STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS MATRICES PAULINA LEWANDOWSKA1 LUKASZ PAWELA1 AND ZBIGNIEW PUCHA LA1

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STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS

MATRICES

PAULINA LEWANDOWSKA1, LUKASZ PAWELA1, AND ZBIGNIEW PUCHA LA1

Abstract. The topic of causality has recently gained traction quantum information

research. This work examines the problem of single-shot discrimination between pro-

cess 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 con-

sideration 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 in-

formation 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 (de-

noted by A≺B), Alice cannot signal to Bob (B≺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 intro-

duced a new resource called a process matrix – a generalization of the notion of quantum

1Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul.

Ba ltycka 5, 44-100 Gliwice, Poland

arXiv:2210.14575v1 [quant-ph] 26 Oct 2022

2 STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS MATRICES

state. This new approach has provided a consistent representation of correlations in ca-

sually 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 speak-

ing, 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 [7–9]. 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 oper-

ations. 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 discrimina-

tion 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 [11–13], we need to use entanglement. Considering

multiple-shot discrimination scenarios, researchers have utilized parallel or adaptive ap-

proaches. 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 pro-

gram 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 proba-

bility 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 dis-

crimination 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

STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS MATRICES 3

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 X,Y. By L(X,Y) we denote the collection of all linear mappings

of the form A:X → Y. As a shorthand put L(X):= L(X,X).By Herm(X) we

denote the set of Hermitian operators while the subset of Herm(X) consisting of positive

semideﬁnite operators will be denoted by Pos(X). The set of quantum states, that

is positive semideﬁnite operators ρsuch that tr ρ= 1, will be denoted by Ω(X). An

operator U∈L (X) is unitary if it satisﬁes the equation UU†=U†U= 1lX. The

notation U (X) will be used to denote the set of all unitary operators. We will also

need a linear mapping of the form Φ : L(X)→L(Y) transforming L(X) into L(Y).

The set of all linear mappings is denoted M(X,Y). There exists a bijection between set

M(X,Y) and the set of operators L(Y⊗X) known as the Choi [17] and Jamio lkowski [18]

isomorphism. For a given linear mapping ΦM: L(X)→L(Y) corresponding Choi matrix

M∈L(Y ⊗ X) can be explicitly written as

M:=

dim(X)−1

i,j=0

ΦM(|iihj|)⊗ |iihj|.(1)

We will denote linear mappings by ΦM,ΦN,ΦRetc., whereas the corresponding Choi

matrices as plain symbols: M, N, R etc. Let us consider a composition of mappings

ΦR= ΦN◦ΦMwhere ΦN: L(Z)→L(Y) and ΦM: L(X)→L(Z) with Choi matrices

N∈L(Z ⊗ Y) and M∈L(X ⊗ Z), respectively. Then, the Choi matrix of ΦRis given

by [19]

R= trZ1lY⊗MTZ(N⊗1lX),(2)

where MTZdenotes the partial transposition of Mon the subspace Z. The above result

can be expressed by introducing the notation of the link product of the operators Nand

Mas

N∗M:= trZ1lY⊗MTZ(N⊗1lX).(3)

Finally, we introduce a special subset of all mappings Φ, called quantum channels,

which are completely positive and trace preserving (CPTP). In other words, the ﬁrst

condition reads

(Φ ⊗ IZ)(X)∈Pos(Y ⊗ Z) (4)

for all X∈Pos(X ⊗Z) and IZis an identity channel acts on L(Z) for any Z, while the

second condition reads

tr(Φ(X)) = tr(X) (5)

for all X∈L(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 ΦN: L(XI⊗ YI)→L(XO⊗ YO) is a

4 STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS MATRICES

non-signaling channel if its Choi operator satisﬁes the following conditions

trXO(N) = 1lXI

dim(XI)⊗trXOX1(N),

trYO(N) = 1lYI

dim(YI)⊗trYOY1(N).

(6)

It can be shown [22] that each non-signaling channel is an aﬃne combination of product

channels. More precisely, any non-signaling channel ΦN: L(XI⊗YI)→L(XO⊗YO) can

be written as

ΦN=X

λiΦSi⊗ΦTi,(7)

where ΦSi: L(XI)→L(XO) and ΦTi: L(YI)→L(YO) are quantum channels, λi∈R

such that Piλi= 1. For the rest of this paper, by NS(XI⊗ XO⊗ YI⊗ YO) 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 {ΦMi}iassociated to all mea-

surement outcomes, characterized by the property that PiΦMiis a quantum channel.

We will also consider the concept of quantum network and tester [25]. We say that

ΦR(N)is a deterministic quantum network (or quantum comb) if it is a concatenation of

Nquantum channels and R(N)∈LN2N−1

i=0 Xifulﬁlls the following conditions

R(N)≥0,

trX2k−1R(k)= 1lX2k−2⊗R(k−1),(8)

where R(k−1) ∈LN2k−3

i=0 Xiis the Choi matrix of the reduced quantum comb with

concatenation of k−1 quantum channels, k= 2, . . . , N. We remind that a probabilistic

quantum network ΦS(N)is equivalent to a concatenation of Ncompletely positive trace

non increasing linear maps. Then, the Choi operator S(N)of ΦS(N)satisﬁes 0 ≤S(N)≤

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

nR(N)

ioiwhose sum is a quantum comb, that is PiR(N)

i=R(N), and additionally

dim(X0) = dim(X2N−1) = 1.

We will also use the Moore–Penrose pseudo–inverse by abusing notation X−1∈

L(Y,X) for an operator X∈L(X,Y). Moreover, we introduce the vectorization op-

eration of Xdeﬁned by |Xii =Pdim(X)−1

i=0 (X|ii)⊗ |ii.

3. Process matrices

This section introduces the formal deﬁnition of the process matrix with its character-

ization and intuition. Next, we present some classes of process matrices considered in

this paper.

Let us deﬁne the operator XYas

XY=1lX

dim(X)⊗trXY(9)

STRATEGIES FOR SINGLE-SHOT DISCRIMINATION OF PROCESS MATRICES 5

for every Y∈L(X ⊗Z), where Zis an arbitrary complex Euclidean space. We will also

need the following projection operator

LV(W) = AOW+BOW−AOBOW−BIBOW+AOBIBOW−AIAOW+AOAIBOW. (10)

where W∈Herm(AI⊗ AO⊗ BI⊗ BO).

Deﬁnition 1. We say that W∈Herm(AI⊗ AO⊗ BI⊗ BO)is a process matrix if it

fulﬁlls the following conditions

W≥0, W =LV(W),tr(W) = dim(AO)·dim(BO),(11)

where the projection operator LVis deﬁned by Eq. (10).

The set of all process matrices will be denoted by WPROC. In the upcoming con-

siderations, 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∈WPROC is a process matrix if it fulﬁlls the following

conditions

W≥0,

AIAOW=AOAIBOW,

BIBOW=AOBIBOW, (12)

W=BOW+AOW−AOBOW,

tr(W) = dim(AO)·dim(BO).

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 AIand an output space

AOfor Alice, and analogously BIand BOfor Bob. In general, a label i, denoting

Alice’s measurement outcome, is associated with the CP map ΦMA

iobtained from the

instrument nΦMA

ioi. Analogously, the Bob’s measurement outcome jis associated with

the map ΦMB

jfrom the instrument nΦMB

joj. Finally, the joint probability for a pair of

outcomes iand jcan be expressed as

pij = tr WMA

i⊗MB

j,(13)

where W∈WPROC 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 MA

i⊗MB

jwhich deﬁnes the probability pij given by Eq. (13). If their

action is somehow correlated, we can write the associated instrument in the following

form nΦNAB

ij o. It was observed in [26] that this instrument describes a valid strategy,

that is

tr 

WX

NAB

ij 

= 1 (14)

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STRATEGIESFORSINGLE-SHOTDISCRIMINATIONOFPROCESSMATRICESPAULINALEWANDOWSKA1,LUKASZPAWELA1,ANDZBIGNIEWPUCHALA1Abstract.Thetopicofcausalityhasrecentlygainedtractionquantuminformationresearch.Thisworkexaminestheproblemofsingle-shotdiscriminationbetweenpro-cessmatriceswhichareanuniversalmethoddeningacau...

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