Quantum multiparameter estimation with multi-mode photon catalysis entangled squeezed state Huan Zhang1 Wei Ye2 Shoukang Chang3 Ying Xia1 Liyun Hu4and Zeyang Liao1y

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Quantum multiparameter estimation with multi-mode photon catalysis entangled

squeezed state

Huan Zhang1, Wei Ye2, Shoukang Chang3, Ying Xia1, Liyun Hu4,∗and Zeyang Liao1†

1School of Physics, Sun Yat-sen University, Guangzhou 510275, China

2School of Information Engineering, Nanchang Hangkong University, Nanchang 330063, China

3MOE 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

4Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China

We propose a method to generate the multi-mode entangled catalysis squeezed vacuum states

(MECSVS) by embedding the cross-Kerr nonlinear medium into the Mach-Zehnder interferometer.

This method realizes the exchange of quantum states between diﬀerent modes based on Fredkin

gate. In addition, we study the MECSVS as the probe state of multi-arm optical interferometer

to realize multi-phase simultaneous estimation. The results show that the quantum Cramer-Rao

bound (QCRB) of phase estimation can be improved by increasing the number of catalytic photons

or decreasing the transmissivity of the optical beam splitter using for photon catalysis. In addition,

we also show that even if there is photon loss, the QCRB of our photon catalysis scheme is lower than

that of the ideal entangled squeezed vacuum states (ESVS), which shows that by performing the

photon catalytic operation is more robust against photon loss than that without the catalytic oper-

ation. The results here can ﬁnd important applications in quantum metrology for multiparatmeter

estimation.

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

I. INTRODUCTION

Quantum metrology is one of the most important re-

search ﬁelds in quantum information science which can

provide signiﬁcant quantum advantages over its classical

counterpart. A fundamental task in quantum metrol-

ogy is to improve the estimation precision of parame-

ters to be measured through quantum resources allowing

by the basic principles of quantum mechanics. Gener-

ally speaking, the typical quantum metrology includes

three steps: the preparation of probe states, the evo-

lution of probe states, and the readout of the evolved

states. In quantum metrology, the quantum Cramer-Rao

bound (QCRB) is usually used to quantify the estima-

tion precision oﬀered by quantum metrology, which gives

the lower limit of the estimation precision that can be

achieved using any possible detection methods [1–5]. For

this purpose, prior works are focused on the estimation of

a single parameter with superior quantum resources [6–

12, 14, 16, 17]. For instance, by adopting nonclassical or

entangled states, such as single (two)-mode squeezed vac-

uum state (S(T)SVS) [11–13] and NOON state [14], the

estimation precision can overcome the so-called standard

quantum limit (SQL) scaling as 1/√Nwith Nbeing the

mean photon number of the probe state, and in certain

cases, the precision can even approach to the renowned

Heisenberg limit (HL) with a scaling 1/N. Although the-

oretically we can achieve arbitrary large squeezing pa-

rameter, in practice increasing the squeezing parameter

∗hlyun@jxnu.edu.cn

†liaozy7@mail.sysu.edu.cn

is not an easy task [15]. If we can perform certain non-

Gaussian operations such as photon addition, subtraction

and catalysis on these experimental achievable squeez-

ing states, we may further enhance the precision of the

metrology and achieve the same precision as those by in-

putting a higher squeezing state which is currently hard

to be generated in experiment [16–21].

In the realistic scenarios, such as biological system

measurement [22–24], the optical imaging and the sen-

sor network [25–29], multiparameter quantum metrology

is indispensable and thus has received a lot of increasing

interest in recent years [30–39], as the number of param-

eters aﬀecting a physical process is usually more than

one. For instance, Humphreys et al. treated the phase

imaging problem regarded as a multiparameter estima-

tion process, and showed the advantages of the multipa-

rameter simultaneous estimation using the multi-mode

NOON state, when comparing with the independent esti-

mation scheme [32]. The advantages remain even if there

are photon losses, as studied by Yue et al. [40]. Apart

from the photon losses, Ho et al. estimated three compo-

nents of an external magnetic ﬁeld using the entangled

Greenberger-Horne-Zeilinger state including the dephas-

ing noise and showed that its sensitivity can beat the

SQL [41]. Besides, Yao et al. investigated the multiple

phase estimation problem for a natural parametrization

of arbitrary pure states under the white noise [42]. In

order to further develop the quantum enhanced multipa-

rameter simultaneous estimation, Hong et al. proposed

a method to generate the multi-mode NOON state, and

they experimentally demonstrated that the QCRB can

be saturated using the multi-mode NOON state [43]. In

addition to the NOON state, entangled coherent state

is also widely used in the ﬁeld of quantum metrology.

arXiv:2210.15381v1 [quant-ph] 27 Oct 2022

To compare the performance of the entangled coher-

ent state can be better with that of the multi-mode

NOON state [3, 10, 30, 44, 45], Liu et al. proposed a

theoretical scheme of quantum enhanced multiparame-

ter metrology with generalized entangled coherent state

and showed that the entangled coherent state can in-

deed give better precision than that of the multi-mode

NOON state [30]. After that, Zhang et al. investigated

the quantum multiparameter estimation with generalized

balanced multimode NOON-like states, including the en-

tangled squeezed vacuum state (ESVS), the entangled

squeezed coherent state, the entangled coherent state,

and the NOON state [46, 54]. Comparing with other

multimode NOON-like states, they found that the ESVS

has the lowest QCRB if the mean photon number is the

same.

In this paper, we propose a scheme to generate the

multi-mode entangled catalysis squeezed vacuum states

(MECSVS). Due to the fact that the multiphoton catal-

ysis operation [47, 48] can improve the ﬁdelity in quan-

tum teleportation [49, 50], extend the transmission dis-

tance in continuous variable quantum key distribution

[51, 52], and enhance the sensitivity of phase estimation

for a single-phase estimation [20] and undo the noise ef-

fect of the channel [53], we also propose ascheme to im-

prove the multiparameter estimation precision by using

the MECSVS. Our results clearly show that the multi-

photon catalytic operation can further improve the pre-

cision of phase estimation compared with the result with

ordinary ESVS as the probe state. Moreover, the usage

of multi-photon catalysis in the multiparameter quantum

metrology is also more robust against the photon losses

which can ﬁnd important applications in the practical

scenarios.

The paper is organized as follows: In Sec. II, we pro-

pose a scheme to generate the MECSVS. In Sec. III and

IV, we evaluate the QCRB of multiparameter estimation

with the symmetric MECSVS under ideal and photon-

loss cases, respectively. In Sec. V, we study the QCRB

when the anti-symmetric MECSVS is used. Finally, we

summarize the results.

II. THE GENERATION OF THE MECSVS

In this section, we propose a scheme to generate the

MECSVS which is the probe quantum state used for the

quantum multiparameter estimation. The schematic dia-

gram is shown in Fig. 1 in which three steps are included.

In the ﬁrst step, we perform the n-photon catalysis (or-

ange box) on the input SSVS |ξi. In the n-photon catal-

ysis, an n-photon Fock state |niin an ancillary mode ∼

ais

injected into one port of a beam splitter with a transmis-

sivity T, and then is detected at the corresponding output

of mode ∼

a. In the meantime, the input squeezed state

|ξiis injected into the other port aof the beam splitter.

If nphotons are detected at the output port ∼

a, the so

called n−photon catalysis is performed and the output

state of port ais given by |ξ0iwhich is the single-mode

n−photon catalyzed squeezed state. This multi-photon

catalyzed process can be regarded as an eﬀective operator

[49], i.e.,

O≡ hn|b

B(T)|ni

=Tn/2

∂n

∂τ nnexp µ(τ)a†a

1−τoτ=0

,(1)

where b

B(T) = e(ea†a−eaa†) arccos √Tand the parameter

µ(τ) = ln √T−τ/√T

1−τ.(2)

Considering that the input state of a-mode is a Gaus-

sian SSVS |ξi=√sechr P∞

l=0 −1

2tanh rlp(2l)!/l!|2li

as the input state, the corresponding single-mode mul-

tiphoton catalysis squeezed vacuum state is thus given

|ξ0i=1

√Nb

O|ξia

√N

Tn/2

∂n

∂τ nhβ(τ)∞

l=0

cl(τ)|2liaiτ=0

,(3)

with β(τ)=1/(1 −τ)√cosh r, cl(τ) =

−1

2e2µ(τ)tanh rlp(2l)!/l! and Nis the normaliza-

tion factor given by

N=Tn

(n!)2

∂2n

∂τ n∂τ∗nℵ(τ, τ ∗)1− =(τ, τ ∗)−1

2τ=τ∗=0

(4)

with

ℵ(τ, τ ∗) = 1

(1 −τ) (1 −τ∗) cosh r,

=(τ, τ ∗) = e2[µ(τ∗)+µ(τ)] tanh2r, (5)

where τ∗is the complex conjugate of τ. In particular,

from Eq. (3), when T= 1, the single-mode multiphoton

catalysis squeezed vacuum state is reduced to the SSVS,

as expected. Although the photon catalysis operation

is probabilistic [55], it can be heralded and we choose to

perform the quantum metrology only when the MECSVS

is successfully generated.

After generating the single-mode multi-photon cataly-

sis squeezed vacuum state as shown in Eq.(3), we next

produce an entangled squeezed state using step 2 as

shown in Fig. 1. For this purpose, we propose to employ

that quantum-optical Fredkin gate which consists of two

Mach-Zehnder interferometers mediated by a cross-Kerr

medium. To be more speciﬁc, the single-mode multipho-

ton catalysis squeezed vacuum states and a vacuum state

|0iare respectively injected into a cross-Kerr medium

based Mach-Zehnder interferometer (MZI-2) from modes

aand b. Simultaneously, a vacuum state |0iuand a single

photon state |1ivare used as inputs of the MZI-1. We

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Quantummultiparameterestimationwithmulti-modephotoncatalysisentangledsqueezedstateHuanZhang1,WeiYe2,ShoukangChang3,YingXia1,LiyunHu4,andZeyangLiao1y1SchoolofPhysics,SunYat-senUniversity,Guangzhou510275,China2SchoolofInformationEngineering,NanchangHangkongUniversity,Nanchang330063,China3MOEKeyLabora...

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Quantum multiparameter estimation with multi-mode photon catalysis entangled squeezed state Huan Zhang1 Wei Ye2 Shoukang Chang3 Ying Xia1 Liyun Hu4and Zeyang Liao1y

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