# Principle of information causality rationalizes quantum composition

Ram Krishna Patra  Department of Physics of Complex Systems, S. N. Bose
National Center for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata
700106, India.  Sahil Gopalkrishna Naik  Department of Physics of Complex
Systems, S. N. Bose National Center for Basic Sciences, Block JD, Sector III,
Salt Lake, Kolkata 700106, India.  Edwin Peter Lobo  Laboratoire d’Information
Quantique, Université libre de Bruxelles (ULB), Av. F. D. Roosevelt 50, 1050
Bruxelles, Belgium  Samrat Sen  Department of Physics of Complex Systems, S.
N. Bose National Center for Basic Sciences, Block JD, Sector III, Salt Lake,
Kolkata 700106, India.  Govind Lal Sidhardh  Department of Physics of Complex
Systems, S. N. Bose National Center for Basic Sciences, Block JD, Sector III,
Salt Lake, Kolkata 700106, India.  Mir Alimuddin  Department of Physics of
Complex Systems, S. N. Bose National Center for Basic Sciences, Block JD,
Sector III, Salt Lake, Kolkata 700106, India.  Manik Banik  Department of
Physics of Complex Systems, S. N. Bose National Center for Basic Sciences,
Block JD, Sector III, Salt Lake, Kolkata 700106, India.

###### Abstract

Principle of information causality, proposed as a generalization of no
signaling principle, has efficiently been applied to outcast beyond quantum
correlations as unphysical. In this letter we show that this principle when
utilized properly can provide physical rationale towards structural derivation
of multipartite quantum systems. In accordance with no signaling condition
state and effect spaces of a composite system can allow different possible
mathematical descriptions even when description for the individual systems are
assumed to be quantum. While in one extreme, namely the maximal tensor product
composition, the state space becomes quite exotic and permits composite states
that are not allowed in quantum theory, the other extreme – minimal tensor
product composition – contains only separable states and the resulting theory
allows only Bell local correlation. As we show, none of these compositions
does commensurate with information causality, and hence get invalidated to be
the bona-fide description of nature. Information causality, therefore,
promises information theoretic derivation of self-duality of state and effect
cones for composite quantum systems.

Introduction.– Quantum mechanics is the most effective theory to describe
almost all the natural phenomena. Its successful marriage with information
theory leads to numerous novel technologies and promises a lot more [1].
However, the theory, starting from its inception, prompts huge debates
regarding its interpretation [2, 3, 4, 5, 6, 7, 8, 9] that persists till date
[10, 11]. Quantum formalism starts with abstract mathematical description of
Hilbert space and cries for its physical justification. The celebrated no-
signaling (NS) principle, that prohibits instantaneous communication between
distant parties, cannot serve the purpose alone. It allows a broad variety of
mathematical models as the possible candidate of the theory of nature.
Interestingly, inspired by the studies in quantum information theory, during
the recent past, several novel principles have been proposed to circumvent the
limitation of NS principle [12, 13, 14, 15, 16, 17, 18]. These new principles
quite efficiently identify some beyond quantum NS correlations as unphysical
and thus adduce physical justification(s) to quantum correlations.

In this letter we analyze one of the intriguing principles called information
causality (IC), proposed nearly a decade back [14]. IC can be envisaged as a
generalization of the NS condition. It limits the information gain that a
receiver (say Bob) can reach about a previously unknown to him data set of a
sender (say Alice), by using two types of resources: (i) all his local
resources that might be correlated with the sender which in this work we will
call Type-I resource, and (ii) some physical system carrying bounded amount of
information from Alice to Bob, Type-II resource. Both these resources can
further be of different kinds – classical, quantum, and beyond quantum; and IC
principle provides a way to test their physicality. Although the correlated
resources by themselves have no communication utility, as shown in the seminal
superdense coding paper [19], a quantum correlation viz. entanglement can
double up the communication capacity of a quantum channel. The power of
entanglement, however, is limited in a way as it cannot enhance communication
capacity of a classical channel. Principle of IC generalizes this no-go by
limiting Bob’s information gain to be at most $m$ bits when $m$ classical bits
are communicated by Alice to him and he is allowed to use any of his local
resources that might be correlated with Alice. Quite interestingly several NS
correlations violate this principle and thus considered as unphysical [20, 21,
22, 23]. In essence, restricting the Type-II resources to be classical, the IC
principle discards some of the Type-I resources as unphysical.

Here we study the reverse scenario, i.e., Type-II resource is varied among
several possibilities while restricting the Type-I resource to be classical.
In particular, we ask the question whether unphysicality of some Type-II
resources can be established through information principles. Interestingly, as
we will show, principle of IC becomes effective in this case too. We consider
the scenarios where individual systems are assumed to be quantum, but their
composition can be modelled by any theory that satisfy the NS condition. Even
for two quantum systems several consistent compositions are possible among
which quantum theory is one of the examples. The state space of the resulting
system lies in between two extremes – maximal tensor product state space and
minimal tensor product state space [24]. While the maximal one grants exotic
joint states that are not allowed in quantum theory, the minimal one allows
only separable states. It turns out that the system obtained through maximal
tensor product of two elementary quantum violates the IC principle. This is
quite remarkable as all the NS correlations obtained from beyond quantum
states are in fact quantum simulable and hence cannot yield beyond quantum
nonlocal correlation [25]. We then show that minimal tensor product
composition is also not compatible with IC principle and hence gets excluded.
This is even more striking as the resulting theory allows only separable
states and hence in Bell type experiments no nonlocal correlation is possible.
Our work establishes a novel yet unexplored proficiency of the IC principle as
it discards extreme descriptions of composite quantum systems, and thus
promises physical rationale for quantum composition.

Compositions of elementary quantum.– Within the mathematical framework of
generalized probability theory (GPT) an elementary system $\mathcal{S}$ is
specified by the tuple of normalized state and effect spaces, i.e.
$\mathcal{S}\equiv(\Omega,\mathcal{E})$ [26, 27, 28, 29, 30]. Sometimes it is
convenient to deal with unnormalized states and effects that form convex cones
embedded in some $\mathbb{R}^{n}$. A GPT also captures the description of the
composite system $\mathcal{S}^{AB}\equiv(\Omega^{AB},\mathcal{E}^{AB})$
consisting of component subsystems
$\mathcal{S}^{A}\equiv(\Omega^{A},\mathcal{E}^{A})$ and
$\mathcal{S}^{B}\equiv(\Omega^{B},\mathcal{E}^{B})$. Under the restriction of
NS and local tomography [31] the composite state space $\Omega^{AB}$ lies in
between two extremes – (i) the maximal tensor product state space and (ii) the
minimal tensor product state space [24]. For instance, the state cone of a
quantum system associated with a Hilbert space $\mathcal{H}$ is the set of
positive semidefinite operators
$\mathcal{P}(\mathcal{H})\subset\mathcal{L}(\mathcal{H})$ acting on
$\mathcal{H}$, whereas the normalized states are the set of density operators
$\mathcal{D}(\mathcal{H})$; here $\mathcal{L}(\mathcal{H})$ denotes the set of
all linear operators acting on $\mathcal{H}$. For two quantum systems
associated with Hilbert spaces $\mathcal{H}^{A}$ and $\mathcal{H}^{B}$
respectively, the state cone for maximal tensor product system is given by

$\displaystyle\Omega^{AB}_{+}[\text{max}]$
$\displaystyle:=\\{\mathcal{W}\in\mathcal{L}(\mathcal{H}^{A}\otimes\mathcal{H}^{B})|\operatorname{Tr}[\mathcal{W}(\pi_{A}\otimes\pi_{B})]\geq
0$
$\displaystyle~{}\forall\pi_{A}\in\mathcal{P}(\mathcal{H}^{A})~{}\text{and}~{}~{}\forall\pi_{B}\in\mathcal{P}(\mathcal{H}^{B})\\}.$
(1)

States in this theory are called positive on pure tensors (POPT) states [25].
All the quantum states are POPT, but there are POPT states that are not
allowed quantum states. The effect cone is constructed in accordance with the
no-restriction hypothesis that allows all mathematically consistent effects in
the theory [32]:

$\displaystyle\mathcal{E}^{AB}_{+}[\text{max}]$
$\displaystyle:=\\{\pi~{}|~{}\pi=\sum_{i}\pi_{i}^{A}\otimes\pi_{i}^{B};~{}\pi^{A}_{i}\in\mathcal{P}(\mathcal{H}^{A})~{}$
$\displaystyle\&~{}\pi^{B}_{i}\in\mathcal{P}(\mathcal{H}^{B})\\}.$ (2)

As it turns out $\mathcal{E}^{AB}_{+}[\text{max}]$ also forms a cone which is
dual to the state cone $\Omega^{AB}_{+}[\text{max}]$. In the other extreme,
minimal tensor product contains only separable states, but the effect space
gets enlarged here. More particularly, the role of state and effect cones of
maximal tensor product are interchanged in the minimal case, i.e.,

$\displaystyle\Omega^{AB}_{+}[\text{min}]:=\mathcal{E}^{AB}_{+}[\text{max}]~{}\&~{}\mathcal{E}^{AB}_{+}[\text{min}]:=\Omega^{AB}_{+}[\text{max}].$
(3)

Quantum composition lies in between the maximal and mininal case, and the
state and effect cones become self dual, i.e.,
$\Omega^{AB}_{+}[Q]=\mathcal{P}(\mathcal{H}^{A}\otimes\mathcal{H}^{B})=\mathcal{E}^{AB}_{+}[Q]$.
An instructive image for theses different composite structures is shown in
Fig. 1.

Figure 1: Different possible compositions of two elementary quantum systems.
Left: minimal tensor product composition – allows only separable states but
effect cone is bigger than quantum effect cone. Right: maximal tensor product
composition – allows only separable effect but state cone is bigger than
quantum. Middle: quantum composition, state and effect cones are identical
(self dual).

Here we recall the concept of information capacity [33] (see also [34, 35,
36]), which is relevant for studying communication utility of a GPT system.

######  Definition 1 (CMP 316, 441 (2012)).

Information capacity $\mathcal{I}(\mathcal{S})$ of a system $\mathcal{S}$ is
the maximum number of states that can be perfectly distinguished in a single
shot measurement.

In standard information-theoretic unit information capacity can be quantified
in terms of bit. If maximum $d$ number of states are perfectly distinguishable
by single-shot measurement then $\mathcal{I}(\mathcal{S})=\log d$ bit.
Information capacity is closely related to the amount of classical information
that can be transferred perfectly by sending one copy of the system. For
instance, the state space of a two level classical and quantum systems are the
$1$-simplex and unit sphere in $\mathbb{R}^{3}$ respectively, and both have
information capacity $2$ (or $\log 2$ bit). However, it might happen that the
Holevo capacity $\Theta(\mathcal{S})$ of a system $\mathcal{S}$ exceeds its
information capacity. By Holevo capacity of a system we mean the optimal rate
of transfer of classical information when asymptotically many copies of the
system are in use. While finding Holevo capacity, the optimization is
performed over all possible encodings and decodings allowed in the composite
theory, i.e.,
$\Theta(\mathcal{S}):=\lim_{k\to\infty}\frac{1}{k}\log\left[\mathcal{I}(\mathcal{S}^{\otimes
k})\right]$. It turns out to be same as information capacity whenever
information capacity is additive, which follows from the results in [37]. For
quantum composition it is known to be additive. On the other hand, as shown in
[38], for the toy models with polygonal state space [39] have information
capacity $1$ bit, but for odd gon cases the Holevo capacity is larger than $1$
bit. Our next result shows that Holevo capacity in minimal tensor product
composition is additive.

######  Proposition 1.

The information capacity of the minimal composition of $k$ local quantum
systems, each of dimension $d$, is $d^{k}$, i.e.,
$\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\min}k}\right)=d^{k}$.

###### Proof.

Assume that $\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\min}k}\right)=N$.
Clearly, $N\geq d^{k}$ since there are $d^{k}$ product states in
$(\mathbb{C}^{d})^{\otimes_{\min}k}$ that can be perfectly distinguished by
product measurements. Furthermore, since we have assumed
$\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\min}k}\right)=N$, there exist
$N$ pure product states $\\{\ket{a_{1}},\cdots,\ket{a_{N}}\\}$ and a
measurement
$\\{E_{1},\cdots,E_{N}~{}|~{}\sum_{i=1}^{N}E_{i}=\mathbf{1}_{d}^{\otimes_{k}}\\}$
such that
$\operatorname{Tr}(E_{i}\ket{a_{j}}\bra{a_{j}})=\delta_{ij}~{}~{}\forall i,j$.
Let $P_{i}$ be the projector on the orthogonal support of
$\ket{a_{i}}\bra{a_{i}}$, i.e.,
$P_{i}=\mathbf{1}_{d}^{\otimes_{k}}-\ket{a_{i}}\bra{a_{i}}$. It is easy to
show that $P_{i}$ is a separable operator. To see this, first introduce an
orthonormal basis for $\mathcal{H}_{d}^{\otimes_{k}}$ consisting of pure
product states such that $\ket{a_{i}}\bra{a_{i}}$ is one of the elements of
the basis set. Then, $P_{i}$ can be written as the sum of projectors on all
the basis vectors except $\ket{a_{i}}\bra{a_{i}}$. We have,

$\displaystyle d^{k}$
$\displaystyle=\operatorname{Tr}(\mathbf{1}_{d}^{\otimes_{k}})=\sum_{i=1}^{N}\operatorname{Tr}(E_{i})=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}\mathbf{1}_{d}^{\otimes_{k}}]$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}(\ket{a_{i}}\bra{a_{i}}+P_{i})]$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}\ket{a_{i}}\bra{a_{i}}]+\sum_{i=1}^{N}\operatorname{Tr}[E_{i}P_{i}]$
$\displaystyle\geq\sum_{i=1}^{N}\operatorname{Tr}[E_{i}\ket{a_{i}}\bra{a_{i}}]=\sum_{i=1}^{N}\delta_{ii}=N.$

The inequality follows from the fact that $P_{i}$ is a separable operator, and
thus $\operatorname{Tr}[E_{i}P_{i}]\geq 0$. We, therefore, conclude that
$N=\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\min}k}\right)=d^{k}$. ∎

As an immediate implication it follows that the Holevo capacity of minimal
tensor product of local quantum systems are same as their information
capacity, i.e.,
$\Theta(\mathbb{C}^{d}_{\min}):=\lim_{k\to\infty}\frac{1}{k}\log\left[\mathcal{I}((\mathbb{C}^{d})^{\otimes_{\min}k})\right]=\lim_{k\to\infty}\frac{1}{k}\log(d^{k})=\log
d$. A similar result, as stated in the next proposition, also holds for
maximal composition of locally quantum systems.

######  Proposition 2.

The information capacity of the maximal composition of $k$ local quantum
systems, each of dimension $d$, is $d^{k}$, i.e.,
$\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\max}k}\right)=d^{k}$.

Proof of this proposition follows a similar reasoning as of Proposition 1.
However, for the sake of completeness we detail the proof in Appendix A. The
objective of the present work is to see whether these extreme compositions can
be discarded from some physical ground. In the following we will invoke the IC
principle to this aim. Before proceeding to our next results we briefly recall
the IC principle.

Information Causality Game.–

Figure 2: (IC-N game) Referee provides a randomly chosen N-bit string
$\mathbf{a}\in\\{0,1\\}^{N}$ to Alice and a random value
$b\in\\{1,\cdots,N\\}$ to Bob. Bob’s aim is to correctly guess the $b^{th}$
bit of Alice’s string, i.e., to yield the outcome $\beta=a_{b}$. Alice and Bob
can share classical Type-I resources, i.e., classical correlation.
Additionally she can send some physical system (Type-II resource) carrying
bounded amount of information to Bob.

It is instructive to understand the IC principle through the distributed
version of random access code task [40], which we recall from the original IC
paper (see Fig.2). Alice receives a string of $N$ independent bits,
$\mathbf{a}=(a_{1},\cdots,a_{N})$ randomly sampled from $\\{0,1\\}^{N}$, and
Bob receives a random value of $b\in\\{1,\cdots,N\\}$. Bob’s aim is to
correctly guess the value of the $b^{th}$ bit of Alice i.e, $a_{b}$. Alice and
Bob may possess some classical shared randomness, but unlike the original
paper, they do not share any quantum entanglement or beyond quantum
correlation (eg. the Popescu-Rohrlich correlation [41]). We call this IC-N
game, and the efficiency of the collaborative strategy is quantified through
the quantity

$\displaystyle I_{N}=\sum_{k=1}^{N}I\left(a_{k}:\beta~{}|~{}b=k\right),$ (4)

where $I\left(a_{k}:\beta~{}|~{}b=k\right)$ is the Shannon’s mutual
information between the $k^{th}$ bit of Alice and Bob’s guess $\beta$,
computed under the condition that Bob has received $b=k$. If Alice
communicates $m$-cbits to Bob then the efficiency is upper bounded by
$I_{N}\leq m$. Furthermore, $I_{N}\leq m$ turns out to be a necessary
condition for IC in an arbitrary theory where mutual information satisfies
three abstract properties – (a) consistency, (b) data processing inequality,
and (c) chain rule [14]. As pointed out by the authors in [14], IC holds true
even if the quantum bits are transmitted provided that they are disentangled
from the systems of the receiver – a consequence of Holevo bound, which limits
information gain after transmission of $m$ such qubits to $m$ classical bits.
While playing the IC-N game by sending some GPT system from Alice to Bob in
presence of classical correlation shared in advance, the precise formulation
of the IC, that we will use in this work, reads as $I_{N}\leq\Theta$, where
$\Theta$ is the Holevo capacity of the communicated system in consideration.
In Appendix B we discuss an example of toy theory that leads to violation of
IC principle.

Unphysicality of extremal quantum compositions.– Let us consider the IC-$3$
game. If Alice communicates $2$-cbits or $2$-qubits (in quantum composition)
to a classically correlated Bob then the necessary condition of IC is
satisfied, i.e., efficiency of their collaborative strategy $I_{3}$ satisfies
the bound $I_{3}\leq 2$. Our next results show that this is not true if one
considers extremal tensor product compositions between two qubits.

######  Theorem 1.

Minimal tensor product composition of two elementary quantum violates IC
principle.

###### Proof.

In minimal tensor product theory Alice can use only the product states of
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ system for encoding her strings. The
exact encodings is shown in Table 1. Since
$\mathcal{E}^{AB}[\text{min}]>\mathcal{E}^{AB}[Q]>\mathcal{E}^{AB}[\text{max}]$,
Bob’s decodings can comprise of exotic measurements that are not allowed in
quantum theory. We consider the decoding measurements
$\mathcal{M}_{i}:=\\{E_{i},\bar{E}_{i}:=\mathbf{1}_{4}-E_{i}\\}$, where

$\displaystyle E_{1}:=\begin{pmatrix}1&0&0&\frac{1}{2}\\\ 0&0&\frac{1}{2}&0\\\
0&\frac{1}{2}&0&0\\\ \frac{1}{2}&0&0&1\end{pmatrix},~{}$ $\displaystyle
E_{2}:=\begin{pmatrix}0&0&0&\frac{1}{2}\\\ 0&1&\frac{1}{2}&0\\\
0&\frac{1}{2}&1&0\\\ \frac{1}{2}&0&0&0\end{pmatrix},$ $\displaystyle
E_{3}:=\mathbf{1}_{2}$
$\displaystyle\otimes\frac{1}{2}(\mathbf{1}_{2}+\sigma_{z}),$

and the bit value is guessed to be $0$ if the effect $E_{i}$ clicks and $1$ if
$\bar{E}_{i}$ clicks. Being a valid quantum measurement, $\mathcal{M}_{3}$ is
also an allowed measurement in minimal tensor product theory. Furthermore, on
an arbitrary two-qubit product state
$\rho_{mn}=\frac{1}{2}(\mathbf{1}_{2}+\vec{m}.\sigma)\otimes\frac{1}{2}(\mathbf{1}_{2}+\vec{n}.\sigma)$
we have,
$\operatorname{Tr}[E_{j}\rho_{mn}]=\frac{1}{2}[1+m_{x}n_{x}+(-1)^{(j+1)}m_{z}n_{z}]\geq
0$ and
$\operatorname{Tr}[\bar{E}_{j}\rho_{mn}]=\frac{1}{2}[1-m_{x}n_{x}-(-1)^{(j+1)}m_{z}n_{z}]\geq
0$ for $j\in\\{1,2\\}$, that in turn assure
$\mathcal{M}_{1}~{}\&~{}\mathcal{M}_{2}$ to be bona-fide measurements in
minimal tensor product theory. According to Proposition 1 the Holevo capacity
of this system is 2 bits, and hence the theory violates IC principle since the
aforesaid encoding-decoding strategy yields $I_{3}\approx 2.19>2$. This
concludes the proof. ∎

Input | Alice’s |  Bob’s decoding
---|---|---
string | encoding | $1^{st}\to\mathcal{M}_{1}$ | $2^{nd}\to\mathcal{M}_{2}$ | $3^{rd}\to\mathcal{M}_{3}$
000 | $\ket{+}\ket{+}$ |  $0$ |  $0$ |  random
001 | $\ket{-}\ket{-}$ |  $0$ |  $0$ |  random
010 | $\ket{0}\ket{0}$ |  $0$ |  $1$ |  $0$
011 | $\ket{1}\ket{1}$ |  $0$ |  $1$ |  $1$
100 | $\ket{1}\ket{0}$ |  $1$ |  $0$ |  $0$
101 | $\ket{0}\ket{1}$ |  $1$ |  $0$ |  $1$
110 | $\ket{+}\ket{-}$ |  $1$ |  $1$ |  random
111 | $\ket{-}\ket{+}$ |  $1$ |  $1$ |  random
$I(a_{k}:\beta~{}|~{}b=k)$ |  $1$ |  $1$ |  $\approx 0.19$
$I_{3}$ | $:=\sum_{k=1}^{3}I(a_{k}:\beta~{}|~{}b=k)\approx 2.19>2$
Table 1: Violation of IC principle with minimal tensor product composition of
two elementary qubits. Since the theory allows only product states it cannot
yield any nonlocal correlation. Here
$\ket{\pm}:=(\ket{0}\pm\ket{1})/\sqrt{2}$. The bit values in rightmost three
columns denote Bob’s guess of the corresponding bit, where ‘random’ means he
guesses the bit value randomly and be correct in half of the time.

At this point it should be noted that the strategy does not invoke any
classical correlation which can be shared between Alice and Bob in advance.
Therefore the obtained success in Theorem 1 might not be the optimal one, and
the question of optimality we leave here for future. However, by considering
maximal tensor product composition between two qubits our next result provides
a perfect strategy for IC-3 game.

######  Theorem 2.

Maximal tensor product composition of two elementary quantum violates IC
principle.

###### Proof.

For any $\ket{\psi}_{AB}\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ we have
$\Gamma(\psi_{AB}):=\mathbb{I}_{A}\otimes\mathrm{T}_{B}(\psi_{AB})\in\Omega[\mathbb{C}^{2}\otimes_{\max}\mathbb{C}^{2}]$,
where $\mathbb{I}$ is the identity map, $\mathrm{T}$ is the transpose map, and
$\psi:=\ket{\psi}\bra{\psi}$. Alice can use any of these states for encoding
her strings. The explicit encodings used by Alice are shown in Table 2. For
decoding, Bob is allowed to perform any separable measurement on the composite
state received from Alice. By $ZZ$ we denote the separable measurement
$\\{\ket{00}_{AB}\bra{00}+\ket{11}_{AB}\bra{11},\ket{01}_{AB}\bra{01}+\ket{10}_{AB}\bra{10},\\}\equiv\\{\pi^{z}_{c}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{z}\otimes\sigma_{z}\right),\pi^{z}_{ac}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{z}\otimes\sigma_{z}\right)\\}$.
Similarly we have the separable measurements
$XX\equiv\\{\pi^{x}_{c}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{x}\otimes\sigma_{x}\right),\pi^{x}_{ac}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{x}\otimes\sigma_{x}\right)\\}$
and
$YY\equiv\\{\pi^{y}_{c}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{y}\otimes\sigma_{y}\right),\pi^{y}_{ac}:=\frac{1}{2}\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{y}\otimes\sigma_{y}\right)\\}$.
For decoding the $1^{st}$, $2^{nd}$, and $3^{rd}$ bit Bob performs $XX$, $YY$,
and $ZZ$ measurements respectively, and guesses the bit value as $0~{}(1)$ if
the effect $\pi^{\star}_{c}~{}(\pi^{\star}_{ac})$ clicks. Expressing Alice’s
encoding states in Hilbert–Schmidt form:

$\displaystyle\Gamma(\phi^{+})$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{x}\otimes\sigma_{x}+\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\psi^{+}$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{x}\otimes\sigma_{x}+\sigma_{y}\otimes\sigma_{y}-\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\phi^{+}$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\Gamma(\psi^{+})$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}+\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}-\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\phi^{-}$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{x}\otimes\sigma_{x}+\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\Gamma(\psi^{-})$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{x}\otimes\sigma_{x}+\sigma_{y}\otimes\sigma_{y}-\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\Gamma(\phi^{-})$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z}\right)/4,$
$\displaystyle\psi^{-}$
$\displaystyle=\left(\mathbf{1}\otimes\mathbf{1}-\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y}-\sigma_{z}\otimes\sigma_{z}\right)/4,$

it becomes evident that this particular encoding-decoding strategy yields a
perfect winning condition for the game IC-3. Thus we have $I_{3}=3$,
establishing a violation of IC principle as according to Proposition 2 Holevo
capacity of the corresponding system is bounded by $2$. ∎

Input | Alice’s |  Bob’s decoding
---|---|---
string | encoding |  $1^{st}\to$ XX |  $2^{nd}\to$ YY |  $3^{rd}\to$ ZZ
000 | $\Gamma(\phi^{+})$ |  $0$ |  $0$ |  $0$
001 | $\psi^{+}$ |  $0$ |  $0$ |  $1$
010 | $\phi^{+}$ |  $0$ |  $1$ |  $0$
011 | $\Gamma(\psi^{+})$ |  $0$ |  $1$ |  $1$
100 | $\phi^{-}$ |  $1$ |  $0$ |  $0$
101 | $\Gamma(\psi^{-})$ |  $1$ |  $0$ |  $1$
110 | $\Gamma(\phi^{-})$ |  $1$ |  $1$ |  $0$
111 | $\psi^{-}$ |  $1$ |  $1$ |  $1$
$I(a_{k}:\beta~{}|~{}b=k)$ |  $1$ |  $1$ |  $1$
$I_{3}$ | $:=\sum_{k=1}^{3}I(a_{k}:\beta~{}|~{}b=k)=3>2$
Table 2: Prefect strategy for IC-$3$ game with communication of two qubits in
maximal tensor product. Here,
$\ket{\phi^{\pm}}:=(\ket{00}\pm\ket{11})/\sqrt{2}$ and
$\ket{\psi^{\pm}}:=(\ket{01}\pm\ket{10})/\sqrt{2}$. Rightmost three columns
show that Bob’s guess is always correct. Thus the strategy yields
$I(a_{k}:\beta~{}|~{}b=k)=1~{}\forall~{}k\in\\{1,2,3\\}$, and as a result we
have $I_{3}=3$.

Discussion.– While we have excluded the two extreme composition through the
violation of IC principle, it would be interesting to exclude all other
intermediate compositions except the quantum one in a similar approach. A
first step to this direction is to generalize our Proposition 1 and 2 for such
an arbitrary composition which we conjecture to hold true. Then an encoding-
decoding strategy in such a composite model yielding a better than quantum
success in IC-N game would invalidate that composition to be the physical one.
For instance, any composition allowing states and effects used for encoding
and decoding either in Theorem 1 or in Theorem 2 immediately gets excluded as
unphysical.

The work of Ref.[25] is also worth mentioning, where it has been shown that
all the NS correlations possible in any of the bipartite composition of local
quantum systems are quantum simulable hence beyond quantum nonlocal
correlation is not possible there (see also [42]). More precisely, the work in
[25] does not rule out the existence of POPT states, but shows that they do
not generate stronger than quantum correlations in Bell kind of experiment. In
fact minimal tensor product composition allows only local correlation in Bell
scenario. Despite this, our result show that different such exotic
compositions can be rejected as unphysical with the help of IC principles. The
power emerges from the fact that IC deals with communication tasks involving
both the state space for encodings and effect space for decoding. The beyond
quantum exotic nature of state and/or effect space exhibit their implication
in the violation of IC principle.

Some recent studies by the present authors [35, 36] are also worth noting
here. By considering the pairwise distinguishability game, where it has been
shown that minimal and maximal tensor product compositions allow stronger than
quantum correlations in timelike scenario. The feature of ‘dimension mismatch’
[43] plays the crucial role there. In that respect, the approach of the
present work is logically independent as it can be argued that ‘dimension
mismatch’ is not necessary for violation of IC principle (see Appendix C).

Conclusion.– The notion of composition is one of the guiding tools to
fabricate our worldview – while complex objects are composed of elementary
parts, some compositions deem implausible [44]. The idea becomes important
even while constructing theories in Physics [45, 46, 47]. In this work, we
study this particular aspect while considering multiple quantum systems.
Interestingly, we show that the principle of Information Causality plays a
crucial role in selecting the quantum composition among different mathematical
possibilities. In the process Information Causality alone can discard a theory
that admits only Bell local correlation. This might make Information Causality
champion over the other principles [15, 16, 17]. Present work thus brings some
physical justifying towards quantum composition structure from information
principles. The potentiality arises from the communication aspect of those
principles since they invoke preparations (for encoding) and measurements (for
decoding) of the involved systems and thus becomes more structure sensitive.
As for future works, it would be quite interesting to see what other
structural aspects of multipartite quantum systems can be rationalized through
information principles as the study in the present work is limited to
bipartite compositions only.

###### Acknowledgements.

SGN acknowledges support from the CSIR project 09/0575(15951)/2022-EMR-I. GLS
acknowledges support from the CSIR project 09/0575(15830)/2022-EMR-I. MA and
MB acknowledge funding from the National Mission in Interdisciplinary Cyber-
Physical systems from the Department of Science and Technology through the
I-HUB Quantum Technology Foundation (Grant no: I-HUB/PDF/2021-22/008). MB
acknowledges support through the research grant of INSPIRE Faculty fellowship
from the Department of Science and Technology, Government of India and the
start-up research grant from SERB, Department of Science and Technology (Grant
no: SRG/2021/000267).

##  I Appendix: A

Proof of Proposition 2.

###### Proof.

Assume that $\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\max}k}\right)=N$.
Clearly, $N\geq d^{k}$ since there are $d^{k}$ product states in
$(\mathbb{C}^{d})^{\otimes_{\max}k}$ that can be perfectly distinguished by
product measurements. Furthermore, since we have assumed
$\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\max}k}\right)=N$, there exist
$N$ POPT states $\\{W_{1},\cdots,W_{N}\\}$ and a separable measurement
$\\{E_{1},\cdots,E_{N}~{}|~{}\sum_{i=1}^{N}E_{i}=\mathbf{1}_{d}^{\otimes_{k}}\\}$
such that $\operatorname{Tr}(E_{i}W_{j})=\delta_{ij}~{}~{}\forall i,j$. Let
$Y_{i}:=\mathbf{1}_{d}^{\otimes_{k}}-W_{i}$. We will first show that $Y_{i}$
is an (unnormalized) POPT state, i.e., $\bra{a_{1\cdots
k}}Y_{i}\ket{a_{1\cdots k}}\geq 0$ for any choice of product state
$\ket{a_{1\cdots k}}:=\ket{a_{1}\otimes a_{2}\otimes\cdots\otimes a_{k}}$.
Indeed, $\bra{a_{1\cdots k}}Y_{i}\ket{a_{1\cdots k}}=1-\bra{a_{1\cdots
k}}W_{i}\ket{a_{1\cdots k}}$, and we only need to show that $\bra{a_{1\cdots
k}}W_{i}\ket{a_{1\cdots k}}\leq 1$. But this follows from the fact that
$\bra{a_{1\cdots k}}W_{i}\ket{a_{1\cdots k}}$ must be a valid probability
since $W_{i}$ is a POPT state and $\ket{a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{k}}\bra{a_{1}\otimes a_{2}\otimes\cdots\otimes a_{k}}$ is a valid effect in
maximal composition. Now we can proceed in a similar manner as in the proof of
Proposition 1, i.e.

$\displaystyle d^{k}$
$\displaystyle=\operatorname{Tr}(\mathbf{1}_{d}^{\otimes_{k}})=\sum_{i=1}^{N}\operatorname{Tr}(E_{i})=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}\mathbf{1}_{d}^{\otimes_{k}}]$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}(W_{i}+Y_{i})]$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Tr}[E_{i}W_{i}]+\sum_{i=1}^{N}\operatorname{Tr}[E_{i}Y_{i}]$
$\displaystyle\geq\sum_{i=1}^{N}\operatorname{Tr}[E_{i}W_{i}]=\sum_{i=1}^{N}\delta_{ii}=N.$

The inequality follows from the fact that $Y_{i}$ is POPT, and thus
$\operatorname{Tr}[E_{i}Y_{i}]\geq 0$. Since we already showed that $N\geq
d^{k}$, we conclude that
$N=\mathcal{I}\left((\mathbb{C}^{d})^{\otimes_{\max}k}\right)=d^{k}$.
Consequently we have
$\Theta(\mathbb{C}^{d}_{\max}):=\lim_{k\to\infty}\frac{1}{k}\log\left[\mathcal{I}((\mathbb{C}^{d})^{\otimes_{\max}k})\right]=\lim_{k\to\infty}\frac{1}{k}\log(d^{k})=\log
d$. ∎

##  II Appendix: B

Figure 3: Perfect strategy of the IC-$2$ with communication of a square-bit
from Alice to Bob. Alice’s encodings are shown in the figure. For decoding the
first bit Bob performs the measurement $\mathcal{M}_{1}:=\\{e_{0},e_{2}\\}$
and guesses the bit value as $0$ if the effect $e_{0}$ clicks, else he guesses
$1$. For the second bit he performs the measurement
$\mathcal{M}_{2}:=\\{e_{1},e_{3}\\}$ and guesses the bit value as $0$ if the
effect $e_{3}$ clicks, else he guesses $1$. Since Bob always guesses the bit
correctly, $I(a_{k}:\beta|b=k)=1$ for $k\in\\{1,2\\}$, which in turn results
in $I_{2}=2$, a violation of the IC principle.

We first briefly recall a toy system having a square shaped state space [39].
Its four extremal states are given by
$\omega_{0}=(1,0,1)^{\mathrm{T}},~{}\omega_{1}=(0,1,1)^{\mathrm{T}},~{}\omega_{2}=(-1,0,1)^{\mathrm{T}},~{}\&~{}\omega_{3}=(0,-1,1)^{\mathrm{T}}$;
and four of its ray extremal effects are
$e_{0}=\frac{1}{2}(1,1,1)^{\mathrm{T}},~{}e_{1}=\frac{1}{2}(-1,1,1)^{\mathrm{T}},~{}e_{2}=\frac{1}{2}(-1,-1,1)^{\mathrm{T}},~{}\&~{}e_{3}=\frac{1}{2}(1,-1,1)^{\mathrm{T}}$,
while $u=(0,0,1)^{\mathrm{T}}$ is the unit effect. An interesting feature of
this theory is that, unlike quantum theory, it is not self dual, i.e., the
state cone and effect cone are not identical to each other. Although all the
four states of this square-bit theory are pairwise distinguishable, its
information capacity as well as Holevo capacity is the same as that of a
qubit.

However, it turns out that the square-bit theory violates the IC principle. To
see this, consider the IC-$2$ game. With a classical bit or quantum bit
communication from Alice to Bob one has $I_{2}=1$, which satisfies the
necessary condition of IC. But with a square-bit communication one can have
$I_{2}=2$, a clear cut violation of the IC principle. The prefect strategy of
the IC-$2$ game with a square bit is shown in Fig. 3. For more detail we refer
to the work of Ref. [38]. The violation of IC by square-bit model possibly
caused by the lack of ‘consistency’ in mutual information as required in the
derivation of the necessary condition of IC. A consistent notion of entropy is
not well defined in such a model [48, 49].

##  III Appendix: C

Figure 4: Perfect strategy of the IC-$2$ game with communication of an
octagon-bit from Alice to Bob. Alice’s encodings and Bob’s decodings are
shown. A straightforward calculation yields
$I_{2}:=\sum_{k=1}^{2}I(a_{k}:\beta|b=k)=2-\frac{1}{2\sqrt{2}}+(1-1/\sqrt{2})\log_{2}(1-1/\sqrt{2})\approx
1+0.1275$, a violation of the IC principle.

While information capacity $\mathcal{I}(\mathcal{S})$ of a system
$\mathcal{S}$ is the the maximum number of states that can be perfectly
distinguished in a single shot measurement, we can also introduce another
concept called information dimension of a system [43]. Information dimension
$\mathcal{I}_{nf}(\mathcal{S})$ of a system $\mathcal{S}$ is the maximum
number of states of the system that are perfectly distinguishable pairwise.
Arguably it follows that
$\Delta\mathcal{I}:=\mathcal{I}_{nf}(\mathcal{S})-\mathcal{I}(\mathcal{S})\geq
0$, and whenever $\Delta\mathcal{I}>0$ we say the system exhibits ‘dimension
mismatch’. Importantly, ‘dimension mismatch’ is not necessary for violation of
IC principle. To see this, we recall the polygon system with eight extremal
states $\omega_{i}=\left(r_{8}\cos\frac{2\pi i}{8},r_{8}\sin\frac{2\pi
i}{8},1\right)^{\mathrm{T}}$ and eight extremal effects
$e_{i}=\frac{1}{2}\left(r_{8}\cos\frac{2\pi(2i+1)}{8},r_{8}\sin\frac{2\pi(2i+1)}{8},1\right)^{\mathrm{T}}$,
with $r_{8}=\sqrt{\sec\frac{\pi}{n}}~{}\&~{}i\in\\{0,\cdots,7\\}$ [39].
Clearly, this system does not exhibit any dimension mismatch. However, as
noted in [38], it violates the IC principle. For the sake of completeness we
depict the explicit protocol in Fig. 4.

## References

  * [1] I. H. Deutsch; Harnessing the Power of the Second Quantum Revolution, PRX Quantum 1, 020101 (2020).
  * [2] D. Bohm; A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I, Phys. Rev. 85, 166 (1952).
  * [3] H. Everett; “Relative State” Formulation of Quantum Mechanics, Rev. Mod. Phys. 29, 454 (1957).
  * [4] J. A. Wheeler; Assessment of Everett’s “Relative State” Formulation of Quantum Theory, Rev. Mod. Phys. 29, 463 (1957).
  * [5] L. E. Ballentine; The Statistical Interpretation of Quantum, Rev. Mod. Phys. 42, 358 (1970).
  * [6] R. B. Griffiths; Consistent histories and the interpretation of quantum mechanics, J. Stat. Phys. 36, 219 (1984).
  * [7] G. C. Ghirardi, A. Rimini, and T. Weber; Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D 34, 470 (1986).
  * [8] R. Omnès; Consistent interpretations of quantum mechanics, Rev. Mod. Phys. 64, 339 (1992).
  * [9] C. A. Fuchs and R. Schack; Quantum-Bayesian coherence, Rev. Mod. Phys. 85, 1693 (2013).
  * [10] M. Tegmark; The Interpretation of Quantum Mechanics: Many Worlds or Many Words? Fortsch. Phys. 46, 855 (1998).
  * [11] M. Schlosshauer, J. Kofler, and A. Zeilinger; A Snapshot of Foundational Attitudes Toward Quantum Mechanics, Stud. Hist. Phil. Mod. Phys. 44, 222 (2013).
  * [12] G. Brassard, H. Buhrman, N. Linden, A. A. Méthot, A. Tapp, and F. Unger; Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial, Phys. Rev. Lett. 96, 250401 (2006).
  * [13] W. van Dam; Implausible consequences of superstrong nonlocality, Nat. Comput. 12, 9 (2013).
  * [14] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski; Information causality as a physical principle, Nature 461, 1101 (2009).
  * [15] M. Navascués and H. Wunderlich; A glance beyond the quantum model, Proc. R. Soc. London A 466, 881 (2010).
  * [16] T. Fritz, A.B. Sainz, R. Augusiak, J Bohr Brask, R. Chaves, A. Leverrier, and A. Acín; Local orthogonality as a multipartite principle for quantum correlations, Nat Commun 4, 2263 (2013).
  * [17] B. Amaral, M. T. Cunha, and A. Cabello; Exclusivity principle forbids sets of correlations larger than the quantum set, Phys. Rev. A 89, 030101(R) (2014).
  * [18] M. Dall’Arno, S. Brandsen, A. Tosini, F. Buscemi, and V. Vedral; No-Hypersignaling Principle, Phys. Rev. Lett. 119, 020401 (2017).
  * [19] C. H. Bennett and S. J. Wiesner; Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  * [20] A. Ahanj, S. Kunkri, A. Rai, R. Rahaman, and P. S. Joag; Bound on Hardy’s nonlocality from the principle of information causality, Phys. Rev. A 81, 032103 (2010)
  * [21] S. Das, M. Banik, M. R. Gazi, A. Rai, S. Kunkri, and R. Rahaman; Bound on tri-partite Hardy’s nonlocality respecting all bi-partite principles, Quantum Inf Process 12, 3033 (2013).
  * [22] Md. R. Gazi, M. Banik, S. Das, A. Rai, and S. Kunkri; Macroscopic locality with equal bias reproduces with high fidelity a quantum distribution achieving the Tsirelson’s bound, Phys. Rev. A 88, 052115 (2013).
  * [23] N. Miklin and M. Pawłowski; Information Causality without Concatenation, Phys. Rev. Lett. 126, 220403 (2021).
  * [24] I. Namioka and R. R. Phelps; Tensor products of compact convex sets, Pac. J. Math. 31, 469 (1969).
  * [25] H. Barnum, S. Beigi, S. Boixo, M. B. Elliott, and S. Wehner; Local Quantum Measurement and No-signaling Imply Quantum Correlations, Phys. Rev. Lett. 104, 140401 (2010).
  * [26] J. Barrett; Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007).
  * [27] G. Chiribella, G. M. D’Ariano, and P. Perinotti; Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011).
  * [28] H. Barnum and A. Wilce; Information processing in convex operational theories, Electron. Notes Theor. Comput. Sci. 70, 3 (2011).
  * [29] L. Masanes and M. P. Müller; A derivation of quantum theory from physical requirements, New J. Phys. 13, 063001 (2011).
  * [30] M. Plávala; General probabilistic theories: An introduction, arXiv:2103.07469 [quant-ph].
  * [31] H. Barnum and A. Wilce; Local Tomography and the Jordan Structure of Quantum Theory, Found. Phys. 44, 192 (2014).
  * [32] G. Chiribella, G. M. D’Ariano, and P. Perinotti; Probabilistic theories with purification, Phys. Rev. A 81, 062348 (2010).
  * [33] M. P. Müller, O. C. O. Dahlsten, and V. Vedral; Unifying Typical Entanglement and Coin Tossing: on Randomization in Probabilistic Theories, Commun. Math. Phys. 316, 441 (2012).
  * [34] H. Arai, Y. Yoshida, and M. Hayashi; Perfect Discrimination of Non-Orthogonal Separable Pure States on Bipartite System in General Probabilistic Theory, J. Phys. A: Math. Theo. 52, 465304 (2019).
  * [35] S. G. Naik, E. P. Lobo, S. Sen, R. K. Patra, M. Alimuddin, T. Guha, S. S. Bhattacharya, and M. Banik; Composition of Multipartite Quantum Systems: Perspective from Timelike Paradigm, Phys. Rev. Lett. 128, 140401 (2022).
  * [36] S. Sen, E. P. Lobo, R. K. Patra, S. G. Naik, M. Alimuddin, A. D. Bhowmik, and M. Banik; Timelike correlations and quantum tensor product structure, Phys. Rev. A 106, 062406 (2022).
  * [37] P.E. Frenkel and M. Weiner; Classical Information Storage in an n-Level Quantum System, Commun. Math. Phys. 340, 563 (2015).
  * [38] S. Massar and M. K. Patra; Information and communication in polygon theories, Phys. Rev. A 89, 052124 (2014).
  * [39] P. Janotta, C. Gogolin, J. Barrett, and N. Brunner; Limits on nonlocal correlations from the structure of the local state space, New J. Phys. 13, 063024 (2011).
  * [40] A. Ambainis, A. Nayak, A. Ta-Shma, and U. Vazirani; Quantum dense coding and quantum finite automata, J. ACM 49, 496 (2002).
  * [41] S. Popescu and D. Rohrlich; Quantum nonlocality as an axiom, Found. Phys. 24, 379 (1994).
  * [42] E. P. Lobo, S. G. Naik, S. Sen, R. K. Patra, M. Banik, and M. Alimuddin; Certifying beyond quantumness of locally quantum no-signaling theories through a quantum-input Bell test, Phys. Rev. A 106, L040201 (2022).
  * [43] N. Brunner, M. Kaplan, A. Leverrier, and P. Skrzypczyk; Dimension of physical systems, information processing, and thermodynamics, New J. Phys. 16, 123050 (2014).
  * [44] L. S. Penrose and R. Penrose; Impossible objects: A special type of visual illusion, Br. J. Psychol. 49, 31 (1958).
  * [45] L. Hardy; On the theory of composition in physics, in Computation, Logic, Games, and Quantum Foundations, The Many Facets of Samson Abramsky, edited by B. Coecke, L. Ong, and P. Panangaden, Lecture Notes in Computer Science Vol. 7860 (Springer, Berlin, Heidelberg, 2013), 10.1007/978-3-642-38164-5-7.
  * [46] L. Hardy; Operational General Relativity: Possibilistic, Probabilistic, and Quantum, arXiv:1608.06940 [gr-qc].
  * [47] A. D. Bhowmik, P. Parashar, G. Kar, and M. Banik; From no causal loop to absoluteness of cause: discarding the quantum NOT logic, arXiv:2109.09953 [quant-ph].
  * [48] M. Krumm, H. Barnum, J. Barrett, and M. P. Mueller; Thermodynamics and the structure of quantum theory, New J. Phys. 19, 043025 (2017).
  * [49] R. Takakura; Entropy of mixing exists only for classical and quantum-like theories among the regular polygon theories, J. Phys. A: Math. Theor. 52, 465302 (2019).