Quantum-improved phase estimation with a displacement-assisted SU11 interferometer Wei Ye1 Shoukang Chang2 Shaoyan Gao2 Huan Zhang3Ying Xia3yand Xuan Rao2z 1School of Information Engineering Nanchang Hangkong University Nanchang 330063 China

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Quantum-improved phase estimation with a displacement-assisted SU(1,1) interferometer

Wei Ye1, Shoukang Chang2, Shaoyan Gao2, Huan Zhang3,∗Ying Xia3,†and Xuan Rao2‡

1School of Information Engineering, Nanchang Hangkong University, Nanchang 330063, China

2MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,

Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,

School of Physics, Xi’an Jiaotong University, 710049, People’s Republic of China

3School of Physics, Sun Yat-sen University, Guangzhou 510275, China

By performing two local displacement operations (LDOs) inside an SU(1,1) interferometer, called as

the displacement-assisted SU(1,1) [DSU(1,1)], both the phase sensitivity based on homodyne detection

and quantum Fisher information (QFI) with and without photon losses are investigated in this paper. In

this DSU(1,1) interferometer, we focus our attention on the extent to which the introduced LDO affects

the phase sensitivity and the QFI, even in the realistic scenario. Our analyses show that the estimation

performance of DSU(1,1) interferometer is always better than that of SU(1,1) interferometer without

the LDO, especially the phase precision of the former in the ideal scenario gradually approaching to

the Heisenberg limit via the increase of the LDO strength. More signiﬁcantly, different from the latter,

the robustness of the former can be enhanced markedly by regulating and controlling the LDO. Our

ﬁndings would open an useful view for quantum-improved phase estimation of optical interferometers.

PACS: 03.67.-a, 05.30.-d, 42.50,Dv, 03.65.Wj

I. INTRODUCTION

Quantum metrology is an excellent candidate of pa-

rameter estimation theory to serve as the high-precision

requirement of various quantum information tasks [1–

6], such as quantum sensor [1, 2] and quantum imag-

ing [3, 4]. Thus, how to achieve the higher precision

of quantum metrology has become a general consensus

among scientists. To this end, the optical interferom-

eters, e.g., Mach-Zehnder interferometer (MZI) [7–13]

and SU(1,1) interferometer [14–21], are often used for

understanding the subtle phase variations thoroughly.

Generally, in optical-interferometer systems, it is possi-

ble to obtain the higher precision of phase estimation us-

ing three probe strategies: generation, modiﬁcation and

readout [22]. In the probe generation stage, nonclassi-

cal quantum resources as the inputs of the MZI have been

proven to more effectively enhance the precise measure-

ment than its classical counterpart [23–27]. In partic-

ular, when using the NOON states [25, 28], the two-

mode squeezed vacuum states (TMSVS) [26] and the

twin Fock states [27], the standard quantum limit (SQL)

[13] that is not exceeded by only exploiting the classi-

cal resources can be easily beaten, even inﬁnitely reach-

ing at the famed Heisenberg limit (HL) [18, 26]. These

states, however, are extremely sensitive to noisy environ-

ments [29], so that non-Gaussian resources [12, 30–33]

as an alternative that can be produced by taking advan-

tage of non-Gaussian operations on an arbitrary initial

state play an important role in improving the estimation

performance of the MZI, even in the presence of noisy

scenarios [12, 33]. Apart from the generation stage,

∗Corresponding author. zhangh739@mail2.sysu.edu.cn

†Corresponding author. xiay78@mail2.sysu.edu.cn

‡Corresponding author. raoxuancom@163.com

many efforts have devoted to conceiving the probe mod-

iﬁcation by replacing the conventional beam splitters in

the conventional MZI with the optical parametric ampli-

ﬁers (OPAs) [14–18, 21, 22, 34], which is also called

as SU(1,1) interferometer proposed ﬁrst by Yurke [16].

In this SU(1,1) interferometer with two OPAs, the ﬁrst

OPA (denoted as OPA1) is used not only to obtain the en-

tangled resources but also to eliminate ampliﬁed noise;

while the usage of the second OPA (denoted as OPA2)

can result in the signal enhancement [21, 22], which

paves a feasible way to achieve the higher precision of

phase estimation. Taking advantage of these features, an

SU(1,1) interferometer scheme with the phase shift in-

duced by a kerr medium was suggested by Chang [22],

pointing out that the signiﬁcant improvement of both the

phase sensitivity and quantum Fisher information can be

achieved even in the presence of photon losses. In addi-

tion, the noiseless quantum ampliﬁcation of parameter-

dependent processes was used to SU(1,1) interferome-

ter, indicating how this process results in the HL [35].

More interestingly, by using the non-Gaussian operations

inside the SU(1,1) interferometer, both the phase sen-

sitivity and the robustness of this interferometer system

against the photon losses can be further enhanced [36].

From works [12, 23, 33, 36], we also notice that the us-

age of non-Gaussian operations can signiﬁcantly improve

the estimation performance of the optical interferome-

ters, but at the expense of the high cost of implementing

these operations.

To solve the above problem, the local operations con-

taining the local squeezing operation (LSO) [37–39] and

the local displacement operation (LDO) [40] are one of

the most promising choices. In particular, J. Sahota and

D. F. V. James suggested a quantum-enhanced phase es-

timation scheme by applying the LSO into the MZI [39].

However, it should be mentioned that the LSO plays a

key role in quantum metrology [39], quantum key distri-

bution [38] and entanglement distillation [37], but the

arXiv:2210.02645v1 [quant-ph] 6 Oct 2022

OPA2Hom

Pump





Generation Modification Readout

OPA1

FIG. 1: (Color online) Schematic diagram of the DSU(1,1) interferometer together with homodyne detection, in which a squeezed

vacuum state |ξiaand a coherent state |βibare respectively used as the inputs of DSU(1,1) interferometer in paths aand b.

OPA1and OPA2: the ﬁrst and second optical parametric ampliﬁer. LDO is a local displacement operation. φis a phase shift to

be measured. Hom: an homodyne detection. a0(b0)and a2(b2): the input and output operators of DSU(1,1) interferometer,

respectively.

degree of the LSO is not inﬁnite, e.g., its maximum at-

tainable degree for the TMSVS about 1.19 (10.7dB) [41].

For this reason, here we suggest a quantum-improved

phase estimation of the SU(1,1) interferometer based

on the LDO, which can be called as the displacement-

assisted SU(1,1) [DSU(1,1)] interferometer. Under the

framework of this DSU(1,1)] interferometer, we not only

derive its explicit forms of both the quantum Fisher infor-

mation (QFI) and the phase sensitivity based on homo-

dyne detection, but also consider the effects of photon

losses on its estimation performance. Our analyses man-

ifest that the increase of the LDO strength is conducive to

the improvement of both the QFI and the phase sensitiv-

ity, even in the presence of photon losses. In particular,

this increasing LDO can narrow the gap for the phase

sensitivity between with and without photon losses. This

implies that the usage of the sufﬁciently large LDO can

make the SU(1,1) interferometer systems more robust

against photon losses.

The remainder of this paper is arranged as follows.

In section II, we ﬁrst describe the theoretical model of

DSU(1,1) interferometer, and then give the relationship

between the output and input operators for this interfer-

ometer. In sections III, for the ideal scenario, we analyze

and discuss both the QFI and the phase sensitivity based

on homodyne detection in DSU(1,1) interferometer, be-

fore making a comparison about phase sensitivities con-

taining the SQL, the HL and the DSU(1,1) interferometer

scheme in section VI. Subsequently, we also consider the

effects of photon losses on both the QFI and the phase

sensitivity of DSU(1,1) interferometer in section V. Fi-

nally, our main conclusions are drawn in the last section.

II. THE DSU(1,1) INTERFEROMETER AND ITS

RELATIONSHIP BETWEEN THE OUTPUT AND INPUT

OPERATORS

Now, let us begin with introducing the theoretical

model of DSU(1,1) interferometer, whose structure is

comprised of two OPAs, two LDOs and a linear phase

shift, as depicted in Fig. 1. For simplicity, here we only

consider both a squeezed vacuum state |ξia=ˆ

S(ξ)|0ia

with the squeezing operator ˆ

S(ξ) = exp[(ξ∗ˆa2−ξˆa†2)/2]

(ξ=reiθξ) on the vacuum state |0iaand a coherent state

|βibwith β=|β|eiθβas the inputs of DSU(1,1) interfer-

ometer in paths aand b, respectively. After these input

states pass through the OPA1, paths aand brespectively

experience the same LDO process, denoted as ˆ

Da(γ) =

eγˆa†−γ∗ˆaand ˆ

Db(γ) = eγˆ

b†−γ∗ˆ

bwith γ=|γ|eiθγ, so that

the probe state |ψγican be achieved. Then, we also as-

sume that path aserves as the reference path, while path

bundergoes a linear phase shifter for producing a phase

shift φto be estimated. Finally, after paths aand brecom-

bine in the OPA2, we can extract the phase information

about the value of φby implementing the homodyne de-

tection in path a. Indeed, the relationship between the

output and input operators for DSU(1,1) interferometer

can be given by

ˆa2=W1+Yˆa0−Zˆ

b†

b2=W2+eiφ(Yˆ

b0−Zˆa†

0),(1)

where W1and W2are caused by the LDO process, and

Y= cosh g1cosh g2+ei(θ2−θ1−φ)sinh g1sinh g2,

Z=eiθ1sinh g1cosh g2+ei(θ2−φ)cosh g1sinh g2,

W1=γcosh g2−γ∗ei(θ2−φ)sinh g2,

W2=γeiφ cosh g2−γ∗eiθ2sinh g2,(2)

with g1(g2)and θ1(θ2)respectively representing the gain

factor and the phase shift in the OPA1(OPA2). Accord-

ing to Eq. (1), one can further derive the explicit form

of phase sensitivity, which is a prerequisite for our anal-

ysis and discussion about the estimation performance of

DSU(1,1) interferometer in the following sections.

(a)

|γ|=2|γ|=1

|γ|=0

|γ|=3

0.0 0.5 1.0 1.5 2.0

0.5

1.5

2.5

3.5

4.5

Log10F

|β|=1|β|=2

|β|=3

(b)

012345

3.5

4.0

4.5

5.0

5.5

|γ|

Log10F

(c)

|γ|=0

|γ|=1

|γ|=2

|γ|=3

0.0 0.5 1.0 1.5 2.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Log10ΔϕQCRB

(d)

|β|=3

|β|=2

|β|=1

0 1 2 3 4 5

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-2.0

|γ|

Log10ΔϕQCRB

FIG. 2: (Color online) Both (a)-(b) the QFI log10 Fand the QCRB log10 ∆φQCRB as a function of gat (a) and (c) with ﬁxed values

|β|=r= 1 for different |γ|= 0,1,2,3,and of |γ|at (b) and (d) with ﬁxed values g= 2, r = 1 for different |β|= 1,2,3.Other

parameters are as following: φ=θξ= 0 and θβ=θγ=π/2.

III. THE QFI AND PHASE SENSITIVITY OF DSU(1,1)

INTERFEROMETER IN AN IDEAL SCENARIO

So far, we have described the schematic of DSU(1,1)

interferometer in detail. In this section, we shall present

and analyze the estimation performance of DSU(1,1)

interferometer from the perspective of both quantum

Fisher information and phase sensitivity in an ideal sce-

nario. Moreover, for the sake of discussion, in the fol-

lowing sections, we also assume that the DSU(1,1) inter-

ferometer is in the balanced case, i.e., θ2−θ1=πand

g1=g2=g(set θ1= 0 and θ2=πfor simplicity).

A. The QFI

To directly assess the estimation performance of a un-

known phase parameter without any detection strate-

gies, it is an enormous success for utilizing the QFI of

the probe state since the quantum Cram´

er-Rao bound

(QCRB) ∆φQCRB representing the ultimate precision is

in inverse proportion to the QFI (denoted as F). In addi-

tion, the increased value of the QFI indicates that the es-

timation precision becomes more excellent. In this con-

text, the QFI for an arbitrary pure state in the ideal sce-

nario can be expressed as [12, 22, 32, 42]

F= 4[ψ0

φ|ψ0

φ− |ψ0

φ|ψφ|2],(3)

where |ψφi=eiφˆ

b†ˆ

b|ψγiis the state vector prior to the

OPA2and ψ0

φE=∂|ψφi/∂φ. Thus, if the probe state is

obtained, Eq. (3) can be rewritten as

F= 4[hψγ|ˆn2|ψγi−hψγ|ˆn|ψγi2],(4)

where ˆn=ˆ

b†ˆ

bis the photon number operator of path b.

As a consequence, based on Eq. (4), when inputting the

state |ψini=|ξia⊗|βib, one can obtain the explicit form

of the QFI of DSU(1,1) interferometer (see Appendix A

for more details), i.e.,

F= 4(Γ2+ Γ1−Γ2

1),(5)

where Γm(m= 1,2) are the average value of opera-

tors ˆ

b†mˆ

bmwith respect to the probe state. By using Eq.

(5), one also can obtain the QCRB providing the ultimate

phase precision of DSU(1,1) interferometer regardless of

detection schemes [43, 44], i.e.,

∆φQCRB =1

√νF ,(6)

with the number of trials ν(for simplicity, set ν= 1).

From Eq. (6), it is obvious that, the larger the value of F,

the smaller the ∆φQCRB, which implies the attainabil-

ity of the higher phase sensitivity. In order to see this

point, Fig. 2 shows both the QFI and the QCRB chang-

ing with the gain factor gand the LDO strength |γ|. As

we can see from Fig. 2(a), compared to SU(1,1) inter-

ferometer without the LDO (the black solid line), with

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Quantum-improvedphaseestimationwithadisplacement-assistedSU(1,1)interferometerWeiYe1,ShoukangChang2,ShaoyanGao2,HuanZhang3,YingXia3,yandXuanRao2z1SchoolofInformationEngineering,NanchangHangkongUniversity,Nanchang330063,China2MOEKeyLaboratoryforNonequilibriumSynthesisandModulationofCondensedMatter,S...

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Quantum-improved phase estimation with a displacement-assisted SU11 interferometer Wei Ye1 Shoukang Chang2 Shaoyan Gao2 Huan Zhang3Ying Xia3yand Xuan Rao2z 1School of Information Engineering Nanchang Hangkong University Nanchang 330063 China

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