Interplay of nonlocality and incompatibility breaking qubit channels Swati Kumari1Javid Naikoo2ySibasish Ghosh3zand A. K. Pan4x 1Department of Physics and Center for Quantum Frontiers of Research

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Interplay of nonlocality and incompatibility breaking qubit channels

Swati Kumari,1, ∗Javid Naikoo,2, †Sibasish Ghosh,3, ‡and A. K. Pan4, §

1Department of Physics and Center for Quantum Frontiers of Research &

Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan

2Centre for Quantum Optical Technologies, Centre of New Technologies,

University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland

3Optics and Quantum Information Group, The Institute of Mathematical Sciences,

HNBI, CIT Campus, Taramani, Chennai 600113, India

4Department of physics, Indian Institute of Technology Hyderabad Kandi,Telengana, India

Incompatibility and nonlocality are not only of foundational interest but also act as important resources for

quantum information theory. In the Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a

pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the

context of qubit channels. The Bell-CHSH inequality has a greater perspective—compared to any genuine

tripartite nonlocality scenario—while determining the interplay between nonlocality breaking qubit channels

and incompatibility breaking qubit channels. In the Bell-CHSH scenario, we prove that if the conjugate of a

channel is incompatibility breaking, then the channel is itself nonlocality breaking and vice versa. However, this

equivalence is not straightforwardly generalized to multipartite systems, due to the absence of an equivalence

relation between incompatibility and nonlocality in the multipartite scenario. We investigate this relation in

the tripartite scenario by considering some well-known states like Greenberger-Horne-Zeilinger and Wstates

and using the notion of Mermin and Svetlichny nonlocality. By subjecting the parties in question to unital

qubit channels, we identify the range of state and channel parameters for which incompatibility coexists with

nonlocality. Further, we identify the set of unital qubit channels that is Mermin or Svetlichny nonlocality

breaking irrespective of the input state.

I. INTRODUCTION

Nonlocality is one of the most profound notions in quantum

mechanics [1] and is often discussed in conjunction with in-

compatibility of observables. Recent developments in quan-

tum information theory have found nonlocality a useful phe-

nomenon underpinning many advantages aﬀorded by various

quantum information processing tasks [2]. Nonlocality can

also be considered as a potential quantum resource for in-

formation processing, such as in developing quantum proto-

cols to reduce the amount of communication needed in cer-

tain computational tasks [2] and providing secure quantum

communications [3,4]. Incompatibility, like nonlocality, is

not merely of theoretical interest but of practical utility, for

example, in order to explore the advantage of entanglement

shared by two parties in a cryptography task, each party needs

to carry out measurements that are incompatible, in the sense

that these cannot be carried out simultaneously by a single

measurement device. Incompatibility should not be confused

with noncommutativity or the related concept of uncertainty

principle. The notion of incompatibility is best understood

in terms of joint measurability [5]. A collection of quantum

measurements is jointly measurable, if it can be simulated by

a single common quantum measurement device. If such a sin-

gle common device cannot be constructed by a given set of

quantum measurements, it then enables the set to be used as a

quantum resource. This was ﬁrst noted in [6] in the context of

∗swatipandey084@gmail.com

†j.naikoo@cent.uw.edu.pl

‡sibasish@imsc.res.in

§akp@phy.iith.ac.in

Clauser-Horne-Shimony-Holt (CHSH) inequalities and later

in the Einstein-Podolsky-Rosen steering, which is more ex-

plicit, when incompatibility appears as a quantum resource.

Incompatibility is necessary and suﬃcient for the violation of

the steering inequalities [7,8]. The relation between incom-

patibility and contextuality has also been studied in references

[9,10]. Further, a set of observables that is pairwise incom-

patible, but not triplewise can violate the Liang-Spekkens-

Wiseman noncontextuality inequality [11]. Recently, the con-

nection between steerability and measurement incompatibility

was studied in [12] in the context of the so-called steerability

equivalent observables. Thus, both nonlocality and incompat-

ibility can be considered as quantum resources whose under-

standing is of utmost importance in view of emerging quan-

tum technologies.

The interplay of nonlocality and incompatibility has been a

subject matter of various studies. It is well known that any in-

compatible local measurements, performed by the constituent

parties of a system, lead to the violation of Bell inequality

provided they share a pure entangled state [1,2]. Absence of

either of them (i.e., entanglement or incompatibility) will not

allow the system to exhibit nonlocality. It is important to men-

tion here that the notion of quantum nonlocality without en-

tanglement has been proposed in [13] which is diﬀerent from

Bell nonlocality [1] and amounts to the inability of discrimi-

nating a set of product states by local operations and classical

communication, while mutual orthogonality of the states as-

sures their perfect global discrimination.

Further, for any pair of dichotomic incompatible observables,

there always exists an entangled state which enables the vi-

olation of a Bell inequality [6]. The relationship of incom-

patibility and nonlocality is sensitive to the dimension of the

system; for example, increasing the dimension beyond 2, the

arXiv:2210.02744v3 [quant-ph] 18 Feb 2023

incompatible observables do not necessarily lead to the vi-

olation of Bell-type inequalities, implying that the measure-

ment incompatibility can not guarantee nonlocality in gen-

eral [14,15]. Here, we probe the interplay between incom-

patibility and nonlocality in the tripartite case by using the

well-known Mermin and Svetlichny inequalities [16]. The

Svetlichny inequality, unlike the Mermin inequality, is a gen-

uine measure of nonlocality that assumes nonlocal correla-

tions between two parties which are locally related to a third

party and is known to provide a suitable measure to detect

tripartite nonlocality for Wand Greenberger-Horne-Zeilinger

(GHZ) classes of states [17]. We refer the interested reader to

[2,18] for various facets of the multipartite nonlocality.

The extent to which a system can exhibit nonlocal correla-

tions is also sensitive to its interaction with the ambient en-

vironment. Such interaction is usually accompanied with a

depletion of various quantum features like coherence, entan-

glement and nonlocality. The reduced dynamics of the system

in such cases is given by completely positive and trace pre-

serving maps, also known as quantum channels (QCs). On

the other hand, the action of conjugate channels on projec-

tive measurements turns them into unsharp positive operator-

valued measures (POVMs) which may be biased, in general.

In light of the above discussion, a study of the open system

eﬀects on the interplay of nonlocality and incompatibility nat-

urally leads to the notions of nonlocality breaking and incom-

patibility breaking quantum channels [5,19]. A nonlocality

breaking channel (NBC) can be deﬁned as a channel which

when applied to a system (or part of it) leads to a state which

is local [19], while the incompatibility breaking channel (IBC)

is the one that turns incompatible observables into compatible

ones [20,21]. An IBC that renders any set of nincompati-

ble observables compatible would be denoted by n-IBC. The

notion of the NBC has been introduced in a similar spirit of

well-studied entanglement breaking channels [22]. Every en-

tanglement breaking channel is nonlocality breaking but the

converse is not true. As an example, the qubit depolarizing

channel Eρ:=pρ+(1 −p)I/2 is CHSH nonlocality break-

ing for all 1

3≤p≤1

2, but not entanglement breaking [19].

Hence, based on this classiﬁcation, nonlocality and entangle-

ment emerge as diﬀerent resources.

The equivalence of the steerability breaking channels and the

incompatibility breaking channels was reported in [23] and

CHSH nonlocality breaking channels were shown to be a strict

subset of the steerability breaking channels [24]. The con-

nection between Bell nonlocality and incompatibility of two

observable is well understood, however, the question of the

equivalence between NBC [19] and IBC [21] is rarely dis-

cussed. This motivates us to explore the relation between

CHSH nonlocality breaking channels (CHSH-NBC) and 2-

IBC, such that the action of one may be replaced by the other.

The tripartite nonlocality has much richer and complex struc-

ture and less is known about its synergy with incompatibility

as compared to its bipartite counterpart. Mermin inequality

assumes local-realistic correlations among all the three qubits;

hence a violation would be a signature of the tripartite nonlo-

cality shared among the qubits. However, biseparable states

were shown to also violate the Mermin inequality [3,25]. This

motivated Svetlichny to introduce the notion of genuine tripar-

tite nonlocality [16] and provide a set of inequalities suﬃcient

to witness it. We make use of these notions of absolute and

genuine nonlocality to ﬁgure out the ranges of state and chan-

nels parameters in which NBC and 2-IBC coexist.

The paper is organized as follows. In Sec. II, we revisit

some basic notions and deﬁnitions used in this paper. Sec-

tion III is devoted to results and their discussion where we

prove an equivalence between NBCs and 2-IBCs in the CHSH

scenario. This is followed by an analysis of these notions in

the tripartite scenario, where we identify the state and channel

parameters in which NBCs and 2-IBCs co-exist. A conclusion

is given in Sec. IV.

II. PRELIMINARIES

In this section, we discuss the notion of incompatibility in the

context of observables and quantum channels and look at spe-

ciﬁc cases of bipartite and tripartite scenarios.

A. Incompatibility

1. Incompatibility of observables

A ﬁnite collection of observables A1,...,Anassociated with

the respective outcome spaces ΩA1,...,ΩAn, is said to be com-

patible (or jointly measurable) if there exists a joint observ-

able G, deﬁned over the product outcome space ΩA1×···×ΩAn,

such that for all X1⊂ΩA1,...,Xn⊂ΩAn, the following

marginal relations hold [26]:

ai,i=1,...,n;i,k

G(a1,...,an)=Ak(ak) (1)

where akis the outcome associated to observable Akand the

summation is carried out over all outcomes aiexcept for i=

k. The notion of the incompatibility of observables can be

illustrated by a simple example of Pauli matrices σxand σz

which are noncommuting and can not be measured jointly.

However, consider the unsharp observables Sx(±)=1

2(I±

√2σx) and Sz(±)=1

2(I±1

√2σz), with Ibeing the 2×2 identity

matrix. The joint observable G(i,j)=1

4(I+i

√2σx+j

√2σz) with

i,j=±1 jointly determines the probabilities of generalized

measurements Sxand Sz, since the later can be obtained as

marginals Sx(±)=PjG(±,j) and Sz(±)=PiG(i,±).

2. Incompatibility breaking quantum channel

A QC, in the Schr¨

odinger picture, is a completely positive

trace preserving map E:L(HA)→ L(HB), where L(Hi)

is the set of bounded linear operators on Hilbert space Hi(i=

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InterplayofnonlocalityandincompatibilitybreakingqubitchannelsSwatiKumari,1,JavidNaikoo,2,ySibasishGhosh,3,zandA.K.Pan4,x1DepartmentofPhysicsandCenterforQuantumFrontiersofResearch&Technology(QFort),NationalChengKungUniversity,Tainan701,Taiwan2CentreforQuantumOpticalTechnologies,CentreofNewTechnologi...

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Interplay of nonlocality and incompatibility breaking qubit channels Swati Kumari1Javid Naikoo2ySibasish Ghosh3zand A. K. Pan4x 1Department of Physics and Center for Quantum Frontiers of Research

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