Constructing Local Models for General Measurements on Bosonic Gaussian States

2025-04-24 0 0 691.24KB 6 页 10玖币

侵权投诉

Michael G. Jabbour∗and Jonatan Bohr Brask†

Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark

We derive a simple suﬃcient criterion for the locality of correlations obtained from given mea-

surements on a Gaussian quantum state. The criterion is based on the construction of a local-

hidden-variable model that works by passing part of the inherent Gaussian noise of the state onto

the measurements. We illustrate our result in the setting of displaced photodetection on a two-mode

squeezed state. Here, our criterion exhibits the existence of a local-hidden-variable model for a range

of parameters where the state is still entangled.

Introduction.–Quantum mechanics allows for correla-

tions than are impossible classically and which can be ex-

ploited in a variety of applications. In particular, entan-

gled quantum states are a key resource for quantum infor-

mation science, enabling advantages in computing, com-

munication, and sensing [1–4]. Furthermore, as shown

by Bell [5], measurements on certain entangled states can

lead to observations that violate a so-called Bell inequal-

ity and are then incompatible with local causal explana-

tions. This phenomenon, known as nonlocality, demon-

strates a profound departure from classical physics and

is a cornerstone of modern understanding of quantum

physics [6]. Nonlocal correlations also enable advantages

for communication [7,8] and information processing at

an unprecedented level of security [9–11].

Entanglement and nonlocality, however, are not equiv-

alent. While entanglement is a prerequisite for non-

locality, in general only carefully chosen measurements

on a given entangled state will produce nonlocal obser-

vations, and while such measurements can always be

found for pure entangled states [12], there exist mixed

entangled states that are local for any possible mea-

surements [13,14]. Deciding whether given states can

give rise to nonlocality is desirable both for applications

and fundamentally. This is, for instance, crucial in the

context of device-independent (DI) quantum key distri-

bution (QKD), the strongest form of quantum crypto-

graphic protocols [15,16]. In DIQKD and other DI pro-

tocols, security relies on the violation of a Bell inequality

and hence requires the use of entangled states that enable

nonlocality.

Certifying whether an entangled state exhibits nonlo-

cality is far from trivial. To demonstrate nonlocality,

it is suﬃcient to ﬁnd a particular set of measurements

that leads to violation of a particular Bell inequality.

Demonstrating that a state cannot give rise to nonlo-

cality is much harder because there are inﬁnitely many

possible measurements and Bell inequalities. It requires

the construction of local-hidden-variable (LHV) models

that can reproduce the observations for any combination

of measurements. Constructing such models is challeng-

ing, even for particular classes of measurements. A num-

ber of methods for constructing LHV models have nev-

ertheless been developed [13,14,17–22], applicable to a

variety of entangled states and measurements. Very of-

ten, a clear connection between the introduction of noise

and the vanishing of nonlocality can be identiﬁed in these

models, e.g., in [13,18].

While most previous work is concerned mainly with

systems of ﬁnite dimension, another relevant class is that

of so-called continuous-variables systems [23]. Most par-

ticularly, Gaussian bosonic states and transformations

are ubiquitous in quantum theory and in experiments in

e.g. optical, superconducting, and mechanical platforms.

At the same time, Gaussian systems are relatively easy

to model. Their entanglement properties have been ex-

tensively studied [24,25] and their nonlocality [26–35]

and steering [36] have also been explored. The relation

between noise and nonlocality has also been investigated

[37,38]. For Gaussian measurements on Gaussian states,

the resulting observations are always local, because the

positive Wigner function of such states enables the con-

struction of an LHV model for any set of Gaussian mea-

surements (as explained in more detail below). However,

little is known about the existence of LHV models for

Gaussian states subject to non-Gaussian measurements.

Here, we develop a suﬃcient criterion for the existence

of LHV models for general measurements on Gaussian

states. Given a state and a candidate family of measure-

ments, the criterion enables one to certify that they will

never lead to nonlocal correlations. The idea behind our

result follows the lines of Werner and Wolf’s criterion for

the separability of Gaussian states [24]. Furthermore, we

provide an interesting interpretation in terms of the role

of noise for the vanishing of nonlocality, separating the

inherent quantum noise resulting from the uncertainty

relations from additional classical Gaussian noise. Be-

fore presenting our main result, we review some elements

of the theory of bosonic systems and nonlocality.

Bosonic systems and Bell nonlocality.–A bosonic sys-

tem [23] is described by Nmodes, where each mode is as-

sociated with an inﬁnite-dimensional Hilbert space and a

pair of bosonic ﬁeld operators ˆak,ˆa†

k, where k= 1, . . . , N

denotes the mode. The total system Hilbert space is

the tensor product over the modes. The ﬁeld operators

satisfy the bosonic commutation relations [ˆai,ˆa†

j] = δij ,

[ˆai,ˆaj] = 0, [ˆa†

i,ˆa†

j] = 0. Alternatively, the system can

be described using the quadrature operators {ˆqk,ˆpk}N

k=1

arXiv:2210.05474v3 [quant-ph] 18 Nov 2024

deﬁned as ˆqk:= ˆak+ ˆa†

k, ˆpk:=i(ˆa†

k−ˆak) (we take

ℏ= 2 throughout), which can also be arranged in the

vector ˆ

r:= (ˆq1,ˆp1,...,ˆqN,ˆpN)T. The quadratures sat-

isfy [ˆrk,ˆrl]=2iΩkl, where Ω:=LN

k=1 0 1

−1 0is the

symplectic form.

Any positive Hermitian operator in state space can

equivalently be completely described by its real-valued

Wigner function in phase space. If the operator is of

unit trace (e.g. the density matrix ρof a quantum state),

its Wigner function integrates to unity. Two quantities of

particular interest are the two ﬁrst statistical moments:

the mean of the quadratures ¯

r:= Tr[ˆ

rˆρ] and the co-

variance matrix Vwith Vij := Tr[{∆ˆri,∆ˆrj}ˆρ]/2, where

∆ˆri:= ˆri−¯riand {·,·} is the anticommutator. Whenever

ρis a genuine quantum state, the 2N×2Nreal, symmet-

ric covariance matrix satisﬁes the uncertainty principle

V+iΩ≥0, which also implies V≥0.

As already mentioned, Gaussian states [39] are ubiq-

uitous in quantum experiments. These are states whose

Wigner function is a multivariate Gaussian distribution.

As such, they are completely described by their ﬁrst two

statistical moments, and their Wigner function can be

written as

W(r) = 1

(2π)N√det Ve−1

2(r−¯

r)TV−1(r−¯

r).(1)

The entanglement in a Gaussian state is determined by

its covariance matrix alone. A bipartite Gaussian state

with covariance matrix VAB will be separable if and only

if there exist genuine covariance matrices γAand γBof

parties Aand Bsuch that V≥γA⊕γB[24].

A stronger form of correlations, Bell nonlocality is de-

ﬁned at the level of the observed input-output distribu-

tion in an experiment with multiple observers. In par-

ticular, a bipartite experiment with observers Aand B

is characterized by the distribution p(ab|xy), where x,

ylabel the choice of input (measurement setting) of A

and B, respectively, and a,blabel their outputs (mea-

surement outcomes). The distribution is called nonlocal

if it does not admit an LHV model, i.e., if it cannot be

written as

p(ab|xy) = Zdλ q(λ)p(a|x, λ)p(b|y, λ),(2)

where the integral is over the (hidden) variable λ, which

is distributed according to a probability density q(λ) and

where p(a|x, λ) and p(b|y, λ) are local response functions.

Entanglement is necessary but not suﬃcient for the

generation of nonlocal correlations [6]. In a general bipar-

tite quantum experiment, Aand Bshare a state ˆρAB and

each perform a generalized measurement with positive-

operator-valued-measure (POVM) elements Qa|xand

Rb|y, respectively. The corresponding probabilities are

p(ab|xy) = Tr[ˆρAB Qa|x⊗Rb|y]. If the quantum state

and all the POVM elements have positive Wigner func-

tions, p(ab|xy) is necessarily local. Indeed, if ˆρAB ,Qa|x

and Rb|yhave respective Wigner functions W,Qa|xand

Rb|y, we have

p(ab|xy) = ZdrW(r)Qa|x(rA)

(4π)−NARb|y(rB)

(4π)−NB,(3)

with r= (rA,rB), where rAand rBare the phase-space

variables and NAand NBare the number of modes of

party Aand B, respectively. This can be understood

as an LHV model (2) with ras the hidden variable.

Wis normalized and is hence a probability density over

r. Since PaQa|x=I, with Ithe identity operator, the

Wigner functions fulﬁll PaQa|x(rA) = (4π)−NAfor all

xand rA, because the Wigner function of the identity

on Nmodes is the constant (4π)−Nin our convention

and similarly for Rb|y. It follows that the last two terms

in (3) are probability distributions over aand b, respec-

tively, and can be interpreted as local response functions.

Hence (3) is of the form (2). An immediate consequence

is that correlations obtained by Gaussian measurements

on a Gaussian state will never be nonlocal.

Constructing the LHV model –We denote by G¯

s,γ the

multivariate Gaussian distribution with mean ¯

sand co-

variance matrix γ, and by f∗gthe convolution of func-

tions fand g, which is deﬁned as

(f∗g)(r):=Zdr′f(r′)g(r−r′).(4)

We also deﬁne 0:= (0,...,0)T. The following statement

provides a suﬃcient criterion for the existence of LHV

models for Gaussian states subject to speciﬁc measure-

ments.

Theorem 1. Let ¯

rbe the mean and Vthe covariance

matrix of a Gaussian state ˆρAB and let Qa|xand Rb|ybe

the Wigner functions of the POVM elements Qa|xand

Rb|y. If there exist matrices γA≥0and γB≥0such

that

V≥γA⊕γB,(5)

and

Qa|x∗ G0,γA≥0 and Rb|y∗ G0,γB≥0,(6)

for all a, x and b, y, then the probabilities p(ab|xy) =

Tr[ˆρAB Qa|x⊗Rb|y]exhibit an LHV model.

Proof. Let ω=V−γA⊕γB≥0. Since γA≥0

and γB≥0, one can deﬁne genuine Gaussian probabil-

ity distributions G0,γAand G0,γB, and similarly for G¯

r,ω.

A useful property of Gaussian distributions is that con-

volving two such distributions results in a Gaussian dis-

tribution, i.e., G¯

s1,γ1∗ G¯

s2,γ2=G¯

s,γ , with ¯

s=¯

s1+¯

and γ=γ1+γ2. Exploiting this and the symmetries of

Gaussian distributions, we have

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：

ConstructingLocalModelsforGeneralMeasurementsonBosonicGaussianStatesMichaelG.Jabbour∗andJonatanBohrBrask†DepartmentofPhysics,TechnicalUniversityofDenmark,2800KongensLyngby,DenmarkWederiveasimplesufficientcriterionforthelocalityofcorrelationsobtainedfromgivenmea-surementsonaGaussianquantumstate.Thecr...

展开>> 收起<<

Constructing Local Models for General Measurements on Bosonic Gaussian States.pdf

共6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！

Constructing Local Models for General Measurements on Bosonic Gaussian States

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: