Synchronized Bell protocol for detecting non-locality between modes of light Madhura Ghosh Dastidar1Gniewomir Sarbicki2and Vidya Praveen Bhallamudi3 1Department of Physics Indian Institute of Technology Madras Chennai 600036 Tamil Nadu India

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Synchronized Bell protocol for detecting non-locality between modes of light

Madhura Ghosh Dastidar,1Gniewomir Sarbicki,2and Vidya Praveen Bhallamudi3

1Department of Physics, Indian Institute of Technology Madras, Chennai 600036, Tamil Nadu, India

2Institute of Physics, Faculty of Physics, Astronomy and Informatics,

Nicolaus Copernicus University, Grudzi¸adzka 5/7, 87-100 Toru´n, Poland

3Quantum Center of Excellence for Diamond and Emerging Materials (QuCenDiEM) Group,

Departments of Physics and Electrical Engineering,

Indian Institute of Technology Madras, Chennai 600036, India.

(Dated: May 30, 2023)

In the following paper, we discuss a possible detection of non-locality in two-mode light states

in the Bell protocol, where the local observables are constructed using displacement operators,

implemented by Mach-Zender Interferometers fed by strong coherent states. We report numerical

results showing that maximizing the Braunstein-Caves Chained Bell (BCCB) inequalities requires

equal phases of displacements. On the other hand, we prove that non-locality cannot be detected

if the phases of displacements are unknown. Hence, the Bell experiment has to be equipped with a

synchronization mechanism. We discuss such a mechanism and its consequences.

I. INTRODUCTION

Entangled quantum systems have grown in importance

for technological as well as fundamental scientiﬁc ap-

plications. The advantage of quantum non-locality has

been proved in various ﬁelds such as quantum communi-

cation [1,2], metrology [3,4] and computation [5,6].

Entangled modes of light typically, are useful in pho-

tonic quantum metrology schemes [7], where the pur-

pose is to achieve the quantum limit of measurement [8].

These states of light are multiphotonic, i.e., combina-

tions of superpositions of Fock states. Thus, the exper-

imental veriﬁcation of entanglement in such states re-

quires many measurements with complex experimental

setups. For example, recent works [9,10] show that ex-

perimental veriﬁcation of entanglement in certain impor-

tant classes of two-mode entangled states require multi-

ple single photon detectors or photon-counting electron-

multiplying charge-coupled-device (EMCCD) camera. In

this approach, the density matrix is reconstructed in the

process of full-state tomography which requires restric-

tion to an eﬀective Fock space of dimension nand the

number of observables to be measured grows fast with n.

As an alternative to performing such intricate exper-

imental schemes, one can perform a Bell-CHSH [11,12]

experiment as proposed in [13] for entanglement detec-

tion in two-mode light states. This work describes using

Mach-Zehnder Interferometers (MZI) fed with a strong

coherent state at one input port and having a photodetec-

tor at one output port. The photodetector can measure

zero or non-zero intensities of the incoming pulse. This

experimental unit (MZI + coherent state + photodetec-

tor) is possessed by each of two parties. The above is

relatively simpler compared to the existing schemes for

veriﬁcation of entanglement in two modes of light.

A CHSH inequality is deﬁned for two parties with

two measurement settings (n= 2) per party. The

Braunstein-Caves chained Bell (BCCB) inequalities [14]

generalise the CHSH inequality to nmeasurement set-

tings per party. A particular expression for the quantum

bound of BCCB inequalities has been reported in [15].

It is also shown there that the diﬀerence between quan-

tum and classical bound grows with nfor n > 2. Thus,

in an experiment, the violation of the classical bound by

a two-mode entangled state can be resolved better with

n > 2.

In the following paper we check, whether the CHSH

inequality in the mentioned experimental scheme can be

improved by using BCCB inequality when the parties

again use observables implemented by MZI + coherent

state + photodetector.

In this paper, we intend to check if such a generaliza-

tion can be extended to the proposed setup in [13]. We

observe that the Mach-Zehnder interferometric setup in-

volved in entanglement detection requires phase synchro-

nization of the two inputs to the interferometer. We re-

port that without a constant phase diﬀerence between the

two inputs, the measurement observables get restricted

to the classical regime. Thus, entanglement detection

is only possible when there is a known and ﬁxed phase

diﬀerence between the two inputs of the MZI. We also

discuss the two-mode light states for which this setup is

best for the experimental detection of entanglement.

Further, the entanglement detection in the scheme

should be also analysed under restriction to experimen-

tally accessible classes of entangled two-mode light states.

We check whether the proposed experimental scheme de-

tects entanglement for certain important states of light

useful for quantum metrology, namely, entangled coher-

ent states (ECS) [16] and two-mode squeezed vacuum

(TMSV) [17].

The paper is organized as follows: Sec. II describes the

formulation of the BCCBI inequality for nmeasurement

settings per party for our proposed experimental setting.

In Sec. III, we report our numerical results of maximal

violation obtained by the n-MZI settings and comment

on the phase synchronization issues. We also give a brief

description of the states that correspond to this maximal

violation. In Sec. IV, we consider entanglement detection

for two important classes of light: entangled coherent

arXiv:2210.05341v3 [quant-ph] 29 May 2023

states and two-mode squeezed vacuum, and discuss the

values of parameters maximizing the violation obtained

by n-MZI settings. We summarise our observations in

Sec. V.

II. THE BRAUNSTEIN-CAVES CHAINED BELL

(BCCB) INEQUALITY

In general, for ndichotomic observables (of output val-

ues ±1) per party, the following inequality holds under

the assumption of the existence of underlying probability

space (local hidden variable model):

|E(

i=1

Xi⊗Yi+

n−1

i=1

Xi+1 ⊗Yi−X1⊗Yn)| ≤ 2n−2.

(1)

where {X1, ..., Xn}and {Y1, ..., Yn}are dichotomic ob-

servables employed by Lab X and Y, respectively, corre-

sponding to their nindependent measurement settings.

The above inequality is known as the Braunstein-Caves

chained Bell (BCCB) inequality and the maximum of the

LHS over all quantum states is 2ncosπ

2n[15]. The

bound is saturated for the qubit singlet state (|00⟩+

|11⟩)/√2 and observables:

Xi= cos(αi)σx+ sin(αi)σy=0e−iαi

eiαi0(2)

Yi= cos(βi)σx+ sin(βi)σy=0e−iβi

eiβi0(3)

where αk=kπ/n,βk=−kπ/n.

For n= 2 the BCCB inequality becomes the famous

CHSH inequality.

Let us assume, that each party performs intensity-

based measurements on its mode using a photodetector

at the output of Mach-Zehnder interferometer (MZI),

where its ﬁrst input is fed by the possessed mode and

the second by a strong coherent state of light (Fig-

ure 1). Such interferometer setting implements a dis-

placement operator ˆ

D(α) on the input mode and, to-

gether with the photodetector, a projective measure-

ment: {|α⟩⟨α|, I − |α⟩⟨α|}. Prescribing output values

±1, we obtain a hermitian observable:

A(α) = I−2|α⟩⟨α|.(4)

The nmeasurement settings on each side correspond to

ndisplacements.

Let the measurement settings or displacements imple-

mented by MZIs in Lab X and Y be {β1, ..., βn}and

{γ1, ..., γn}, respectively. The corresponding observables

are {A(β1), ..., A(βn)}and {A(γ1), ..., A(γn)}.

Therefore, the observables in [Eq. 1] are:

Xi=A(βi) = I−2|βi⟩⟨βi|

Yi=A(γi) = I−2|γi⟩⟨γi|(5)

Thus, we can write the LHS of BCCB inequality (S)

for n-MZI settings from [Eq. 1] as:

i=1

A(βi)⊗A(γi)+

n−1

i=1

A(βi+1)⊗A(γi)−A(β1)⊗A(γn)

(6)

In [13], it has been proven, that for n= 2, the max-

imal violation of the CHSH inequality can be achieved

by an appropriate choice of displacements in both

(MZI+photodetector) settings possessed by Lab X and

Lab Y. Now, to detect entanglement by a larger (n > 2)

number of settings, the classical bound (2n−2) must be

violated, i.e., (2n−2) <E(S)max ≤2ncos (π/2n). Fur-

ther, we check the MZI settings in both labs maximiz-

ing the violation and how close to the maximal violation

2ncos (π/2n)−(2n−2) can it be.

III. RESULTS

In this section, we discuss a number of optimization

results we have obtained analysing the BCCB inequality

with observables originating from MZI setups and for var-

ious families of experimentally accessible states. We have

obtained the results numerically and the Appendices A

and B 1 describe the details of our codes.

A. Maximal Eigenvalues of BCCB matrix

As discussed earlier, the BCCB inequality is maximally

violated in a pure state represented by an eigenvector of

S(6) related to its maximal eigenvalue. The maximum

possible violation is equal to 2ncos ( π

2n)−(2n−2), and

in particular for n= 2, we have obtained 2√2−2−the

maximal violation for standard CHSH inequality. First,

we perform optimization for n= 3,...,8 with respect to

the parameters {βi},{γi}, for i∈ {1, . . . , n}. We mini-

mize the probability of getting stuck in a local maximum

by repeating the procedure multiple times, with a num-

ber of randomly chosen starting points. At this stage,

for n > 10 the method typically gets stuck in a local

minimum and shows no violation.

The optimal sequences of {βi}and {γi}are shown in

Fig. 2. We observe that the sequences {βi}and {γi}

each behave co-linearly on the complex plane. The com-

mon phase of βican be made zero by applying a local

unitary transformation. Similarly, the ﬁrst displacement

(measurement setting) can be made zero by applying a

displacement operator, which is a local unitary transfor-

mation as well. The same applies to co-linear complex

numbers γi. Hence both sequences are real and start from

0. In this way, we reduced the number of optimization

parameters from 4nto 2n−2.

Moreover, in each sequence, we observe almost equal

spacing between displacements except for the ﬁrst/last

one, being signiﬁcantly bigger [see Fig. 2]. We con-

ﬁrm this observation in the optimization over a reduced

Source

(Two-mode light)

Laser (LX)Laser (LY)

PDXPDY

Mode 1

Mode 1 Mode 2

Mode 2

BSX1

BSX2

BSY1

BSY2

MX2MY2

MY1

MX1

Party X Party Y

Block Block

FIG. 1. Schematic of experimental arrangement for entanglement detection: The proposed setting with a source

producing a two-mode entangled state of light and 2 laboratories (X and Y) involved in a Bell-type experiment. Each party

uses a Mach-Zehnder interferometer (MZI), comprising 50:50 beam-splitters BSj

iand mirrors Mj

i(i= X or Y, j = 1 or 2). Each

MZI is fed with a coherent state |α⟩⟨α|iat one input and is terminated by a photodetector PDi(i= X or Y) which measures

zero or non-zero intensities. The two paths in the MZI have a relative phase diﬀerence ϕi(i= X or Y).

FIG. 2. Complex-plane representation of {βi}and {γi}:Numerically generated plots for the optimized {βi}and {γi}

(i= 1 to n, n ∈[3,5]) for which the corresponding maximal violation of BCCB inequality [Eq. 1] is achieved. A co-linear

trend is observed in the complex-plane representation of these {βi}and {γi}and thus, each may be written as approximate

arithmetic sequences.

number of parameters, obtaining almost perfect match-

ing with a two-parameter optimization, where β2−β1=

γn−γn−1= ∆′,βi+1 −βi= ∆ for i > 1 and βi+1 −βi= ∆

for i<n.

The diﬀerence between the results from the above op-

timization schemes starts to be visible for n≈8. Hence,

the two-parameter assumption is only a good approxima-

tion of the optimal pattern of displacements. Using it,

we have improved our general optimization scheme: ﬁrst,

we perform a quick two-parameter optimization, repeat-

ing it a large number of times to avoid local minima.

Then, we use the ﬁrst stage result as a starting point

for a single 2n−2-parameter optimization, reaching the

global optimum.

After the ﬁrst stage of the optimization the last n−1

displacements βiand the ﬁrst n−1 displacements γi

form arithmetic sequences. In the second stage of the

optimization, this linear dependence obtains a sine-like

component. Fig. 3shows the comparison of the results

of the optimization after the ﬁrst and second stages for

n= 19.

Fig. 4shows the optimization results. The blue trian-

gles are the ﬁrst-stage (two-parameter optimization) re-

sults. The black dots are the results of the second stage.

These values can be surprisingly well-ﬁtted by an expo-

nential function: D(n) = C−Aexp(−Bn). As the max-

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SynchronizedBellprotocolfordetectingnon-localitybetweenmodesoflightMadhuraGhoshDastidar,1GniewomirSarbicki,2andVidyaPraveenBhallamudi31DepartmentofPhysics,IndianInstituteofTechnologyMadras,Chennai600036,TamilNadu,India2InstituteofPhysics,FacultyofPhysics,AstronomyandInformatics,NicolausCopernicusUni...

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Synchronized Bell protocol for detecting non-locality between modes of light Madhura Ghosh Dastidar1Gniewomir Sarbicki2and Vidya Praveen Bhallamudi3 1Department of Physics Indian Institute of Technology Madras Chennai 600036 Tamil Nadu India

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