# Strategies for single-shot discrimination of process matrices

Paulina Lewandowska1 ,  Łukasz Pawela1 and  Zbigniew Puchała1 1Institute of
Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Bałtycka
5, 44-100 Gliwice, Poland

###### Abstract.

The topic of causality has recently gained traction quantum information
research. This work examines the problem of single-shot discrimination between
process matrices which are an universal method defining a causal structure. We
provide an exact expression for the optimal probability of correct
distinction. In addition, we present an alternative way to achieve this
expression by using the convex cone structure theory. We also express the
discrimination task as semidefinite programming. Due to that, we have created
the SDP calculating the distance between process matrices and we quantify it
in terms of the trace norm. As a valuable by-product, the program finds an
optimal realization of the discrimination task. We also find two classes of
process matrices which can be distinguished perfectly. Our main result,
however, is a consideration of the discrimination task for process matrices
corresponding to quantum combs. We study which strategy, adaptive or non-
signalling, should be used during the discrimination task. We proved that no
matter which strategy you choose, the probability of distinguishing two
process matrices being a quantum comb is the same.

##  1\. Introduction

The topic of causality has remained a staple in quantum physics and quantum
information theory for recent years. The idea of a causal influence in quantum
physics is best illustrated by considering two characters, Alice and Bob,
preparing experiments in two separate laboratories. Each of them receives a
physical system and performs an operation on it. After that, they send their
respective system out of the laboratory. In a causally ordered framework,
there are three possibilities: Bob cannot signal to Alice, which means the
choice of Bob’s action cannot influence the statistics Alice records (denoted
by $A\prec B$), Alice cannot signal to Bob ($B\prec A$), or neither party can
influence the other $(A||B)$. A causally neutral formulation of quantum theory
is described in terms of quantum combs [1].

One may wonder if Alice’s and Bob’s action can influence each other. It might
seem impossible, except in a world with closed time-like curves (CTCs) [2].
But the existence of CTCs implies some logical paradoxes, such as the
grandfather paradox [3]. Possible solutions have been proposed in which
quantum mechanics and CTCs can exist and such paradoxes are avoided, but
modifying quantum theory into a nonlinear one [4]. A natural question arises:
is it possible to keep the framework of linear quantum theory and still go
beyond definite causal structures?

One such framework was proposed by Oreshkov, Costa and Brukner [5]. They
introduced a new resource called a process matrix – a generalization of the
notion of quantum state. This new approach has provided a consistent
representation of correlations in casually and non-causally related
experiments. Most interestingly, they have described a situation that two
actions are neither causally ordered and one cannot say which action
influences the second one. Thanks to that, the term of causally non-separable
(CNS) structures started to correspond to superpositions of situations in
which, roughly speaking, Alice can signal to Bob, and Bob can signal to Alice,
jointly. A general overview of causal connection theory is described in [6].

The indefinite causal structures could make a new aspect of quantum
information processing. This more general model of computation can outperform
causal quantum computers in specific tasks, such as learning or discriminating
between two quantum channels [7, 8, 9]. The problem of discriminating quantum
operations is of the utmost importance in modern quantum information science.
Imagine we have an unknown operation hidden in a black box. We only have
information that it is one of two operations. The goal is to determine an
optimal strategy for this process that achieves the highest possible
probability of discrimination. For the case of a single-shot discrimination
scenario, researchers have used different approaches, with the possibility of
using entanglement in order to perform an optimal protocol. In [10], Authors
have shown that in the task of discrimination of unitary channels, the
entanglement is not necessary, whereas for quantum measurements [11, 12, 13],
we need to use entanglement. Considering multiple-shot discrimination
scenarios, researchers have utilized parallel or adaptive approaches. In the
parallel case, we establish that the discrimination between operations does
not require pre-processing and post-processing. One example of such an
approach is distinguishing unitary channels [10], or von Neumann measurements
[14]. The case when the black box can be used multiple times in an adaptive
way was investigated by the authors of [15, 16], who have proven that the use
of adaptive strategy and a general notion of quantum combs can improve
discrimination.

In this work, we study the problem of discriminating process matrices in a
single-shot scenario. We obtain that the probability of correct distinction
process matrices is strictly related to the Holevo-Helstrom theorem for
quantum channels. Additionally, we write this result as a semidefinite program
(SDP) which is numerically efficient. The SDP program allows us to find an
optimal discrimination strategy. We compare the effectiveness of the obtained
strategy with the previously mentioned strategies. The problem gets more
complex in the case when we consider the non-causally ordered framework. In
this case, we consider the discrimination task between two process matrices
having different causal orders.

This paper is organized as follows. In Section 2 we introduce necessary
mathematical framework. Section 3 is dedicated to the concept of process
matrices. Section 4 presents the discrimination task between pairs of process
matrices and calculate the exact probability of distinguishing them. Some
examples of discrimination between different classes of process matrices are
presented in Section 5. In Section 5.1, we consider the discrimination task
between free process matrices, whereas in Section 5.2 we consider the
discrimination task between process matrices being quantum combs. In Section
5.3, we show a particular class of process matrices having opposite causal
structures which can be distinguished perfectly. Finally, Section 6 and
Section 7 are devoted to semidefinite programming, thanks to which, among
other things, we obtain an optimal discrimination strategy. In Section 8, we
analyze an alternative way to achieve this expression using the convex cone
structure theory. Concluding remarks are presented in the final Section 9. In
the Appendix A, we provide technical details about the convex cone structure.

##  2\. Mathematical preliminaries

Let us introduce the following notation. Consider two complex Euclidean spaces
and denote them by $\mathcal{X},\mathcal{Y}$. By
$\mathrm{L}(\mathcal{X},\mathcal{Y})$ we denote the collection of all linear
mappings of the form $A:\mathcal{X}\rightarrow\mathcal{Y}$. As a shorthand put
$\mathrm{L}(\mathcal{X})\coloneqq\mathrm{L}(\mathcal{X},\mathcal{X}).$ By
$\mathrm{Herm}(\mathcal{X})$ we denote the set of Hermitian operators while
the subset of $\mathrm{Herm}(\mathcal{X})$ consisting of positive semidefinite
operators will be denoted by $\mathrm{Pos}(\mathcal{X})$. The set of quantum
states, that is positive semidefinite operators $\rho$ such that
$\operatorname{tr}\rho=1$, will be denoted by $\Omega(\mathcal{X})$. An
operator $U\in\mathrm{L}\left(\mathcal{X}\right)$ is unitary if it satisfies
the equation $UU^{\dagger}=U^{\dagger}U={\rm 1l}_{\mathcal{X}}$. The notation
$\mathrm{U}\left(\mathcal{X}\right)$ will be used to denote the set of all
unitary operators. We will also need a linear mapping of the form
$\Phi:\mathrm{L}(\mathcal{X})\rightarrow\mathrm{L}(\mathcal{Y})$ transforming
$\text{L}(\mathcal{X})$ into $\text{L}(\mathcal{Y})$. The set of all linear
mappings is denoted $\mathrm{M}(\mathcal{X},\mathcal{Y})$. There exists a
bijection between set $\mathrm{M}(\mathcal{X},\mathcal{Y})$ and the set of
operators $\mathrm{L}(\mathcal{Y}\otimes\mathcal{X})$ known as the Choi [17]
and Jamiołkowski [18] isomorphism. For a given linear mapping
$\Phi_{M}:\mathrm{L}(\mathcal{X})\rightarrow\mathrm{L}(\mathcal{Y})$
corresponding Choi matrix $M\in\mathrm{L}(\mathcal{Y}\otimes\mathcal{X})$ can
be explicitly written as

$M\coloneqq\sum_{i,j=0}^{\dim(\mathcal{X})-1}\Phi_{M}(|i\rangle\\!\langle
j|)\otimes|i\rangle\\!\langle j|.$ (1)

We will denote linear mappings by $\Phi_{M},\Phi_{N},\Phi_{R}$ etc., whereas
the corresponding Choi matrices as plain symbols: $M,N,R$ etc. Let us consider
a composition of mappings $\Phi_{R}=\Phi_{N}\circ\Phi_{M}$ where
$\Phi_{N}:\mathrm{L}(\mathcal{Z})\rightarrow\mathrm{L}(\mathcal{Y})$ and
$\Phi_{M}:\mathrm{L}(\mathcal{X})\rightarrow\mathrm{L}(\mathcal{Z})$ with Choi
matrices $N\in\mathrm{L}(\mathcal{Z}\otimes\mathcal{Y})$ and
$M\in\mathrm{L}(\mathcal{X}\otimes\mathcal{Z})$, respectively. Then, the Choi
matrix of $\Phi_{R}$ is given by [19]

$R=\operatorname{tr}_{\mathcal{Z}}\left[\left({\rm 1l}_{\mathcal{Y}}\otimes
M^{T_{\mathcal{Z}}}\right)\left(N\otimes{\rm 1l}_{\mathcal{X}}\right)\right],$
(2)

where $M^{T_{\mathcal{Z}}}$ denotes the partial transposition of $M$ on the
subspace $\mathcal{Z}$. The above result can be expressed by introducing the
notation of the link product of the operators $N$ and $M$ as

$N*M\coloneqq\operatorname{tr}_{\mathcal{Z}}\left[\left({\rm
1l}_{\mathcal{Y}}\otimes M^{T_{\mathcal{Z}}}\right)\left(N\otimes{\rm
1l}_{\mathcal{X}}\right)\right].$ (3)

Finally, we introduce a special subset of all mappings $\Phi$, called quantum
channels, which are completely positive and trace preserving (CPTP). In other
words, the first condition reads

$(\Phi\otimes\mathcal{I_{\mathcal{Z}}})(X)\in\mathrm{Pos}(\mathcal{Y}\otimes\mathcal{Z})$
(4)

for all $X\in\mathrm{Pos}(\mathcal{X}\otimes\mathcal{Z})$ and
$\mathcal{I_{\mathcal{Z}}}$ is an identity channel acts on
$\mathrm{L}(\mathcal{Z})$ for any $\mathcal{Z}$, while the second condition
reads

$\operatorname{tr}(\Phi(X))=\operatorname{tr}(X)$ (5)

for all $X\in\mathrm{L}(\mathcal{X})$.

In this work we will consider a special class of quantum channels called non-
signaling channels (or causal channels) [20, 21]. We say that
$\Phi_{N}:\mathrm{L}(\mathcal{X}_{I}\otimes\mathcal{Y}_{I})\rightarrow\mathrm{L}(\mathcal{X}_{O}\otimes\mathcal{Y}_{O})$
is a non-signaling channel if its Choi operator satisfies the following
conditions

$\begin{split}&\operatorname{tr}_{\mathcal{X}_{O}}(N)=\frac{{\rm
1l}_{\mathcal{X}_{I}}}{\dim(\mathcal{X}_{I})}\otimes\operatorname{tr}_{\mathcal{X}_{O}\mathcal{X}_{1}}(N),\\\
&\operatorname{tr}_{\mathcal{Y}_{O}}(N)=\frac{{\rm
1l}_{\mathcal{Y}_{I}}}{\dim(\mathcal{Y}_{I})}\otimes\operatorname{tr}_{\mathcal{Y}_{O}\mathcal{Y}_{1}}(N).\end{split}$
(6)

It can be shown [22] that each non-signaling channel is an affine combination
of product channels. More precisely, any non-signaling channel
$\Phi_{N}:\mathrm{L}(\mathcal{X}_{I}\otimes\mathcal{Y}_{I})\rightarrow\mathrm{L}(\mathcal{X}_{O}\otimes\mathcal{Y}_{O})$
can be written as

$\Phi_{N}=\sum_{i}\lambda_{i}\Phi_{S_{i}}\otimes\Phi_{T_{i}},$ (7)

where
$\Phi_{S_{i}}:\mathrm{L}(\mathcal{X}_{I})\rightarrow\mathrm{L}(\mathcal{X}_{O})$
and
$\Phi_{T_{i}}:\mathrm{L}(\mathcal{Y}_{I})\rightarrow\mathrm{L}(\mathcal{Y}_{O})$
are quantum channels, $\lambda_{i}\in\mathbb{R}$ such that
$\sum_{i}\lambda_{i}=1$. For the rest of this paper, by
$\mathbf{NS}(\mathcal{X}_{I}\otimes\mathcal{X}_{O}\otimes\mathcal{Y}_{I}\otimes\mathcal{Y}_{O})$
we will denote the set of Choi matrices of non-signaling channels.

The most general quantum operations are represented by quantum instruments
[23, 24], that is, collections of completely positive (CP) maps
$\left\\{\Phi_{M_{i}}\right\\}_{i}$ associated to all measurement outcomes,
characterized by the property that $\sum_{i}\Phi_{M_{i}}$ is a quantum
channel.

We will also consider the concept of quantum network and tester [25]. We say
that $\Phi_{R^{(N)}}$ is a deterministic quantum network (or quantum comb) if
it is a concatenation of $N$ quantum channels and
$R^{(N)}\in\mathrm{L}\left(\bigotimes_{i=0}^{2N-1}\mathcal{X}_{i}\right)$
fulfills the following conditions

$\begin{split}R^{(N)}&\geq 0,\\\
\operatorname{tr}_{\mathcal{X}_{2k-1}}\left(R^{(k)}\right)&={\rm
1l}_{\mathcal{X}_{2k-2}}\otimes R^{(k-1)},\end{split}$ (8)

where
$R^{(k-1)}\in\mathrm{L}\left(\bigotimes_{i=0}^{2k-3}\mathcal{X}_{i}\right)$ is
the Choi matrix of the reduced quantum comb with concatenation of $k-1$
quantum channels, $k=2,\ldots,N$. We remind that a probabilistic quantum
network $\Phi_{S^{(N)}}$ is equivalent to a concatenation of $N$ completely
positive trace non increasing linear maps. Then, the Choi operator $S^{(N)}$
of $\Phi_{S^{(N)}}$ satisfies $0\leq S^{(N)}\leq R^{(N)}$, where $R^{(N)}$ is
Choi matrix of a quantum comb. Finally, we recall the definition of a quantum
tester. A quantum tester is a collection of probabilistic quantum networks
$\left\\{R_{i}^{(N)}\right\\}_{i}$ whose sum is a quantum comb, that is
$\sum_{i}R^{(N)}_{i}=R^{(N)}$, and additionally
$\dim(\mathcal{X}_{0})=\dim(\mathcal{X}_{2N-1})=1$.

We will also use the Moore–Penrose pseudo–inverse by abusing notation
$X^{-1}\in\mathrm{L}(\mathcal{Y},\mathcal{X})$ for an operator
$X\in\mathrm{L}(\mathcal{X},\mathcal{Y})$. Moreover, we introduce the
vectorization operation of $X$ defined by
$|X\rangle\rangle=\sum_{i=0}^{\dim(\mathcal{X})-1}(X|i\rangle)\otimes|i\rangle$.

##  3\. Process matrices

This section introduces the formal definition of the process matrix with its
characterization and intuition. Next, we present some classes of process
matrices considered in this paper.

Let us define the operator $\prescript{}{\mathcal{X}}{Y}$ as

$\prescript{}{\mathcal{X}}{Y}=\frac{{\rm
1l}_{\mathcal{X}}}{\dim(\mathcal{X})}\otimes\operatorname{tr}_{\mathcal{X}}Y$
(9)

for every $Y\in\mathrm{L}(\mathcal{X}\otimes\mathcal{Z})$, where $\mathcal{Z}$
is an arbitrary complex Euclidean space. We will also need the following
projection operator

$L_{V}(W)=\prescript{}{\mathcal{A}_{O}}{W}+\prescript{}{\mathcal{B}_{O}}{W}-\prescript{}{\mathcal{A}_{O}\mathcal{B}_{O}}{W}-\prescript{}{\mathcal{B}_{I}\mathcal{B}_{O}}{W}+\prescript{}{\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}{W}-\prescript{}{\mathcal{A}_{I}\mathcal{A}_{O}}{W}+\prescript{}{\mathcal{A}_{O}\mathcal{A}_{I}\mathcal{B}_{O}}{W}.$
(10)

where
$W\in\mathrm{Herm}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.

######  Definition 1.

We say that
$W\in\mathrm{Herm}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$
is a process matrix if it fulfills the following conditions

$W\geq 0,\,\,\,W=L_{V}(W),\,\,\
\operatorname{tr}(W)=\dim(\mathcal{A}_{O})\cdot\dim(\mathcal{B}_{O}),$ (11)

where the projection operator $L_{V}$ is defined by Eq. (10).

The set of all process matrices will be denoted by $\mathbf{W^{PROC}}$. In the
upcoming considerations, it will be more convenient to work with the
equivalent characterization of process matrices which can be found in [26].

######  Definition 2.

We say that $W\in\mathbf{W^{PROC}}$ is a process matrix if it fulfills the
following conditions

$W\geq 0,$
$\prescript{}{\mathcal{A}_{I}\mathcal{A}_{O}}{W}=\prescript{}{\mathcal{A}_{O}\mathcal{A}_{I}\mathcal{B}_{O}}{W},$
$\prescript{}{\mathcal{B}_{I}\mathcal{B}_{O}}{W}=\prescript{}{\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}{W},$
(12)
$W=\prescript{}{\mathcal{B}_{O}}{W}+\prescript{}{\mathcal{A}_{O}}{W}-\prescript{}{\mathcal{A}_{O}\mathcal{B}_{O}}{W},$
$\operatorname{tr}(W)=\dim(\mathcal{A}_{O})\cdot\dim(\mathcal{B}_{O}).$

The concept of process matrix can be best illustrated by considering two
characters, Alice and Bob, performing experiments in two separate
laboratories. Each party acts in a local laboratory, which can be identified
by an input space $\mathcal{A}_{I}$ and an output space $\mathcal{A}_{O}$ for
Alice, and analogously $\mathcal{B}_{I}$ and $\mathcal{B}_{O}$ for Bob. In
general, a label $i$, denoting Alice’s measurement outcome, is associated with
the CP map $\Phi_{M^{A}_{i}}$ obtained from the instrument
$\left\\{\Phi_{M^{A}_{i}}\right\\}_{i}$. Analogously, the Bob’s measurement
outcome $j$ is associated with the map $\Phi_{M^{B}_{j}}$ from the instrument
$\left\\{\Phi_{M^{B}_{j}}\right\\}_{j}$. Finally, the joint probability for a
pair of outcomes $i$ and $j$ can be expressed as

$p_{ij}=\operatorname{tr}\left[W\left(M_{i}^{A}\otimes
M_{j}^{B}\right)\right],$ (13)

where $W\in\mathbf{W^{PROC}}$ is a process matrix that describes the causal
structure outside of the laboratories. The valid process matrix is defined by
the requirement that probabilities are well defined, that is, they must be
non-negative and sum up to one. These requirements give us the conditions
present in Definition 1 and Definition 2.

In the general case, the Alice’s and Bob’s strategies can be more complex than
the product strategy $M_{i}^{A}\otimes M_{j}^{B}$ which defines the
probability $p_{ij}$ given by Eq. (13). If their action is somehow correlated,
we can write the associated instrument in the following form
$\left\\{\Phi_{N_{ij}^{AB}}\right\\}$. It was observed in [26] that this
instrument describes a valid strategy, that is

$\operatorname{tr}\left(W\sum_{ij}N_{ij}^{AB}\right)=1$ (14)

for all process matrix $W\in\mathbf{W^{PROC}}$ if and only if

$\sum_{ij}N_{ij}^{AB}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}).$
(15)

In this paper, we will consider different classes of process matrices.
Initially, we define the subset of process matrices known as free objects in
the resource theory of causal connection [27]. Such process matrices will be
defined as follows.

######  Definition 3.

We say that $W^{A||B}\in\mathbf{W^{PROC}}$ is a free process matrix if it
satisfies the following condition

$W^{A||B}=\rho_{\mathcal{A}_{I}\mathcal{B}_{I}}\otimes{\rm
1l}_{\mathcal{A}_{O}\mathcal{B}_{O}},$ (16)

where
$\rho_{\mathcal{A}_{I}\mathcal{B}_{I}}\in\Omega(\mathcal{A}_{I}\otimes\mathcal{B}_{I})$
is an arbitrary quantum state and ${\rm
1l}_{\mathcal{A}_{O}\mathcal{B}_{O}}\in\mathrm{L}(\mathcal{A}_{O}\otimes\mathcal{B}_{O})$.
The set of all process matrices of this form will be denoted by
$\mathbf{W^{A||B}}$.

We often consider process matrices corresponding to quantum combs[19]. For
example, a quantum comb $A\prec B$ (see in Fig. 1) shows that Alice’s and
Bob’s operations are performed in causal order. This means that Bob cannot
signal to Alice and the choice of Bob’s instrument cannot influence the
statistics Alice records. Such process matrices are formally defined in the
following way.

######  Definition 4.

We say that $W^{A\prec B}\in\mathbf{W^{PROC}}$ is a process matrix
representing a quantum comb $A\prec B$ if it satisfies the following
conditions

$\begin{split}W^{A\prec
B}=W^{\prime}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}}\otimes{\rm
1l}_{\mathcal{B}_{O}},\\\
\operatorname{tr}_{\mathcal{B}_{I}}W^{\prime}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}}=W^{\prime\prime}_{\mathcal{A}_{I}}\otimes{\rm
1l}_{\mathcal{A}_{O}}.\end{split}$ (17)

The set of all process matrices of this form will be denoted by
$\mathbf{W^{A\prec B}}$.

Figure 1. A schematic representation of a process matrix $W^{A\prec B}$
representing a quantum comb $A\prec B$.

One can easily observe that the set $\mathbf{W^{A||B}}$ is an intersection of
the sets $\mathbf{W^{A\prec B}}$ and $\mathbf{W^{B\prec A}}$. Finally, the
definition of the set $\mathbf{W^{A\prec B}}$, together with
$\mathbf{W^{B\prec A}}$ allow us to provide their convex hull which is called
as causally separable process matrices.

######  Definition 5.

We say that $W^{SEP}\in\mathbf{W^{PROC}}$ is a causally separable process
matrix if it is of the form

$W^{\text{SEP}}=pW^{A\prec B}+(1-p)W^{B\prec A},$ (18)

where $W^{A\prec B}\in\mathbf{W^{A\prec B}}$, $W^{B\prec
A}\in\mathbf{W^{B\prec A}}$ for some parameter $p\in[0,1]$. The set of all
causally separable process matrices will be denoted by $\mathbf{W^{SEP}}$.

There are however process matrices that do not correspond to a causally
separable process and such process matrices are known as causally non-
separable (CNS). The examples of such matrices were provided in [5, 28]. The
set of all causally non-separable process matrices will be denoted by
$\mathbf{W^{CNS}}$. In Fig. 2 we present a schematic plot of the sets of
process matrices.

Figure 2. A schematic representation of the sets of process matrices
$\mathbf{W^{PROC}}$.

##  4\. Discrimination task

This section presents the concept of discrimination between pairs of process
matrices. It is worth emphasizing that the definition of a process matrix is a
generalization of the concept of quantum states, channels, superchannels [29]
and even generalized supermaps [30, 31]. The task of discrimination between
process matrices poses a natural extension of discrimination of quantum states
[32], channels [33] or measurements [13]. The process matrices discrimination
task can be described by the following scenario.

Let us consider two process matrices $W_{0},W_{1}\in\mathbf{W^{PROC}}$. The
classical description of process matrices $W_{0},W_{1}$ is assumed to be known
to the participating parties. We know that one of the process matrices,
$W_{0}$ or $W_{1}$, describes the actual correlation between Alice’s and Bob’s
laboratories, but we do not know which one. Our aim is to determine, with the
highest possible probability, which process matrix describes this correlation.
For this purpose, we construct a discrimination strategy $S$. In the general
approach, such a strategy $S$ is described by an instrument
$S=\\{S_{0},S_{1}\\}$. Due to the requirement given by Eq. (14), the
instrument $S$ must fulfill the condition
$S_{0}+S_{1}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
The result of composing a process matrix $W$ with the discrimination strategy
$S$ results in a classical label which can take values zero or one. If the
label zero occurs, we decide to choose that the correlation is given by
$W_{0}$. Otherwise, we decide to choose $W_{1}$. In this setting the maximum
success probability $p_{\text{succ}}(W_{0},W_{1})$ of correct discrimination
between two process matrices $W_{0}$ and $W_{1}$ can be expressed by

$p_{\text{succ}}(W_{0},W_{1})=\frac{1}{2}\max_{S=\\{S_{0},S_{1}\\}}\left[\operatorname{tr}(W_{0}S_{0})+\operatorname{tr}(W_{1}S_{1})\right].$
(19)

The following theorem provides the optimal probability of process matrices
discrimination as a direct analogue of the Holevo–Helstrom theorem for quantum
states and channels.

######  Theorem 1.

Let $W_{0},W_{1}\in\mathbf{W^{PROC}}$ be two process matrices. For every
choice of discrimination strategy $S=\\{S_{0},S_{1}\\}$, it holds that

$\begin{split}&\frac{1}{2}\operatorname{tr}(S_{0}W_{0})+\frac{1}{2}\operatorname{tr}(S_{1}W_{1})\leq\\\
&\frac{1}{2}+\frac{1}{4}\max\left\\{\|\sqrt{N}(W_{0}-W_{1})\sqrt{N}\|_{1}:N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\right\\},\end{split}$
(20)

where
$\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$
is the set of Choi matrices of non-signaling channels. Moreover, there exists
a discrimination strategy $S$, which saturates the inequality Eq. (20).

###### Proof.

Let us define the sets

$\mathbf{A}\coloneqq\left\\{(S_{0},S_{1}):S_{0}+S_{1}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),\,\,S_{0},\,S_{1}\geq
0\right\\}.$ (21)

and

$\begin{split}\mathbf{B}\coloneqq\\{(\sqrt{N}Q_{0}\sqrt{N},\sqrt{N}Q_{1}\sqrt{N}):&\,N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),\\\
&Q_{0},\,Q_{1}\geq 0,\\\ &Q_{0}+Q_{1}={\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}\\}.\end{split}$
(22)

We prove the equality between sets $\mathbf{A}$ and $\mathbf{B}$. To show
$\mathbf{B}\subseteq\mathbf{A}$, it is suffices to observe that
$\sqrt{N}Q_{0}\sqrt{N}+\sqrt{N}Q_{1}\sqrt{N}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
To prove $\mathbf{A}\subseteq\mathbf{B}$ let us take $N\coloneqq S_{0}+S_{1}$.
It implies that

$\Pi_{\text{im}(N)}=\sqrt{N}^{-1}N\sqrt{N}^{-1}=\sqrt{N}^{-1}S_{0}\sqrt{N}^{-1}+\sqrt{N}^{-1}S_{1}\sqrt{N}^{-1}.$
(23)

Let us fix $\widetilde{Q_{0}}\coloneqq\sqrt{N}^{-1}S_{0}\sqrt{N}^{-1}$ and
$\widetilde{Q_{1}}\coloneqq\sqrt{N}^{-1}S_{1}\sqrt{N}^{-1}$. Then, we have
${\rm 1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}={\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}-\Pi_{\text{im}(N)}+\Pi_{\text{im}(N)}={\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}-\Pi_{\text{im}(N)}+\widetilde{Q_{0}}+\widetilde{Q_{1}}$.
Finally, it is suffices to take

$Q_{0}\coloneqq{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}-\Pi_{\text{im}(N)}+\widetilde{Q_{0}}$
(24)

and $Q_{1}\coloneqq\widetilde{Q_{1}}$. It implies that
$\mathbf{A}=\mathbf{B}$. In conclusion, we obtain

$\begin{split}&\frac{1}{2}\operatorname{tr}(S_{0}W_{0})+\frac{1}{2}\operatorname{tr}(S_{1}W_{1})\\\
&=\frac{1}{2}\operatorname{tr}\left(\sqrt{N}Q_{0}\sqrt{N}W_{0}\right)+\frac{1}{2}\operatorname{tr}\left(\sqrt{N}Q_{1}\sqrt{N}W_{1}\right)\\\
&=\frac{1}{2}\operatorname{tr}\left(Q_{0}\sqrt{N}W_{0}\sqrt{N}\right)+\frac{1}{2}\operatorname{tr}\left(Q_{1}\sqrt{N}W_{1}\sqrt{N}\right)\\\
&\leq\frac{1}{2}+\frac{1}{4}\max\left\\{||\sqrt{N}(W_{0}-W_{1})\sqrt{N}||_{1}:N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\right\\}.\end{split}$
(25)

Moreover, from Holevo-Helstrom theorem [33] there exists a projective binary
measurement $Q=\\{Q_{0},Q_{1}\\}$ such that the last inequality is saturated,
which completes the proof. ∎

######  Corollary 1.

The maximum probability $p_{\text{succ}}(W_{0},W_{1})$ of correct
discrimination between two process matrices $W_{0}$ and $W_{1}$ is given by

$\begin{split}&p_{\text{succ}}(W_{0},W_{1})=\\\
&\frac{1}{2}+\frac{1}{4}\max\left\\{\|\sqrt{N}(W_{0}-W_{1})\sqrt{N}\|_{1}:N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\right\\}.\end{split}$
(26)

Figure 3. A schematic representation of the setup for distinguishing between
process matrices $W_{0}$ and $W_{1}$. The discrimination strategy is
constructed by using the quantum channel
$\Phi_{K}:\mathrm{L}(\mathcal{A}_{I}\otimes\mathcal{B}_{I})\rightarrow\mathrm{L}(\mathcal{A}_{O}\otimes\mathcal{B}_{O}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}\otimes\mathcal{X}_{4})$
and the binary measurement $Q=\\{Q_{0},Q_{1}\\}$ defined in the proof of
Theorem 1.

As a valuable by-product of Theorem 1, we receive a realization of process
matrices discrimination scheme. The schematic representation of this setup is
presented in Fig. 3. To distinguish the process matrices $W_{0}$ and $W_{1}$,
Alice and Bob prepare the strategy $S=\\{S_{0},S_{1}\\}$ such that
$S_{0}+S_{1}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
To implement it, let us introduce complex Euclidean spaces
$\mathcal{X}_{1},\ldots,\mathcal{X}_{4}$ such that
$\dim\left(\bigotimes_{i=1}^{4}\mathcal{X}_{i}\right)=\dim(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
Alice and Bob prepare the quantum channel
$\Phi_{K}:\mathrm{L}(\mathcal{A}_{I}\otimes\mathcal{B}_{I})\rightarrow\mathrm{L}(\mathcal{A}_{O}\otimes\mathcal{B}_{O}\otimes\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}\otimes\mathcal{X}_{4})$
with the Choi matrix $K$ given by

$K=\left({\rm 1l}_{\mathcal{X}_{1,2,3,4}}\otimes\sqrt{N}\right)\left(|{\rm
1l}\rangle\rangle\langle\langle{\rm 1l}|\right)\left({\rm
1l}_{\mathcal{X}_{1,2,3,4}}\otimes\sqrt{N}\right),$ (27)

where $|{\rm
1l}\rangle\rangle\in\mathrm{L}(\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}\otimes\mathcal{X}_{4}\otimes\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$
and
$N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$
maximizes the trace norm $\|\sqrt{N}(W_{0}-W_{1})\sqrt{N}\|_{1}$. It is worth
noting that the quantum channel $\Phi_{K}$ is correctly defined due to the
fact that
$\operatorname{tr}_{\mathcal{X}_{1,2,3,4}}K\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
Afterwards, they perform the binary measurement $Q=\\{Q_{0},Q_{1}\\}$, where
the effect
$Q_{0}\in\mathrm{L}(\mathcal{X}_{1}\otimes\mathcal{X}_{2}\otimes\mathcal{X}_{3}\otimes\mathcal{X}_{4})$
is defined by Eq. (24). Next, they decide which process matrix was used during
the calculation assuming $W_{0}$ if the measurement label is $0$. Otherwise,
they assume $W_{1}$.

##  5\. Discrimination between different classes of process matrices

This section presents some examples of discrimination between different
classes of process matrices. We begin our consideration with the problem of
discrimination between two free process matrices $\mathbf{W^{A||B}}$. Next, we
will consider various cases of process matrices discrimination representing a
quantum comb. First, we calculate exact probability of correct discrimination
between two process matrices come from the same class $\mathbf{W^{A\prec B}}$.
Next, we study the discrimination task assuming that one of the process
matrices is of the form $\mathbf{W^{A\prec B}}$ and the other one is of the
form $\mathbf{W^{B\prec A}}$. Finally, we construct a particular class of
process matrices which can be perfectly distinguished.

###  5.1. Free process matrices

The following consideration confirms an intuition that the task of
discrimination between free process matrices reduces to the problem of
discrimination between quantum states.

From definition of $p_{\text{succ}}(W_{0},W_{1})$ we have

$\begin{split}p_{\text{succ}}\left(W_{0},W_{1}\right)=\frac{1}{2}\max_{S=\\{S_{0},
S_{1}\\}}\left[\operatorname{tr}\left(W_{0}S_{0}\right)+\operatorname{tr}\left(W_{1}S_{1}\right)\right],\end{split}$
(28)

Let $W_{0}$ and $W_{1}$ be two process matrices of the form
$W_{0}=\rho\otimes{\rm 1l}_{\mathcal{A}_{O}\mathcal{B}_{O}}$ and
$W_{1}=\sigma\,\otimes\,{\rm 1l}_{\mathcal{A}_{O}\mathcal{B}_{O}}$, where
$\rho,\,\sigma\in\Omega(\mathcal{A}_{I}\otimes\mathcal{B}_{I})$. Then,
$p_{\text{succ}}$ is exactly equal to

$\begin{split}&\max_{S=\\{S_{0},S_{1}\\}}\left[\frac{1}{2}\operatorname{tr}\left(W_{0}S_{0}\right)+\frac{1}{2}\operatorname{tr}\left(W_{1}S_{1}\right)\right]=\\\
&\max_{S=\\{S_{0},S_{1}\\}}\left[\frac{1}{2}\operatorname{tr}\left((\rho\otimes{\rm
1l})S_{0}\right)+\frac{1}{2}\operatorname{tr}\left((\sigma\otimes{\rm
1l})S_{1}\right)\right]=\\\
&\max_{S=\\{S_{0},S_{1}\\}}\left[\frac{1}{2}\operatorname{tr}\left(\rho\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{0}\right)+\frac{1}{2}\operatorname{tr}\left(\sigma\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{1}\right)\right].\end{split}$
(29)

Let us observe
$\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{0}+\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{1}={\rm
1l}_{\mathcal{A}_{I}\mathcal{B}_{I}}$. So,
$\\{\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{0},\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{1}\\}$
is a binary measurement and therefore, from Holevo-Helstrom theorem for
quantum states, we have

$\max_{S=\\{S_{0},S_{1}\\}}\left[\frac{1}{2}\operatorname{tr}\left(\rho\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{0}\right)+\frac{1}{2}\operatorname{tr}\left(\sigma\operatorname{tr}_{\mathcal{A}_{O}\mathcal{B}_{O}}S_{1}\right)\right]\leq\frac{1}{2}+\frac{1}{4}\|\rho-\sigma\|_{1}.$
(30)

Now, assume that $E=\\{E_{0},E_{1}\\}$ is the Holevo-Helstrom measurement (by
taking $E_{0}$ and $E_{1}$ as positive and negative part of $\rho-\sigma$,
respectively). Hence, we obtain

$\max_{E=\\{E_{0},E_{1}\\}}\left[\frac{1}{2}\operatorname{tr}\left(\rho
E_{0}\right)+\frac{1}{2}\operatorname{tr}\left(\sigma
E_{1}\right)\right]=\frac{1}{2}+\frac{1}{4}\|\rho-\sigma\|_{1}.$ (31)

Observe, it is suffices to take $S_{0}\coloneqq
E_{0}\otimes|0\rangle\\!\langle 0|\otimes|0\rangle\\!\langle 0|$ and
$S_{1}\coloneqq E_{1}\otimes|0\rangle\\!\langle 0|\otimes|0\rangle\\!\langle
0|$. Note that $S_{0}+S_{1}={\rm
1l}_{\mathcal{A}_{I}\mathcal{B}_{I}}\otimes|0\rangle\\!\langle
0|\otimes|0\rangle\\!\langle 0|$ is non-signaling channel. Therefore, we have

$p_{\text{succ}}\left(W_{0},W_{1}\right)=\frac{1}{2}+\frac{1}{4}\|\rho-\sigma\|_{1}.$
(32)

which completes the consideration.

Due to the above consideration, we obtain the following corollary.

######  Corollary 2.

Let $\rho,\sigma\in\Omega(\mathcal{A}_{I}\otimes\mathcal{B}_{I})$ be quantum
states and let $W_{0},W_{1}\in\mathbf{W^{A||B}}$ be two free process matrices
of the form $W_{0}=\rho\otimes{\rm 1l}$ and $W_{1}=\sigma\otimes{\rm 1l}$.
Then,

$p_{\text{succ}}\left(W_{0},W_{1}\right)=\frac{1}{2}+\frac{1}{4}\|\rho-\sigma\|_{1}.$
(33)

###  5.2. Process matrices representing quantum combs

Here, we will compare the probability of correct discrimination between two
process matrices being quantum combs of the form $W_{0}^{A\prec
B},W_{1}^{A\prec B}\in\mathbf{W^{A\prec B}}$ by using non-signalling strategy
$S=\\{S_{0},S_{1}\\}$ described by Eq. (54) or an adaptive strategy.

Before that, we will discuss the issue of adaptive strategy. The most general
strategy of quantum operations discrimination is known as an adaptive strategy
[34, 19]. An adaptive strategy is realized by a quantum tester [1]. A
schematic representation of this setup is presented in Fig. 4.

Let us consider a quantum tester $\\{L_{0},L_{1}\\}$. The probability of
correct discrimination between $W_{0}^{A\prec B}$ and $W_{1}^{A\prec B}$ by
using an adaptive strategy is defined by equation

$p_{adapt}\left(W_{0}^{A\prec B},W_{1}^{A\prec
B}\right)\coloneqq\frac{1}{2}\max_{\\{L_{0},L_{1}\\}}L_{0}*W_{0}+L_{1}*W_{1}.$
(34)

Figure 4. A schematic representation of an adaptive strategy discriminating
two process matrices $W_{0},W_{1}\in W^{A\prec B}$ by using a quantum tester
$\\{L_{0},L_{1}\\}$.

It turns out that we do not need adaptation in order to obtain the optimal
probability os distinction. This is stated formally in the following theorem.

######  Theorem 2.

Let $W_{0}^{A\prec B},W_{1}^{A\prec B}\in\mathbf{W^{A\prec B}}$ be two process
matrices representing quantum combs $A\prec B$. Then,

$p_{succ}\left(W_{0}^{A\prec B},W_{1}^{A\prec
B}\right)=p_{adapt}\left(W_{0}^{A\prec B},W_{1}^{A\prec B}\right).$ (35)

###### Proof.

For simplicity, we will omit superscripts ($A\prec B$ and $B\prec A$). The
inequality $p_{succ}\left(W_{0},W_{1}\right)\leq
p_{adapt}\left(W_{0},W_{1}\right)$ is trivial by observing that we calculate
maximum value over a larger set.

To show $p_{succ}\left(W_{0},W_{1}\right)\geq
p_{adapt}\left(W_{0},W_{1}\right)$, let us consider the quantum tester
$\\{L_{0},L_{1}\\}$ which maximizes Eq. (34), that means

$p_{adapt}\left(W_{0},W_{1}\right)=\frac{1}{2}\left(L_{0}*W_{0}+L_{1}*W_{1}\right).$
(36)

Hence, from definition of $W^{A\prec B}$ we have

$W^{A\prec B}=W^{\prime}\otimes{\rm 1l}_{\mathcal{B}_{O}},$ (37)

and then we obtain

$\frac{1}{2}\left(L_{0}*W_{0}+L_{1}*W_{1}\right)=\frac{1}{2}\operatorname{tr}\left(W^{\prime}_{0}\operatorname{tr}_{\mathcal{B}_{O}}L_{0}+W^{\prime}_{1}\operatorname{tr}_{\mathcal{B}_{O}}L_{1}\right).$
(38)

Observe that $\operatorname{tr}_{\mathcal{B}_{O}}(L_{0}+L_{1})={\rm
1l}_{\mathcal{B}_{I}}\otimes J$, where $J$ is a Choi matrix of a channel
$\Phi_{J}:\mathrm{L}(\mathcal{A}_{I})\rightarrow\mathrm{L}(\mathcal{A}_{O})$.
Let us define a strategy $S=\\{S_{0},S_{1}\\}$ such that

$\begin{split}&S_{0}=\operatorname{tr}_{\mathcal{B}_{O}}L_{0}\otimes\frac{{\rm
1l}_{\mathcal{B}_{O}}}{\dim(\mathcal{B}_{O})},\\\
&S_{1}=\operatorname{tr}_{\mathcal{B}_{O}}L_{1}\otimes\frac{{\rm
1l}_{\mathcal{B}_{O}}}{\dim(\mathcal{B}_{O})},\\\ &S=S_{0}+S_{1}.\end{split}$
(39)

It easy to observe that
$S\in\mathbf{NS}(\mathcal{A}_{O}\otimes\mathcal{A}_{I}\otimes\mathcal{B}_{O}\otimes\mathcal{B}_{I})$.
Then, we have

$\frac{1}{2}\left(S_{0}*W_{0}+S_{1}*W_{1}\right)=\frac{1}{2}\left(\operatorname{tr}\left(W^{\prime}_{0}\operatorname{tr}_{\mathcal{B}_{O}}L_{0}+W^{\prime}_{1}\operatorname{tr}_{\mathcal{B}_{O}}L_{1}\right)\right).$
(40)

It implies that

$p_{succ}\left(W_{0},W_{1}\right)\geq p_{adapt}\left(W_{0},W_{1}\right),$ (41)

which completes the proof. ∎

###  5.3. Process matrices of the form $W^{A\prec B}$ and $W^{B\prec A}$

Now, we present some results for discrimination task assuming the one of the
process matrices if of the form $\mathbf{W^{A\prec B}}$ and the other one is
of the form $\mathbf{W^{B\prec A}}$. We will construct a particular class of
such process matrices for which the perfect discrimination is possible.

Let us define a process matrix of the form

$W^{A\prec B}=\rho\otimes|U\rangle\rangle\langle\langle U|\otimes{\rm 1l},$
(42)

where $\rho\in\Omega(\mathcal{A}_{I}),|U\rangle\rangle\langle\langle U|$ is
the Choi matrix of a unitary channel
$\mathrm{Ad}_{U^{\top}}:\mathrm{L}(\mathcal{A}_{O})\rightarrow\mathrm{L}(\mathcal{B}_{I})$
of the form $\mathrm{Ad}_{U^{\top}}(X)=U^{\top}X\,\bar{U}$ and ${\rm
1l}\in\mathrm{L}(\mathcal{B}_{O})$. A schematic representation of this process
matrix we can see in Fig. 5.

Figure 5. A schematic representation of process matrix $W^{A\prec B}$ given by
Eq. (42).

######  Proposition 1.

Let $W^{A\prec B}$ be a process matrix given by Eq. (42). Let us define a
process matrix $W^{B\prec A}$ of the form

$W^{B\prec A}=P_{\pi}W^{A\prec B}P_{\pi},$ (43)

where $P_{\pi}$ is the swap operator replacing the systems
$\mathcal{A}_{I}\rightarrow\mathcal{B}_{I}$ and
$\mathcal{A}_{O}\rightarrow\mathcal{B}_{O}$. Then, the process matrix
$W^{A\prec B}$ is perfectly distinguishable from $W^{B\prec A}$.

###### Proof.

Let us consider the process matrix given by Eq. (42) described by Fig. 5.
W.l.o.g. let $d$ be a dimension of each of the systems. Let
$\rho=\sum_{i=0}^{d-1}\lambda_{i}|x_{i}\rangle\\!\langle x_{i}|$, where
$\lambda_{i}\geq 0$ such that $\sum_{i}\lambda_{i}=1$. Based on the spectral
decomposition of $\rho$ we create the unitary matrix $V$ by taking $i$-th
eigenvector of $\rho$, and the measurement $\Delta_{V}$ (in basis of $\rho$)
given by

$\Delta_{V}(X)=\sum_{i=0}^{d-1}\langle x_{i}|X|x_{i}\rangle|i\rangle\\!\langle
i|\otimes|x_{i}\rangle\\!\langle x_{i}|.$ (44)

Let us also define the permutation matrix
$P_{\sigma}=\sum_{i=0}^{d-1}|x_{{i+1}\text{ mod d}}\rangle\\!\langle x_{i}|$
corresponding to the permutation $\sigma=(0,1,\ldots,d-1)$.

Alice and Bob prepare theirs discrimination strategy. Alice performs the local
channel (see Fig. 6) given by

$\Phi_{A}(\rho)=\left(\left(\mathcal{I}\otimes\mathrm{Ad}_{\bar{U}}\right)\circ\Delta_{V}\right)(\rho),$
(45)

Meanwhile, Bob performs his local channel (see Fig. 7) given by

$\Phi_{B}(\rho)=\left(\left(\mathcal{I}\otimes\mathrm{Ad}_{\bar{U}}\right)\circ\Delta_{V}\circ\mathrm{Ad}_{P_{\sigma}}\right)(\rho).$
(46)

Figure 6. A schematic representation of Alice’s discrimination strategy
described by Eq. (45). Figure 7. A schematic representation of Bobs’
discrimination strategy described by Eq. (46).

Let us consider the case $A\prec B$. The output after Alice’s action is
described by

$\Phi_{A}(\rho)=\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes\bar{U}|x_{i}\rangle\\!\langle x_{i}|U^{\top}.$ (47)

Next, we apply the quantum channel $\mathrm{Ad}_{U^{\top}}$ (see Fig.5), and
hence we have

$(\mathcal{I}\otimes\mathrm{Ad}_{U^{\top}})\circ\Phi_{A}(\rho)=\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes|x_{i}\rangle\\!\langle x_{i}|.$ (48)

In the next step, Bob applies his channel as follows

$(\mathcal{I}\otimes\Phi_{B})\circ(\mathcal{I}\otimes\mathrm{Ad}_{U^{\top}})\circ\Phi_{A}(\rho)=\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes|i+1\rangle\\!\langle i+1|\otimes|x_{i+1}\rangle\\!\langle x_{i+1}|.$
(49)

Finally, we apply partial trace operation on the subspace $\mathcal{B}_{O}$
(see Fig.5), that means

$\operatorname{tr}_{\mathcal{B}_{O}}\left(\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes|i+1\rangle\\!\langle i+1|\otimes|x_{i+1}\rangle\\!\langle
x_{i+1}|\right)=\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes|i+1\rangle\\!\langle i+1|.$ (50)

So, the quantum state obtained after the discrimination scenario in the case
$A\prec B$ is given by

$\sigma^{A\prec B}=\sum_{i}\lambda_{i}|i\rangle\\!\langle
i|\otimes|i+1\rangle\\!\langle i+1|.$ (51)

It implies that if Alice measures her system, she obtains the label $i$ with
probability $\lambda_{i}$ whereas Bob obtains the label $(i+1)\mod d$ with the
same probability. On the other hand, considering the case $B\prec A$, then the
state obtained after the discrimination scenario is given by

$\sigma^{B\prec A}=\sum_{i}\lambda_{i}|i+1\rangle\\!\langle
i+1|\otimes|i+1\rangle\\!\langle i+1|.$ (52)

So, Bob and Alice obtain the same label $(i+1)\mod d$ with probability
$\lambda_{i}$. Then, the quantum channel $\Phi_{K}$ (realizing the
discrimination strategy $S$) is created as a tensor product of Alice’s and
Bob’s local channels, that means $\Phi_{K}=\Phi_{A}\otimes\Phi_{B}$. Due to
that they perform the binary measurement $Q=\\{Q_{0},Q_{1}\\}$, where the
effect $Q_{1}$ is given by $Q_{1}=\sum_{i=0}^{d-1}|i\rangle\\!\langle
i|\otimes|i\rangle\\!\langle i|$. Hence, we have

$p_{\text{succ}}(W^{A\prec B},W^{B\prec
A})=\frac{1}{2}\operatorname{tr}\left(\sigma^{A\prec
B}Q_{0}\right)+\frac{1}{2}\operatorname{tr}\left(\sigma^{B\prec
A}Q_{1}\right)=1.$ (53)

In summary, the process matrices $W^{A\prec B}$ and $W^{B\prec A}$ are
perfectly distinguishable by Alice and Bob which completes the proof.

∎

##  6\. SDP program for calculating the optimal probability of process
matrices discrimination

In the standard approach, we would need to compute the probability of correct
discrimination between two process matrices $W_{0}$ and $W_{1}$. For this
purpose, we use the semidefinite programming (SDP). This section presents the
SDP program for calculating the optimal probability of discrimination between
$W_{0}$ and $W_{1}$.

Recall that the maximum value of such a probability can be noticed by

$p_{\text{succ}}(W_{0},W_{1})=\frac{1}{2}\max_{S=\\{S_{0},S_{1}\\}}\left[\operatorname{tr}(W_{0}S_{0})+\operatorname{tr}(W_{1}S_{1})\right],$
(54)

with requirement that the optimal strategy $S=\\{S_{0},S_{1}\\}$ is a quantum
instrument such that
$S_{0}+S_{1}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
Hence, we arrive at the primal and dual problems presented in the Program 1.
To optimize this problem we used the Julia programming language along with
quantum package QuantumInformation.jl[35] and SDP optimization via SCS solver
[36, 37] with absolute convergence tolerance $10^{-5}$. The code is available
on GitHub [38].

It may happen that the values of primal and dual programs are equal. This
situation is called strong duality. Slater’s theorem provides the set of
conditions which guarantee strong duality [33]. It can be shown that Program 1
fulfills conditions of Slater’s theorem (it is suffices to take
$Y_{0},Y_{1}=0$ and
$\alpha>\frac{1}{2}\max\\{\Lambda^{\text{max}}(W_{0}),\Lambda^{\text{max}}(W_{1})\\}$,
where $\Lambda^{\text{max}}(X)$ is the maximum eigenvalue of $X$). Therefore,
we can consider the primal and the dual problem equivalently.

SDP program for calculating the optimal probability of discrimination between
$W_{0}$ and $W_{1}$

Primal problem

$\begin{split}\text{maximize:}\quad&\frac{1}{2}\operatorname{Tr}(W_{0}S_{0})+\frac{1}{2}\operatorname{Tr}(W_{1}S_{1})\\\\[5.69054pt]
\text{subject
to:}\quad&\operatorname{tr}_{\mathcal{A}_{O}}(S_{0}+S_{1})=\frac{{\rm
1l}_{\mathcal{A}_{I}}}{\dim(\mathcal{A}_{I})}\otimes\operatorname{tr}_{\mathcal{A}_{O}\mathcal{A}_{I}}(S_{0}+S_{1}),\\\
&\operatorname{tr}_{\mathcal{B}_{O}}(S_{0}+S_{1})=\frac{{\rm
1l}_{\mathcal{B}_{I}}}{\dim(\mathcal{B}_{I})}\otimes\operatorname{tr}_{\mathcal{B}_{O}\mathcal{B}_{I}}(S_{0}+S_{1}),\\\
&\operatorname{tr}(S_{0}+S_{1})=\dim(\mathcal{A}_{I})\dim(\mathcal{B}_{I}),\\\
&S_{0}\in\mathrm{Pos}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),\\\
&S_{1}\in\mathrm{Pos}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}).\end{split}$

Dual problem

$\begin{split}\text{minimize:}\quad&\alpha\cdot\dim(\mathcal{A}_{I})\dim(\mathcal{B}_{I})\\\\[5.69054pt]
\text{subject to:}\quad&{\rm 1l}_{\mathcal{A}_{O}}\otimes Y_{0}-\frac{{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}}}{\dim(\mathcal{A}_{I})}\otimes\operatorname{tr}_{\mathcal{A}_{I}}(Y_{0})+{\rm
1l}_{\mathcal{B}_{O}}\otimes Y_{1}+\\\ &-\frac{{\rm
1l}_{\mathcal{B}_{I}\mathcal{B}_{O}}}{\dim(\mathcal{B}_{I})}\otimes\operatorname{tr}_{\mathcal{B}_{I}}Y_{1}+\alpha\cdot{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}\geq\frac{1}{2}W_{0},\\\
&{\rm 1l}_{\mathcal{A}_{O}}\otimes Y_{0}-\frac{{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}}}{\dim(\mathcal{A}_{I})}\otimes\operatorname{tr}_{\mathcal{A}_{I}}(Y_{0})+{\rm
1l}_{\mathcal{B}_{O}}\otimes Y_{1}+\\\ &-\frac{{\rm
1l}_{\mathcal{B}_{I}\mathcal{B}_{O}}}{\dim(\mathcal{B}_{I})}\otimes\operatorname{tr}_{\mathcal{B}_{I}}Y_{1}+\alpha\cdot{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}\geq\frac{1}{2}W_{1},\\\
&Y_{0}\in\text{
Herm}(\mathcal{A}_{I}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),\\\
&Y_{1}\in\text{
Herm}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}),\\\
&\alpha\in\mathbb{R}.\end{split}$

Table 1. Semidefinite program for maximizing the probability of correct
discrimination between two process matrices $W_{0}$ and $W_{1}$.

##  7\. Distance between process matrices

In this section we present the semidefinite programs for calculating the
distance in trace norm between a given process matrix $W\in\mathbf{W^{PROC}}$
and different subsets of process matrices, such that $\mathbf{W^{A||B}}$ ,
$\mathbf{W^{A\prec B}}$, $\mathbf{W^{B\prec A}}$ or $\mathbf{W^{SEP}}$.

For example, let us consider the case $\mathbf{W^{A||B}}$. Theoretically, the
distance between a process matrix $W$ and the set of free process matrices
$\mathbf{W^{A||B}}$ can be expressed by

$\begin{split}&\text{dist}\left(W,\mathbf{W^{A||B}}\right)=\\\
&\min_{\widetilde{W}\in\mathbf{W^{A||B}}}\max\left\\{\|\sqrt{N}(W-\widetilde{W})\sqrt{N}\|_{1}:N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\right\\}.\end{split}$
(55)

Analogously, for the sets $\mathbf{W^{A\prec B}},\mathbf{W^{B\prec A}}$ and
$\mathbf{W^{SEP}}$ with the minimization condition
$\min_{\widetilde{W}\in\mathbf{W^{A\prec B}}}$,
$\min_{\widetilde{W}\in\mathbf{W^{B\prec A}}}$,
$\min_{\widetilde{W}\in\mathbf{W^{SEP}}}$, respectively. Due to the results
obtained from the previous section (see Program 1) and Slater theorem we are
able to note the Eq. (55) to SDP problem presented in the Program 2. We use
the SDP optimization via SCS solver [36, 37] with absolute convergence
tolerance $10^{-8}$ and relative convergence tolerance $10^{-8}$. The
implementations of SDPs in the Julia language are available on GitHub [38].

SDP calculating the distance between a process matrix $W$ and the set
$\Upsilon$.

$\begin{split}\text{minimize:}\quad&4\dim(\mathcal{A}_{I})\dim(\mathcal{B}_{I})\alpha-2\\\\[5.69054pt]
\text{subject to:}\quad&{\rm 1l}_{\mathcal{A}_{O}}\otimes Y_{0}-\frac{{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}}}{\dim(\mathcal{A}_{I})}\otimes\operatorname{tr}_{\mathcal{A}_{I}}(Y_{0})+{\rm
1l}_{\mathcal{B}_{O}}\otimes Y_{1}+\\\ &-\frac{{\rm
1l}_{\mathcal{B}_{I}\mathcal{B}_{O}}}{\dim(\mathcal{B}_{I})}\otimes\operatorname{tr}_{\mathcal{B}_{I}}Y_{1}+\alpha\cdot{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}\geq\frac{1}{2}W,\\\
&{\rm 1l}_{\mathcal{A}_{O}}\otimes Y_{0}-\frac{{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}}}{\dim(\mathcal{A}_{I})}\otimes\operatorname{tr}_{\mathcal{A}_{I}}(Y_{0})+{\rm
1l}_{\mathcal{B}_{O}}\otimes Y_{1}+\\\ &-\frac{{\rm
1l}_{\mathcal{B}_{I}\mathcal{B}_{O}}}{\dim(\mathcal{B}_{I})}\otimes\operatorname{tr}_{\mathcal{B}_{I}}Y_{1}+\alpha\cdot{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}\geq\frac{1}{2}W^{*},\\\
&W^{*}\in\Upsilon,\\\ &Y_{0}\in\text{
Herm}(\mathcal{A}_{I}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),\\\
&Y_{1}\in\text{
Herm}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}),\\\
&\alpha\in\mathbb{R}.\end{split}$

Table 2. Semidefinite program for computation the distance between a process
matrix $W$ and $\Upsilon$, which can be one of the set
$\mathbf{W^{A||B}},\mathbf{W^{A\prec B}},\mathbf{W^{B\prec A}}$ or
$\mathbf{W^{SEP}}$. Depending on the selected set we include additional
constrains to SDP described by Eq. (16) for $\mathbf{W^{A||B}}$, Eq. (17) for
$\mathbf{W^{A\prec B}}$ and $\mathbf{W^{B\prec A}}$ or Eq. (18) for
$\mathbf{W^{SEP}}$.

###  7.1. Example

Let
$\mathcal{A}_{I}=\mathcal{A}_{O}=\mathcal{B}_{I}=\mathcal{B}_{O}=\mathbb{C}^{2}$.
Let us consider a causally non-separable process matrix comes from [5] of the
form

$W^{\text{CNS}}=\frac{1}{4}\left[{\rm
1l}_{\mathcal{A}_{I}\mathcal{A}_{O}\mathcal{B}_{I}\mathcal{B}_{O}}+\frac{1}{\sqrt{2}}\left(\sigma_{z}^{\mathcal{A}_{O}}\sigma_{z}^{\mathcal{B}_{I}}\otimes{\rm
1l}_{\mathcal{A}_{I}\mathcal{B}_{O}}+\sigma_{z}^{\mathcal{A}_{I}}\sigma_{x}^{\mathcal{B}_{I}}\sigma_{z}^{\mathcal{B}_{O}}\otimes{\rm
1l}_{\mathcal{A}_{O}}\right)\right],$ (56)

where $\sigma_{x}^{\mathcal{X}},\sigma_{z}^{\mathcal{X}}$ are Pauli matrices
on space $\mathrm{L}(\mathcal{X})$. We have calculated the distance in trace
norm between $W^{\text{CNS}}$ and different subset of process matrices.
Finally, we obtain

$\text{dist}\left(W^{\text{CNS}},\mathbf{W^{A||B}}\right)\approx
1.00000001\approx 1,$ (57) $\text{dist}\left(W^{\text{CNS}},\mathbf{W^{A\prec
B}}\right)\approx 0.7071068\approx\frac{\sqrt{2}}{2},$ (58)
$\text{dist}\left(W^{\text{CNS}},\mathbf{W^{B\prec A}}\right)\approx
0.7071068\approx\frac{\sqrt{2}}{2},$ (59)
$\text{dist}\left(W^{\text{CNS}},\mathbf{W^{SEP}}\right)\approx
0.2928932\approx 1-\frac{\sqrt{2}}{2}.$ (60)

The numerical computations give us some intuition about the geometry of the
set of process matrices. Those results are presented in Fig. 8. Moreover, by
using $W^{\text{CNS}}$ given by Eq. (56) it can be shown that the set of all
causally non-separable process matrices is not convex. To show this fact, it
suffices to observe that for every
$\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l}\in\\{\sigma_{x},\sigma_{y},\sigma_{z},{\rm
1l}\\}$ the following equation holds

$\left(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\otimes\sigma_{l}\right)W^{\text{CNS}}(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\otimes\sigma_{l})^{\dagger}\in\mathbf{W^{CNS}}.$
(61)

Simultaneously, the average of the process matrices of the form Eq. (61)
distributed uniformly states $\frac{1}{4}{\rm 1l}_{\mathbb{C}^{16}}$, however
$\frac{1}{4}{\rm 1l}_{\mathbb{C}^{16}}\not\in\mathbf{W^{CNS}}$. It implies
that the set $\mathbf{W^{CNS}}$ is not convex.

Figure 8. A schematic representation of the distances between $W^{\text{CNS}}$
defined in Eq. (56) and the sets $\mathbf{W^{A||B}}$, $\mathbf{W^{A\prec B}}$,
$\mathbf{W^{B\prec A}}$ and $\mathbf{W^{SEP}}$.

##  8\. Convex cone structure theory

From geometrical point of view, we present an alternative way to derive of Eq.
(26). It turns out that the task of process matrices discrimination is
strictly connected with the convex cone structure theory. To keep this work
self-consistent, the details of convex cone structure theory are presented in
Appendix A.

Let $\mathcal{V}$ be a finite dimensional real vector space with a proper cone
$\mathcal{C}\subset\mathcal{V}$. A base $B$ of the proper cone $\mathcal{C}$
is a compact convex subset $B\subset\mathcal{C}$ such that each nonzero
element $c\in\mathcal{C}$ has a unique representation in the form
$c=\alpha\cdot b$, where $\alpha>0$ and $b\in B$. The corresponding base norm
in $\mathcal{V}$ is defined by

$||x||_{B}=\\{\alpha+\beta,x=\alpha b_{1}-\beta b_{2},\alpha,\beta\geq
0,b_{1},b_{2}\in B\\}.$ (62)

From [34, Corollary 2] the author showed that the base norm can be written as

$||x||_{B}=\sup_{\widetilde{b}\in\widetilde{B}}||\widetilde{b}^{1/2}x\widetilde{b}^{1/2}||_{1},$
(63)

where

$\widetilde{B}\coloneqq\\{Y\in\mathcal{C}:\operatorname{tr}\left(XY\right)=1,\forall
X\in B\\}.$ (64)

###  8.1. Convex cone structure of process matrices set

Let $\mathcal{V}$ be a Hilbert space given by

$\mathcal{V}=\mathrm{Herm}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}),$
(65)

with proper cone

$\mathcal{C}=\\{W\in\mathcal{V}:W\in\mathrm{Pos}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\\}.$
(66)

Consider the linear subspace $\mathcal{S}\subset\mathcal{V}$ given by

$\begin{split}\mathcal{S}=\\{W\in\mathcal{V}:&\prescript{}{B_{I}B_{O}}{W}=\prescript{}{A_{O}B_{I}B_{O}}{W},\\\
&\prescript{}{A_{I}A_{O}}{W}=\prescript{}{A_{O}A_{I}B_{O}}{W},\\\
&\prescript{}{A_{O}B_{O}}{W}=\prescript{}{B_{O}}{W}+\prescript{}{A_{O}}{W}-W\\}\end{split}$
(67)

together with its proper cone $\mathcal{C}_{\mathcal{S}}$. Observe that if we
fix trace of $W\in\mathcal{C}_{\mathcal{S}}$ such that
$\operatorname{tr}(W)=\dim(\mathcal{A}_{O})\cdot\dim(\mathcal{B}_{O})$, we
achieve the set of all process matrices $\mathbf{W^{PROC}}$. And then,
$\mathbf{W^{PROC}}$ is a base of $\mathcal{C}_{\mathcal{S}}$.

######  Proposition 2.

Let $\mathbf{W^{PROC}}$ be the set of process matrices. Then, the set
$\widetilde{\mathbf{W^{PROC}}}$ is determined by

$\widetilde{\mathbf{W^{PROC}}}=\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}).$
(68)

###### Proof.

We want to prove that

$\operatorname{tr}\left(XW\right)=1\,\,\text{for all
}W\in\mathbf{W^{PROC}}\iff
X\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O}).$
(69)

Let us first take
$X\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$.
Then, from [22, Lemma 1], we note

$X=\sum_{i}\lambda_{i}A_{i}\otimes B_{i},$ (70)

where $A_{i}\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O})$,
$B_{i}\in\mathbf{NS}(\mathcal{B}_{I}\otimes\mathcal{B}_{O})$ and
$\lambda_{i}\in\mathbb{R}$ such that $\sum_{i}\lambda_{i}=1$. From definition
of process matrix and linearity we obtain

$\operatorname{tr}\left(XW\right)=\operatorname{tr}\left(\sum_{i}\lambda_{i}(A_{i}\otimes
B_{i})W\right)=\sum_{i}\lambda_{i}\operatorname{tr}\left((A_{i}\otimes
B_{i})W\right)=\sum_{i}\lambda_{i}=1.$ (71)

To prove opposite implication, let us take $W={\rm
1l}_{\mathcal{A}_{O}}\otimes J$, where $J$ is the Choi matrix of quantum
channel
$\Phi_{J}:\mathrm{L}(\mathcal{B}_{I}\otimes\mathcal{B}_{O})\rightarrow\mathrm{L}(\mathcal{A}_{I})$.
Then, we have

$1=\operatorname{tr}\left(WX\right)=\operatorname{tr}\left(({\rm
1l}_{\mathcal{A}_{O}}\otimes
J)X\right)=\operatorname{tr}\left(J\operatorname{tr}_{\mathcal{A}_{O}}X\right).$
(72)

From [39] we have

$\operatorname{tr}_{\mathcal{A}_{O}}X={\rm 1l}_{\mathcal{A}_{I}}\otimes P,$
(73)

where $P\in\mathrm{Pos}(\mathcal{B}_{I}\otimes\mathcal{B}_{O})$. Similarly, if
we take $W\coloneqq{\rm 1l}_{\mathcal{B}_{O}}\otimes K$, where $K$ is the Choi
matrix of a quantum channel
$\Phi_{K}:\mathrm{L}(\mathcal{A}_{I}\otimes\mathcal{A}_{O})\rightarrow\mathrm{L}(\mathcal{B}_{I})$,
we obtain

$\operatorname{tr}_{\mathcal{B}_{O}}X={\rm 1l}_{\mathcal{B}_{I}}\otimes P,$
(74)

where $P\in\mathrm{Pos}(\mathcal{A}_{I}\otimes\mathcal{A}_{O})$. It implies
that
$X\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})$,
which completes the proof. ∎

Due to Proposition 2, we immediately obtain the following corollary.

######  Corollary 3.

The base norm $||\cdot||_{\mathbf{W^{PROC}}}$ between two process matrices
$W_{1},W_{2}\in\mathbf{W^{PROC}}$ can be expressed as

$||W_{1}-W_{2}||_{\mathbf{W^{PROC}}}=\max\\{||\sqrt{N}(W_{1}-W_{2})\sqrt{N}||_{1}:N\in\mathbf{NS}(\mathcal{A}_{I}\otimes\mathcal{A}_{O}\otimes\mathcal{B}_{I}\otimes\mathcal{B}_{O})\\}.$
(75)

##  9\. Conclusion and discussion

In this work, we studied the problem of single shot discrimination between
process matrices. Our aim was to provide an exact expression for the optimal
probability of correct distinction and quantify it in terms of the trace norm.
This value was maximized over all Choi operators of non-signaling channels and
and poses direct analogues to the Holevo-Helstrom theorem for quantum
channels. In addition, we have presented an alternative way to achieve this
expression by using the convex cone structure theory. As a valuable by-
product, we have also found the optimal realization of the discrimination task
for process matrices that use such non-signalling channels. Additionally, we
expressed the discrimination task as semidefinite programming (SDP). Due to
that, we have created SDP calculating the distance between process matrices
and we expressed it in terms of the trace norm. Moreover, we found an
analytical result for discrimination of free process matrices. It turns out
that the task of discrimination between free process matrices can be reduced
to the task of discrimination between quantum states. Next, we consider the
problem of discrimination for process matrices corresponding to quantum combs.
We have studied which strategy, adaptive or non-signalling, should be used
during the discrimination task. We proved that no matter which strategy you
choose, the optimal probability of distinguishing two process matrices being a
quantum comb is the same. So, it turned out that we do not need to use some
unknown additional processing in this case. Finally, we discovered a
particular class of process matrices having opposite causal order, which can
be distinguished perfectly. This work paves the way toward a complete
description of necessary and sufficient criterion for perfect discrimination
between process matrices. Moreover, it poses a starting point to fully
describe the geometry of the set of process matrices, particularly causally
non-separable process matrices.

## Acknowledgements

This work was supported by the project ,,Near-term quantum computers
Challenges, optimal implementations and applications” under Grant Number
POIR.04.04.00-00-17C1/18-00, which is carried out within the Team-Net
programme of the Foundation for Polish Science co-financed by the European
Union under the European Regional Development Fund and SONATA BIS grant number
2016/22/E/ST6/00062.

PL is a holder of European Union scholarship through the European Social Fund,
grant InterPOWER (POWR.03.05.00-00-Z305).

## References

  * [1] A. Bisio, G. Chiribella, G. D’Ariano, and P. Perinotti, “Quantum networks: general theory and applications,” Acta Physica Slovaca, vol. 61, no. 3, pp. 273–390, 2011.
  * [2] K. Gödel, “An example of a new type of cosmological solutions of einstein’s field equations of gravitation,” Reviews of Modern Physics, vol. 21, no. 3, p. 447, 1949.
  * [3] D. Deutsch and M. Lockwood, “The quantum physics of time travel,” Scientific American, vol. 270, no. 3, pp. 68–74, 1994.
  * [4] N. Gisin, “Weinberg’s non-linear quantum mechanics and supraluminal communications,” Physics Letters A, vol. 143, no. 1-2, pp. 1–2, 1990.
  * [5] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” Nature Communications, vol. 3, no. 1, pp. 1–8, 2012.
  * [6] Č. Brukner, “Quantum causality,” Nature Physics, vol. 10, no. 4, pp. 259–263, 2014.
  * [7] J. Bavaresco, M. Murao, and M. T. Quintino, “Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination,” Physical review letters, vol. 127, no. 20, p. 200504, 2021\.
  * [8] M. T. Quintino and D. Ebler, “Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality,” Quantum, vol. 6, p. 679, 2022.
  * [9] J. Bavaresco, M. Murao, and M. T. Quintino, “Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies,” Journal of Mathematical Physics, vol. 63, no. 4, p. 042203, 2022.
  * [10] R. Duan, Y. Feng, and M. Ying, “Entanglement is not necessary for perfect discrimination between unitary operations,” Physical review letters, vol. 98, no. 10, p. 100503, 2007.
  * [11] G. M. D’Ariano, P. L. Presti, and M. G. Paris, “Using entanglement improves the precision of quantum measurements,” Physical Review Letters, vol. 87, no. 27, p. 270404, 2001.
  * [12] T.-Q. Cao, F. Gao, Z.-C. Zhang, Y.-H. Yang, and Q.-Y. Wen, “Perfect discrimination of projective measurements with the rank of all projectors being one,” Quantum Information Processing, vol. 14, no. 7, pp. 2645–2656, 2015.
  * [13] Z. Puchała, Ł. Pawela, A. Krawiec, and R. Kukulski, “Strategies for optimal single-shot discrimination of quantum measurements,” Physical Review A, vol. 98, no. 4, p. 042103, 2018.
  * [14] Z. Puchała, Ł. Pawela, A. Krawiec, R. Kukulski, and M. Oszmaniec, “Multiple-shot and unambiguous discrimination of von neumann measurements,” Quantum, vol. 5, p. 425, 2021.
  * [15] G. Wang and M. Ying, “Unambiguous discrimination among quantum operations,” Physical Review A, vol. 73, no. 4, p. 042301, 2006.
  * [16] A. Krawiec, Ł. Pawela, and Z. Puchała, “Discrimination of povms with rank-one effects,” Quantum Information Processing, vol. 19, no. 12, pp. 1–12, 2020.
  * [17] M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra and its Applications, vol. 10, no. 3, pp. 285–290, 1975.
  * [18] A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Reports on Mathematical Physics, vol. 3, no. 4, pp. 275–278, 1972.
  * [19] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Theoretical framework for quantum networks,” Physical Review A, vol. 80, no. 2, p. 022339, 2009.
  * [20] D. Beckman, D. Gottesman, M. A. Nielsen, and J. Preskill, “Causal and localizable quantum operations,” Physical Review A, vol. 64, no. 5, p. 052309, 2001.
  * [21] M. Piani, M. Horodecki, P. Horodecki, and R. Horodecki, “Properties of quantum nonsignaling boxes,” Physical Review A, vol. 74, no. 1, p. 012305, 2006\.
  * [22] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, “Quantum computations without definite causal structure,” Physical Review A, vol. 88, no. 2, p. 022318, 2013.
  * [23] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information.  Cambridge University Press, 2010.
  * [24] E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Communications in Mathematical Physics, vol. 17, no. 3, pp. 239–260, 1970.
  * [25] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Quantum circuit architecture,” Physical Review Letters, vol. 101, no. 6, p. 060401, 2008\.
  * [26] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, “Witnessing causal nonseparability,” New Journal of Physics, vol. 17, no. 10, p. 102001, 2015.
  * [27] S. Milz, J. Bavaresco, and G. Chiribella, “Resource theory of causal connection,” Quantum, vol. 6, p. 788, 2022.
  * [28] O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes,” New Journal of Physics, vol. 18, no. 9, p. 093020, 2016.
  * [29] G. Gour, “Comparison of quantum channels by superchannels,” IEEE Transactions on Information Theory, vol. 65, no. 9, pp. 5880–5904, 2019.
  * [30] A. Jenčová, “Generalized channels: channels for convex subsets of the state space,” Journal of Mathematical Physics, vol. 53, no. 1, p. 012201, 2012.
  * [31] A. Jenčová, “Extremality conditions for generalized channels,” Journal of Mathematical Physics, vol. 53, no. 12, p. 122203, 2012.
  * [32] C. Helstrom, “Quantum detection and estimation theory, ser,” Mathematics in Science and Engineering. New York: Academic Press, vol. 123, 1976.
  * [33] J. Watrous, The theory of quantum information.  Cambridge university press, 2018.
  * [34] A. Jenčová, “Base norms and discrimination of generalized quantum channels,” Journal of Mathematical Physics, vol. 55, no. 2, p. 022201, 2014\.
  * [35] P. Gawron, D. Kurzyk, and Ł. Pawela, “QuantumInformation.jl—a julia package for numerical computation in quantum information theory,” PLOS ONE, vol. 13, p. e0209358, dec 2018.
  * [36] B. O’Donoghue, E. Chu, N. Parikh, and S. Boyd, “Conic optimization via operator splitting and homogeneous self-dual embedding,” Journal of Optimization Theory and Applications, vol. 169, pp. 1042–1068, June 2016.
  * [37] B. O’Donoghue, E. Chu, N. Parikh, and S. Boyd, “SCS: Splitting conic solver, version 3.2.1.” https://github.com/cvxgrp/scs, Nov. 2021.
  * [38] https://github.com/iitis/strategies_for_single_shot_discrimination_of_process_matrices.  Permanent link to code/repository, Accessed: 2022-10-21.
  * [39] P. Lewandowska, R. Kukulski, and Ł. Pawela, “Optimal representation of quantum channels,” in International Conference on Computational Science, pp. 616–626, Springer, 2020.
  * [40] W. Rudin, “Functional analysis 2nd ed,” International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991.

##  Appendix A Convex cone structures

To keep this work self-consistent we present in this appendix basic
definitions and properties about convex cone structure theory.

Suppose $\mathcal{X}$ is a finite dimensional real vector space and
$\mathcal{C}\subset\mathcal{X}$ is a closed convex cone. We assume that
$\mathcal{C}$ is pointed, that means $\mathcal{C}\cap-\mathcal{C}=\\{0\\}$. A
closed pointed convex cone is in one-to-one correspondence with partial order
in $\mathcal{X}$, by $x\geq y\iff x-y\in\mathcal{C}$ for each
$x,y\in\mathcal{X}$. If we additionally assume that the cone is generating,
that is for each $x\in\mathcal{X}$ there exists $u,w\in\mathcal{C}$ such that
$x=u-w$, then a nonempty set $\mathcal{C}\subseteq\mathcal{X}$ satisfying all
above properties will be called a proper cone in space $\mathcal{X}$. Let
$\mathcal{X}^{*}$ be a dual space with duality
$\langle{\cdot}|{\cdot}\rangle$. Then, we introduce a partial order in
$\mathcal{X}^{*}$ as well with dual cone
$\mathcal{C}^{*}=\\{f\in\mathcal{X}^{*}:\langle{f}|{z}\rangle\geq 0,\forall
z\in C\\}.$ Observe that the cone $\mathcal{C}^{*}$ is also closed and convex.
Moreover, if $\mathcal{C}$ is generating in space $\mathcal{X}$, then
$\mathcal{C}^{*}$ is pointed, so we can introduce the partial order in
$\mathcal{X}^{*}$ given by

$f\geq g\iff f-g\in\mathcal{C}^{*}$ (76)

for all $f,g\in\mathcal{X}^{*}$.

Next, consider a linear space with fixed inner product. If $\mathcal{X}$ is an
inner product space, then the Riesz representation theorem [40] holds that the
inner product determines an isomorphism between $\mathcal{X}$ and
$\mathcal{X}^{*}$. Therefore, the cone $\mathcal{C}$ is equal to
$\mathcal{C}^{*}$.

An interior point $e\in\mathrm{int}(\mathcal{C})$ of a cone $\mathcal{C}$ is
called an order unit if for each $x\in\mathcal{X}$, there exists $\lambda>0$
such that $\lambda e-x\in\mathcal{C}$. Whereas, a base of $\mathcal{C}$ is
defined as compact and convex subset $B\subset\mathcal{C}$ such that for every
$z\in\mathcal{C}\setminus\\{0\\}$, there exists unique $t>0$ and an element
$b\in B$ such that $z=tb.$ It can be shown that the set

$B=\\{z\in\mathcal{C}:\langle{e}|{z}\rangle=1\\}$ (77)

is the base of $\mathcal{C}$ (determined by element $e$) if and only if an
element $e$ is an order unit and
$e\in\mathrm{int}\left(\mathcal{C}^{*}\right)$. Finally, we define the base
norm as

$||x||_{B}=\\{\alpha+\beta,x=\alpha b_{1}-\beta b_{2},\alpha,\beta\geq
0,b_{1},b_{2}\in B\\}.$ (78)

It can be shown [34] that the base norm is expressed as

$||x||_{B}=\sup_{\widetilde{b}\in\widetilde{B}}||\widetilde{b}^{1/2}x\widetilde{b}^{1/2}||_{1},$
(79)

where
$\widetilde{B}=\\{\widetilde{b}\in\mathcal{C}:\operatorname{tr}(b\widetilde{b})=1,\forall
b\in B\\}.$