Quantum steering in a star network Guangming Jiang1Xiaohua Wu1and Tao Zhou2 3 1College of Physical Science and Technology Sichuan University Chengdu 610064 China

2025-04-29 0 0 278.42KB 7 页 10玖币

侵权投诉

Quantum steering in a star network

Guangming Jiang,1Xiaohua Wu,1, ∗and Tao Zhou2, 3, †

1College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

2Quantum Optoelectronics Laboratory, School of Physical Science and Technology,

Southwest Jiaotong University, Chengdu 610031, China

3Department of Applied Physics, School of Physical Science and Technology,

Southwest Jiaotong University, Chengdu 611756, China

(Dated: April 18, 2024)

In this work, we will consider the star network scenario where the central party is trusted while all the edge

parties (with a number of n) are untrusted. Network steering is deﬁned with an nlocal hidden state model which

can be viewed as a special kind of nlocal hidden variable model. Two different types of sufﬁcient criteria,

nonlinear steering inequality and linear steering inequality will be constructed to verify the quantum steering

in a star network. Based on the linear steering inequality, how to detect the network steering with a ﬁxed

measurement will be discussed.

PACS numbers: 03.65.Ud, 03.65.Ta

I. INTRODUCTION

In 1930s, the concept of steering was introduced by

Schr¨

odinger [1] as a generalization of the Einstein-Podolsky-

Rosen (EPR) paradox [2]. For a bipartite state, steering in-

fers that an observer on one side can affect the state of the

other spatially separated system by local measurements. In

2007, a standard formalism of quantum steering was devel-

oped by Wiseman, Jones and Doherty [3]. In quantum infor-

mation processing, EPR steering can be deﬁned as the task

for a referee to determine whether one party shares entangle-

ment with a second untrusted party [3–5]. Quantum steering

is a type of quantum nonlocality that is logically distinct from

inseparability [6,7] and Bell nonlocality [8].

In the last decade, the investigation of nonlocality has

moved beyond Bell’s theorem to consider more sophisticated

experiments that involve several independent sources which

distribute shares of physical systems among many parties in

a network [9–11]. The discussions, which are about the main

concepts, methods, results and future challenges in the emerg-

ing topic of Bell in networks, can be found in the review ar-

ticle [12]. The independence of various sources leads to non-

convexity in the space of relevant correlations [13–21].

The simplest network scenario is provided by entanglement

swapping [22]. To contrast classical and quantum correlation

in this scenario, the so-called bilocality assumption where the

classical models consist of two independent local hidden vari-

ables (LHV), has been considered [9,10]. The generaliza-

tion of the bilocality scenario to network, is the so-called n-

locality scenario, where the number of independent sources of

states is increased to arbitrary n[14–16,23–25]. Some new

interesting effects, such as the possibility to certify quantum

nonlocality “without inputs”, are offered by the network struc-

ture [10,11,26,27].

The quantum network scenarios, where some of the par-

ties are trusted while the others are untrusted, are natu-

∗wxhscu@scu.edu.cn

†taozhou@swjtu.edu.cn

rally connected to the notion of quantum steering. Though

the notion of multipartite steering has been previously con-

sidered [28,29], the steering in the scenario of network

with independent sources is seldom discussed. In the re-

cent work [30], focusing on the linear network with trusted

end points and intermediated untrusted parties who perform a

ﬁxed measurement, the authors introduced the network steer-

ing and network local hidden state models. Motivated by the

work in Ref. [30], the steering in a star network will be con-

sidered here.

An important example for a multiparty network is a star-

shaped conﬁguration. Such a star network is composed of a

central party that is separately connected, via a number of nin-

dependent bipartite sources, to nedge parties. Correlations in

the network arise through the central party jointly measuring

the nindependent shares received from the sources and edge

parties locally measuring the single shares received from the

corresponding sources. In the present work, we consider the

star network scenario where the central party is trusted while

all the edge parties are untrusted.

In this work, the quantum steering in a star network is de-

ﬁned by introducing of an n-local hidden state (LHS) model.

It will be shown that this n-LHS model can be viewed as a

special kind of n-LHV model developed in [14–16,23–25].

Besides the n-LHS model, we will focus on how to verify

the quantum steering in the star network. Two different types

of sufﬁcient criteria, linear steering inequality and nonlinear

steering inequality, will be designed. Unlike detecting steer-

ing with single source, it will be shown that the network steer-

ing can be demonstrated with a ﬁxed measurement performed

by the trusted central party.

The content of this work is organized as follows. In Sec. II,

we give a brief review on the deﬁnition of steering in bipar-

tite system. In Sec. III, for a star network scenario where the

central party is trusted while all the edge parties are untrusted,

network steering is deﬁned with an n-LHS model. To verify

the network steering, the nonlinear steering inequality, linear

steering inequality, are designed in Sec. IV, and Sec. V, re-

spectively. Finally, we end our work with a short conclusion.

arXiv:2210.01430v2 [quant-ph] 17 Apr 2024

FIG. 1. A star network is composed of a central party and nedge par-

ties, and via a number of nindependent bipartite sources, the central

party is separately connected to the nedge parties. The ﬁgure above

is depicted for a star-shaped network with n=3, where each edge

observer shares a bipartite state W(µ)(µ=1,2,3) with the central

observer.

II. PRELIMINARY

For convenience, in a star network shown in Fig. 1, we call

the observer in the central party as Bob while the observers in

the edge parties as Alices. Before deﬁning quantum steering

in the star network, some necessary conventions are required.

First, for the bipartite state W(µ), the state shared between the

the µth Alice and Bob, the µth Alice can perform Nmeasure-

ments on her side, labelled by xµ=1,2, ..., N, each having

moutcomes aµ=0,1, ..., m−1, and the measurements are

represented by ˆ

Π(µ)

aµ|xµ,Pm−1

aµ=0ˆ

Π(µ)

aµ|xµ

=Id, with Idthe identity

operator for the local d-dimensional Hilbert space. For a bi-

partite state W(µ), the unnormalized postmeasurement states

(UPS) prepared for Bob are given by

˜ρ(µ)

aµ|xµ

=TrAˆ

Π(µ)

aµ|xµ⊗IdW(µ).(1)

The set of the unnormalized states, ˜ρ(µ)

aµ|xµ, is usually called

an assemblage.

In 2007, Wiseman, Jones and Doherty formally deﬁned

quantum steering as the possibility of remotely generating en-

sembles that could not be produced by an LHS model [3]. An

LHS model refers to the case where a source sends a classi-

cal message ξµto the µth Alice, and a corresponding quantum

state ρ(µ)

ξµto Bob. Given that the µth Alice decides to perform

the measurement xµ, the variable ξµinstructs the output aµ

of Alice’s apparatus with the probability p(µ)aµ|xµ, ξµ. The

variable ξµcan also be interpreted as an LHV and chosen ac-

cording to a probability distribution Ω(µ)(ξµ). Bob does not

have access to the classical variable ξµ, and his ﬁnal assem-

blage is composed by

˜ρ(µ)

aµ|xµ

=ZdξµΩ(µ)(ξµ)p(µ)(aµ|xµ, ξµ)ρ(µ)

ξµ,(2)

with the constraints

aµ

p(µ)(aµ|xµ, ξµ)=1,ZdξµΩ(µ)(ξµ)=1.(3)

In this paper, the deﬁnition of steering from the µth Alice

to Bob is directly cited from the review article [31]: An as-

semblage is said to demonstrate steering if it does not admit

a decomposition of the form in Eq. (2). Furthermore, a quan-

tum state W(µ)is said to be steerable from µth Alice to Bob

if the experiments in µth Alice’s part produce an assemblage

that demonstrates steering. On the contrary, an assemblage is

said to be LHS if it can be written as in Eq. (2), and a quan-

tum state is said to be unsteerable if an LHS assemblage is

generated for all local measurements.

Joint measurability, which is a natural extension of com-

mutativity for general measurement, was studied extensively

for a few decades before steering was formulated in its

modern form [32]. Its relation with quantum steering has

been discussed in recent works [33–37]. A set of mea-

surements ˆ

M(µ)

aµ|xµ, where Paµˆ

M(µ)

aµ|xµ

=Iholds for each

setting xµ, is jointly measurable if there exists a set of

positive-operator-valued measures (POVMs) nˆ

M(µ)

λo, such that

M(µ)

aµ|xµ

=Pλπ(λ)paµ|xµ, λˆ

Mλfor all aµand xµ, with

π(λ)and paµ|xµ, λthe probability distributions. Otherwise,

ˆ

M(µ)

aµ|xµis said to be incompatible.

For the bipartite state W(µ)shared by Bob and the µth Alice,

one can denote ρBto be the reduced density matrix on Bob’s

side and the puriﬁcation of ρ∗

Bis denoted by |Ψ(µ)⟩, where ρ∗

Bis

the complex conjugate of ρB. The state W(µ)can be expressed

W(µ)=ε⊗ I|Ψ(µ)⟩⟨Ψ(µ)|,

by introducing a quantum channel ε,ε(ρ)=Pmˆ

Emρˆ

E†

mwith

the Kraus operators nˆ

Emo, and Iis an identity map. With the

deﬁnition that ˆ

M(µ)

aµ|xµ

=Pmˆ

E†

mˆ

Π(µ)

aµ|xµ

Em, it has been shown

that the conditional states in Eq. (1) can be reexpressed as

˜ρ(µ)

aµ|xµ

=ρ∗

Bˆ

M(µ)

aµ|xµρ∗

B†.(4)

If ˜ρ(µ)

aµ|xµadmits an LHS model, by an explicit deﬁnition of

the inverse matrix of ρB, one may ﬁnd that the measurement

ˆ

M(µ)

aµ|xµis jointly measurable [32,34].

For the state W(µ), the measurement performed by the µth

Alice is denoted by ˆ

Π(µ)

aµ|xµ, and if each aµtakes the value 0 or

1, one can introduce an operator

A(µ)

xµ=ˆ

Π(µ)

0|xµ−ˆ

Π(µ)

1|xµ.(5)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

QuantumsteeringinastarnetworkGuangmingJiang,1XiaohuaWu,1,∗andTaoZhou2,3,†1CollegeofPhysicalScienceandTechnology,SichuanUniversity,Chengdu610064,China2QuantumOptoelectronicsLaboratory,SchoolofPhysicalScienceandTechnology,SouthwestJiaotongUniversity,Chengdu610031,China3DepartmentofAppliedPhysics,Schoo...

展开>> 收起<<

Quantum steering in a star network Guangming Jiang1Xiaohua Wu1and Tao Zhou2 3 1College of Physical Science and Technology Sichuan University Chengdu 610064 China.pdf

共7页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Quantum steering in a star network Guangming Jiang1Xiaohua Wu1and Tao Zhou2 3 1College of Physical Science and Technology Sichuan University Chengdu 610064 China

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: