terms are non-zero,

$\displaystyle\langle\\{X,X\\}\rangle=1,\quad\langle\\{Y,Y\\}\rangle=1,\quad$
$\displaystyle\langle\\{Z,Z\\}\rangle=1,$
$\displaystyle\langle\\{\mathbb{I},\mathbb{I}\\}\rangle=1,\quad\langle\\{Z,\mathbb{I}\\}\rangle=1,\quad$
$\displaystyle\langle\\{\mathbb{I},Z\\}\rangle=1.$

Subsequently, the PDM $R_{ex}$ for this sequence of 2 events on a single qubit
system is

$\displaystyle R_{ex}=$
$\displaystyle\frac{1}{4}\sum_{i_{1}}\sum_{i_{2}}\langle\\{\sigma_{i_{1}},\sigma_{i_{2}}\\}\rangle(\sigma_{i_{1}}\otimes\sigma_{i_{2}})$
$\displaystyle=$ $\displaystyle\begin{bmatrix}1&0&0&0\\\ 0&0&1/2&0\\\
0&1/2&0&0\\\ 0&0&0&0\\\ \end{bmatrix}$

with eigenvalues $0,1,\frac{1}{2},-\frac{1}{2}$. We see that $R_{ex}$ has a
negative eigenvalue since the 2 measurement events $E_{1}$ and $E_{2}$ are
causally related by virtue of acting on the same qubit system.

##  Appendix B Semidefinite programming basics

An $n\times n$ matrix $X$ is said to be positive semidefinite (PSD) if and
only if $v^{T}Xv\geq 0$ for any $v\in\mathbb{R}^{n}$, denote PSD matrix $X$ by
$X\succeq 0$. It can be easily checked that the set of PSD matrices forms a
convex cone. Then, a semidefinite program (SDP) is an optimization problem of
the form

$\displaystyle\mathrm{maximize:}\quad$ $\displaystyle\mathrm{Tr}(CX)$
$\displaystyle\mathrm{subject\,to}:\quad$
$\displaystyle\mathrm{Tr}(A_{i}X)=b_{i}\quad,i=1,2,\ldots,m$ (12)
$\displaystyle X\succeq 0,$

where we note that the objective function $\mathrm{Tr}(CX)$ is linear in $X$
and the $m$ linear equation constraints are given by $m$ matrices
$A_{1},\ldots,A_{m}$ and the $m$-vector $b$. Referring to the problem above as
the primal (P) SDP, we can define the dual of it as given below.
Semidefinite programming duality: The dual (D) problem of the above SDP (SDD)
is defined to be

$\displaystyle\mathrm{minimize:}\quad$ $\displaystyle\sum_{i=1}^{m}y_{i}b_{i}$
$\displaystyle\mathrm{subject\,to}:\quad$
$\displaystyle\sum_{i=1}^{m}y_{i}A_{i}+S=C$ (13) $\displaystyle S\succeq 0,$

The following theorem summarizes relevant results from SDP duality theory.

######  Theorem 4.

Consider a pair of primal (P) and dual (D) SDPs. The following holds:

  1. i

(Complementary slackness) Let $X,(y,Z)$ be a pair of primal-dual feasible
solutions for (P) and (D), respectively. Assuming that $p^{*}=d^{*}$ we have
that $X,(y,Z)$ are primal-dual optimal if and only if $\langle X,Z\rangle=0$.

  2. ii

(Strong duality) Assume that $d^{*}>-\infty$ (resp. $p^{*}<+\infty$) and that
(D) (resp. (P)) is strictly feasible. Then $p^{*}=d^{*}$ and furthermore, the
primal (resp. dual) optimal value is attained.

######  Theorem 5.

[54] Let $Z^{*}$ be a dual optimal and nondegenerate solution of a
semidefinite program. Then, there exists a unique primal optimal solution for
that SDP.

The notion of dual nondegeneracy is given by the following definition.

######  Definition 1.

(Dual nondegeneracy) Let $Z^{*}$ be an optimal dual solution and let $M$ be
any symmetric matrix. If the homogeneous linear system

$MZ^{*}=0,$ (14)

$\textrm{Tr}(MA_{i})=0\ (\forall i\in[m]),$ (15)

only admits the trivial solution $M=0$, then $Z^{*}$ is said to be dual
nondegenerate.

######  Theorem 6.

[18] Consider a pair of primal/dual SDPs (P) and (D), where the primal/dual
values are equal and both are attained. Furthermore, assume that the set of
feasible solutions of (P) is contained in some compact subset
$U\subseteq\mathcal{S}^{n}$. Let $\mathcal{P}$ be the set of primal optimal
solutions and $d$ be the singularity degree of (P) defined (in [55]) as the
least number of facial reduction steps required to make (P) strictly feasible,
we have that

$\textrm{dist}(\tilde{X},\mathcal{P})\leq\mathcal{O}(\epsilon^{2^{-d}}),$

for any primal feasible solution $\tilde{X}$ with $p^{*}-\epsilon\leq\langle
C,\tilde{X}\rangle$.

##  Appendix C Proofs

###  C.1 Proofs of Theorems 1 and 2 from Main Text (MT)

####  C.1.1 Proof of Theorem 1

In order to prove Theorem 1, we will first establish a couple of Lemmas.

######  Lemma 2.

The dual optimal solution for the SDP in (3) is given by

$W_{N}=\cos\left(\frac{\pi}{N}\right)\mathbb{I}_{N}+0.5\left(e_{1}e_{N}^{T}+e_{N}e_{1}^{T}\right)-\sum_{i=1}^{N-1}0.5\left(e_{i}e_{i+1}^{T}+e_{i+1}e_{i}^{T}\right),$
(16)

where $\mathbb{I}_{N}$ is an $N\times N$ identity matrix and $e_{i}$ is an
$N\times 1$ column vector with $1$ at the $i$th place and $0$ elsewhere.

###### Proof.

To prove that the matrix in 16 is dual-optimal, we need to prove that

  1. 1.

$W_{N}$ is dual-feasible, and

  2. 2.

Corresponds to the optimal value, i.e., $N\cos\left(\frac{\pi}{N}\right).$

The dual of the SDP in (3) is given by

$\min\text{ }\sum_{i}y_{i}$ (17)

such that

$\sum_{i=1}^{N}y_{i}e_{i}e_{i}^{T}-\Lambda\succcurlyeq 0,$ (18)

where $\Lambda$ is the matrix
$-0.5\left(e_{1}e_{N}^{T}+e_{N}e_{1}^{T}\right)+\sum_{i=1}^{N-1}0.5\left(e_{i}e_{i+1}^{T}+e_{i+1}e_{i}^{T}\right)$.
The matrix in 18 is equal to the matrix in 16 for
$y_{i}=\cos\left(\frac{\Pi}{N}\right)$ $\forall
i\in\left\\{1,2,\cdots,N\right\\}$. Since $\sum y_{i}$ is equal to
$N\cos\left(\frac{\pi}{N}\right)$, it remains to be shown that $W_{N}$ is
positive semidefinite for all $N\geq 3$. We proceed by rewriting $W_{N}$ as

$W_{N}=c_{N}\mathbb{I}_{N}+0.5\underbrace{\begin{bmatrix}0&-1&0&\cdots&\cdots&1\\\
-1&0&-1&\cdots&\cdots&0\\\ 0&-1&0&-1&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&0&\cdots&\cdots&-1&0\end{bmatrix}}_{\textrm{call this matrix }T_{N}}$ (19)

where $c_{N}=\cos{\frac{\pi}{N}}$. Note that $W_{N}$ is PSD if all eigenvalues
of $T_{N}$ are greater than or equal to $-2c_{N}$ which is what we show next.
In order to find the eigenvalues of $T_{N}$, consider the determinant

$|T_{N}-\lambda\mathbb{I}_{N}|=\begin{vmatrix}-\lambda&-1&0&\cdots&\cdots&1\\\
-1&-\lambda&-1&\cdots&\cdots&0\\\ 0&-1&-\lambda&-1&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&0&\cdots&\cdots&-1&-\lambda\end{vmatrix}$ (20)

The matrix above is an ordinary tridiagonal matrix and the determinant can be
evaluated using the following formula [56] involving multiplication of
$2\times 2$ matrices

$\begin{vmatrix}a_{1}&b_{1}&\cdots&c_{0}\\\ c_{1}&a_{2}&b_{2}&\vdots\\\
\vdots&\ddots&\ddots&\vdots\\\
b_{n}&\cdots&c_{n-1}&a_{n}\end{vmatrix}=(-1)^{n+1}(b_{n}\cdots
b_{1}+c_{n-1}\cdots
c_{0})+\textrm{Tr}\left[\begin{pmatrix}a_{n}&-b_{n-1}c_{n-1}\\\
1&0\end{pmatrix}\cdots\begin{pmatrix}a_{2}&-b_{1}c_{1}\\\
1&0\end{pmatrix}\begin{pmatrix}a_{1}&-b_{n}c_{0}\\\ 1&0\end{pmatrix}\right]$
(21)

Plugging the values of matrix elements for our case gives

$|T_{N}-\lambda\mathbb{I}_{N}|=(-1)^{N+1}(2(-1)^{N-1})+\textrm{Tr}\left[\begin{pmatrix}-\lambda&-1\\\
1&0\end{pmatrix}^{N}\right]$ (22)

Note that the matrix $P\equiv\begin{pmatrix}-\lambda&-1\\\ 1&0\end{pmatrix}$
has eigenvalues $\mu=\frac{-\lambda+\sqrt{\lambda^{2}-4}}{2}$ and
$\frac{1}{\mu}$. Also, $P$ is not diagonalizable for $\lambda=2$ which is an
eigenvalue of $T_{N}$ for odd $N$. However, $W_{N}$ remains PSD in this case.
For $\lambda\neq 2$, we have

$\textrm{Tr}\left[\begin{pmatrix}-\lambda&-1\\\
1&0\end{pmatrix}^{N}\right]=\mu^{N}+\frac{1}{\mu^{N}}$ (23)

giving

$|T_{N}-\lambda\mathbb{I}_{N}|=2+\mu^{N}+\frac{1}{\mu^{N}}$ (24)

The expression above vanishes if $(\mu^{N}+1)^{2}=0$. The roots of
$\mu^{N}+1=0$ are $\mu=e^{i(2m+1)\pi/N}$ with $m\in\mathbb{Z}$. Thus,

$\displaystyle\frac{-\lambda+\sqrt{\lambda^{2}-4}}{2}$
$\displaystyle=e^{i(2m+1)\pi/N}$ $\displaystyle\sqrt{\lambda^{2}-4}$
$\displaystyle=\lambda+2e^{i(2m+1)\pi/N}$ $\displaystyle\lambda^{2}-4$
$\displaystyle=\lambda^{2}+4e^{i2(2m+1)\pi/N}+4\lambda e^{i(2m+1)\pi/N}$
$\displaystyle\lambda$
$\displaystyle=-2\cos{\frac{(2m+1)\pi}{N}}\geq-2\cos{\frac{\pi}{N}}\quad\forall\,m\textrm{
and all }N\geq 3$ (25)

Therefore, $W_{N}$ is PSD for all $N\geq 3$. ∎

######  Lemma 3.

The only solution to the system of linear equations

$X_{N}W_{N}=0$

is $X_{N}=0$, where $X_{N}$ is a symmetric $N\times N$ matrix with diagonal
elements equal to zero and $W_{N}$ is the dual optimal for the SDP in (3).

###### Proof.

The proof is trivial and follows from simple linear algebra. There are $N^{2}$
linear equations where the maximum number of variables in a equation is three.
Let us consider the equations with variables corresponding to the first row of
$X_{N}$. There are $N$ such linear equations. Furthermore, there are three
equations with number of variables equal to two. These three equations fix the
value of the variables involved either equal to each other (one such equation)
or a constant times the other variable. Here, the constant is
$2\cos{\frac{\pi}{N}}$. Substituting these constraints in the equations with
three variables, we get new equations with two variables. Following this
approach, we get the value of all the variables corresponding row $1$ of
$X_{N}$ equal to zero as the only self-consistent solution. Same argument
applies for the variables in the other rows. This completes the proof. ∎

Proof of Theorem 1 follows as;

###### Proof.

Lemmas 2 and 3 imply that the dual optimal $W_{N}$ is non-degenerate. Together
with Theorem 5, this implies that the primal optimal for the SDP in (3) is
unique. This completes the proof of Theorem 1. ∎

####  C.1.2 Proof of Theorem 2

###### Proof.

Since the identity matrix belongs to the strictly feasible set of the SDP in
(3), the singularity degree is $0$. Once we substitute $d=0$ in the statement
of Theorem 6, we recover our robustness statement. This completes the proof of
Theorem 2. ∎

###  C.2 Proof of Lemma 1 from MT

###### Proof.

Starting with LHS,

$\displaystyle\mathrm{Tr}\left[(A_{m}\otimes A_{n})R_{AB}\right]$
$\displaystyle=\mathrm{Tr}\left\\{(A_{m}\otimes
A_{n})\left[\left(\rho_{A}\otimes\frac{\mathrm{I}}{2}\right)E_{AB}+E_{AB}\left(\rho_{A}\otimes\frac{\mathrm{I}}{2}\right)\right]\right\\}$
(26) $\displaystyle=\frac{1}{2}\left[\mathrm{Tr}\left(\left(A_{m}\rho\otimes
A_{n}\right)E_{AB}\right)+\mathrm{Tr}\left(\left(A_{m}\otimes
A_{n}\right)E_{AB}(\rho\otimes\mathrm{I})\right)\right]$
$\displaystyle=\frac{1}{2}\left[\sum_{ij}\mathrm{Tr}\left(A_{m}\rho|i\rangle\langle
j|\right)\mathrm{Tr}\left(A_{n}\mathcal{E}_{A|B}\left(|j\rangle\langle
i|\right)\right)+\sum_{ij}\mathrm{Tr}\left(A_{m}|i\rangle\langle
j|\rho\right)\mathrm{Tr}\left(A_{n}\mathcal{E}_{A|B}\left(|j\rangle\langle
i|\right)\right)\right]$ $\displaystyle=\frac{1}{2}\left[\sum_{ij}\langle
j|(A_{m}\rho+\rho
A_{m})|i\rangle\mathrm{Tr}\left(A_{n}\mathcal{E}_{A|B}\left(|j\rangle\langle
i|\right)\right)\right]$ $\displaystyle=\frac{1}{2}\left[\sum_{ij}\langle
j|(A_{m}\rho+\rho
A_{m})|i\rangle\mathrm{Tr}\left(A_{n}\sum_{k}V_{k}|j\rangle\langle
i|V_{k}^{\dagger}\right)\right]$
$\displaystyle=\frac{1}{2}\left[\sum_{ij}\langle j|(A_{m}\rho+\rho
A_{m})|i\rangle\langle i|\sum_{k}V_{k}^{\dagger}A_{n}V_{k}|j\rangle\right]$
$\displaystyle=\frac{1}{2}\left[\sum_{j}\langle j|(A_{m}\rho+\rho
A_{m})\sum_{k}V_{k}^{\dagger}A_{n}V_{k}|j\rangle\right]$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left[\sum_{j}|j\rangle\langle
j|(A_{m}\rho+\rho A_{m})\sum_{k}V_{k}^{\dagger}A_{n}V_{k}\right]$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left[(A_{m}\rho+\rho
A_{m})\mathcal{E}_{A|B}^{\dagger}(A_{n})\right]$ $\displaystyle=\langle
A_{m}A_{n}\rangle_{seq}$

where in the first line,
$E_{AB}=\sum_{ij}(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})(|i\rangle\langle
j|_{A}\otimes|j\rangle\langle i|_{B})$ acting on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is Jamiolkowski isomorphic to
$\mathcal{E}_{A|B}$.

∎

###  C.3 Proof of Theorem 3 from MT

We first prove the following useful result,

######  Lemma 4.

Consider the correlation $\langle\sigma_{k}\sigma_{l}\rangle_{seq}$ in the
generalized scenario with channels with $\sigma_{k/l}$ as the Pauli
observables and $\rho_{A}$ being 2-dimensional. Then,

$\langle\sigma_{k}\sigma_{l}\rangle_{seq}=\frac{1}{2}\Big{[}\langle\sigma_{k}\rangle_{\rho_{A}}\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{0}))+\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{k}))\Big{]}$
(27)

###### Proof.

$\displaystyle\langle\sigma_{k}\sigma_{l}\rangle_{seq}$
$\displaystyle=\mathrm{Tr}[(\sigma_{k}\otimes\sigma_{l})R_{AB}]\quad\mathrm{using\;Lemma\;\ref{lemma_4}}$
$\displaystyle=\mathrm{Tr}\left[(\sigma_{k}\otimes\sigma_{l})(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})\Big{\\{}\rho_{A}\otimes\frac{I}{2},\;\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}\Big{\\}}\right]\quad\mathrm{using\;\eqref{PDM_chn}\;\textrm{from
MT}}$
$\displaystyle=\frac{1}{4}\mathrm{Tr}\left[(\sigma_{k}\otimes\sigma_{l})(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})\sum_{i=0}^{3}\\{\rho_{A},\sigma_{i}\\}\otimes\sigma_{i}\right]$
$\displaystyle=\frac{1}{4}\mathrm{Tr}\left[\sum_{i=0}^{3}\sigma_{k}\\{\rho_{A},\sigma_{i}\\}\otimes\sigma_{l}\mathcal{E}_{A|B}(\sigma_{i})\right]$
$\displaystyle=\frac{1}{4}\sum_{i=0}^{3}\mathrm{Tr}(\sigma_{k}\rho_{A}\sigma_{i}+\sigma_{k}\sigma_{i}\rho_{A}).\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{i}))$
$\displaystyle=\frac{1}{4}\sum_{i=0}^{3}\mathrm{Tr}(\rho_{A}(\sigma_{i}\sigma_{k}+\sigma_{k}\sigma_{i})).\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{i}))$
$\displaystyle=\frac{1}{2}\left[\langle\sigma_{k}\rangle_{\rho_{A}}\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{0}))+\sum_{i=1}^{3}\delta_{ik}\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{i}))\right]$
$\displaystyle=\frac{1}{2}\Big{[}\langle\sigma_{k}\rangle_{\rho_{A}}\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{0}))+\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{k}))\Big{]}.$

∎

Proceeding with the proof of Theorem 3 from MT;

###### Proof.

Take $\rho_{A}=|0\rangle\langle 0|$ since we expect maximal violation to be
obtainable using a pure state and rotational freedom of observables allows us
to take it as an eigenstate of $\sigma_{z}$ . Then using Lemma 4, we get

$\langle\sigma_{k}\sigma_{l}\rangle_{seq}=\begin{cases}\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{k}))/2,\text{
for }k=1,2\\\
\mathrm{Tr}(\sigma_{l}\mathcal{E}_{A|B}(\sigma_{0})+\sigma_{l}\mathcal{E}_{A|B}(\sigma_{3}))/2,\text{
for }k=3\end{cases}$ (28)

Furthermore, it has been established in [57] that all possible quantum
channels acting on 2-dim. states correspond to the convex closure of the maps
parametrized in the Pauli basis $\\{\sigma_{i}\\}_{i}$ using the following
Kraus operators,

$\begin{array}[]{cc}K_{+}&=\left[\cos\frac{v}{2}\cos\frac{u}{2}\right]\sigma_{0}+\left[\sin\frac{v}{2}\sin\frac{u}{2}\right]\sigma_{3}\\\
K_{-}&=\left[\sin\frac{v}{2}\cos\frac{u}{2}\right]\sigma_{1}-i\left[\cos\frac{v}{2}\sin\frac{u}{2}\right]\sigma_{2}\end{array}$
(29)

with $v\in[0,\pi]$, $u\in[0,2\pi]$. Action of the channel
$\mathcal{E}_{A|B}(\sigma_{i})$ can be written using these Kraus operators as,

$\displaystyle K_{+}\sigma_{0}K_{+}^{\dagger}+K_{-}\sigma_{0}K_{-}^{\dagger}$
$\displaystyle=\sigma_{0}+\sin(u)\sin(v)\sigma_{3}$ (30) $\displaystyle
K_{+}\sigma_{1}K_{+}^{\dagger}+K_{-}\sigma_{1}K_{-}^{\dagger}$
$\displaystyle=\cos(u)\sigma_{1}$ $\displaystyle
K_{+}\sigma_{2}K_{+}^{\dagger}+K_{-}\sigma_{2}K_{-}^{\dagger}$
$\displaystyle=\cos(v)\sigma_{2}$ $\displaystyle
K_{+}\sigma_{3}K_{+}^{\dagger}+K_{-}\sigma_{3}K_{-}^{\dagger}$
$\displaystyle=\cos(u)\cos(v)\sigma_{3}.$

Substituting (30) into (28) gives

$\displaystyle\langle\sigma_{k}\sigma_{l}\rangle_{seq}$
$\displaystyle=\delta_{kl}(\delta_{1k}\cos u+\delta_{2k}\cos
v+\delta_{3k}\cos(u-v))$ (31)

Using (31) for $S_{3}$ along with the fact that for 2-dimensional observables
$A_{m/n}=\vec{a}_{m/n}.\vec{\sigma}$ we can write $\langle
A_{m}A_{n}\rangle_{seq}=\sum_{k,l}a_{mk}a_{nl}\langle\sigma_{k}\sigma_{l}\rangle_{seq}$
gives,

$\displaystyle S_{3}$
$\displaystyle=\sum_{k,l=1}^{3}\big{[}(a_{1k}a_{2l}+a_{2k}a_{3l}-a_{3k}a_{1l})\delta_{kl}(\delta_{1k}\cos
u+\delta_{2k}\cos v+\delta_{3k}\cos(u-v))\big{]}$ (32)
$\displaystyle=\sum_{k=1}^{3}\big{[}(a_{1k}a_{2k}+a_{2k}a_{3k}-a_{3k}a_{1k})(\delta_{1k}\cos
u+\delta_{2k}\cos v+\delta_{3k}\cos(u-v))\big{]}$

From (32), it can be seen that the maximal value achieved is $3/2$ in the
cases with $\\{u=v=0\\}$, $\\{u=v=\pi\\}$, $\\{u=0,v=\pi\\}$ and
$\\{u=\pi,v=0\\}$ which correspond to special cases of the Pauli channel.
However, note that the angle between the measurement choices $\\{A_{i}\\}$
would depend on the particular case of the channel $\mathcal{E}_{A|B}$. Hence,
taking a convex combination of channels corresponding to the cases above will
not give maximal violation for a particular choice of 3 measurements
$\\{A_{i}\\}_{i=1,2,3}$. This result again highlights the unitary degree of
freedom in choosing the channel as described in (9). Interestingly, the 4
channels corresponding to the cases above also correspond to the 4 “pseudo-
Bell states”

$\displaystyle R_{AB}^{(1)}$ $\displaystyle=\frac{1}{4}(I+X\otimes X+Y\otimes
Y+Z\otimes Z)$ $\displaystyle R_{AB}^{(2)}$
$\displaystyle=\frac{1}{4}(I+X\otimes X-Y\otimes Y-Z\otimes Z)$ $\displaystyle
R_{AB}^{(3)}$ $\displaystyle=\frac{1}{4}(I-X\otimes X+Y\otimes Y-Z\otimes Z)$
$\displaystyle R_{AB}^{(4)}$ $\displaystyle=\frac{1}{4}(I-X\otimes X-Y\otimes
Y+Z\otimes Z)$

which were used for showing temporal evolution as an analogue of spatial
teleportation in [43]. ∎

##  Appendix D Local isometry in time

Justification for (9) in MT,

$\displaystyle\mathrm{Tr}\left[(A_{m}\otimes A_{n})R_{AB}\right]$
$\displaystyle=\mathrm{Tr}\left[(A_{m}\otimes
A_{n})(\mathcal{I}_{A}\otimes\mathcal{E}_{A|B})\Big{\\{}\rho_{A}\otimes\frac{I}{2},\;\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}\Big{\\}}\right]$
(33) $\displaystyle=\mathrm{Tr}\left[(A_{m}\otimes
A_{n})\frac{1}{2}\sum_{i=0}^{3}(\\{\rho_{A},\sigma_{i}\\}\otimes\mathcal{E}_{A|B}(\sigma_{i})\right]$
$\displaystyle=\frac{1}{2}\sum_{i=0}^{3}\mathrm{Tr}\left[A_{m}(\rho_{A}\sigma_{i}+\sigma_{i}\rho_{A})\right]\mathrm{Tr}\left[A_{n}\mathcal{E}_{A|B}(\sigma_{i})\right]$
$\displaystyle=\frac{1}{2}\sum_{i=0}^{3}\mathrm{Tr}\left[VA_{m}V^{\dagger}(V\rho_{A}V^{\dagger}V\sigma_{i}V^{\dagger}+V\sigma_{i}V^{\dagger}V\rho_{A}V^{\dagger})\right]\mathrm{Tr}\left[UA_{n}U^{\dagger}\tilde{\mathcal{E}}_{A|B}(V\sigma_{i}V^{\dagger})\right]$
$\displaystyle=\frac{1}{2}\sum_{i=0}^{3}\mathrm{Tr}\left[\tilde{A}_{m}(\tilde{\rho}_{A}\sigma_{i}+\sigma_{i}\tilde{\rho}_{A})\right]\mathrm{Tr}\left[\tilde{A}_{n}\tilde{\mathcal{E}}_{A|B}(\sigma_{i})\right]$

where $\tilde{\mathcal{E}}_{A|B}$ has transformed Kraus operators as
$\\{K_{i}\\}\longrightarrow\\{UK_{i}V^{\dagger}\\}$, $U$ is a unitary operator
acting at time point $t_{B}$ and $V$, acting at $t_{A}$, is restricted to
$V\in\\{X,Y,Z\\}$ such that applying the transformation
$V\sigma_{i}V^{\dagger}$ conserves the elements of the set
$\sigma_{i}\in\\{\mathbb{I},X,Y,Z\\}$. Note that we choose this particular
choice for $V$ just to highlight the local isometry in time, there exist other
choices for $V$ such as the Clifford group that would also conserve the set of
Pauli matrices.