Bound for Gaussian-state Quantum illumination using direct photon measurement Su-Yong LeeDong Hwan KimYonggi Jo Taek Jeong Duk Y. Kim and Zaeill Kim Advanced Defense Science Technology Research Institute

2025-04-30 0 0 609.73KB 8 页 10玖币

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Bound for Gaussian-state Quantum illumination using direct photon measurement

Su-Yong Lee,∗Dong Hwan Kim,†Yonggi Jo, Taek Jeong, Duk Y. Kim, and Zaeill Kim

Advanced Defense Science & Technology Research Institute,

Agency for Defense Development, Daejeon 34186, Republic of Korea

(Dated: November 6, 2023)

It is important to ﬁnd feasible measurement bounds for quantum information protocols. We

present analytic bounds for quantum illumination with Gaussian states when using an on-oﬀ de-

tection or a photon number resolving (PNR) detection, where its performance is evaluated with

signal-to-noise ratio. First, for coincidence counting measurement, the best performance is given

by the two-mode squeezed vacuum (TMSV) state which outperforms the coherent state and the

classically correlated thermal (CCT) state. However, the coherent state can beat the TMSV state

with increasing signal mean photon number in the case of the on-oﬀ detection. Second, the per-

formance is enhanced by taking Fisher information approach of all counting probabilities including

non-detection events. In the Fisher information approach, the TMSV state still presents the best

performance but the CCT state can beat the TMSV state with increasing signal mean photon num-

ber in the case of the on-oﬀ detection. Furthermore, we show that it is useful to take the PNR

detection on the signal mode and the on-oﬀ detection on the idler mode, which reaches similar

performance of using PNR detections on both modes.

I. INTRODUCTION

Entanglement is indispensable for quantum teleporta-

tion, quantum sensing, and quantum illumination (QI).

Contrary to the other protocols, QI loses entanglement

but takes quantum advantage by survival of quantum

correlation. The objective of QI is to discriminate the

presence or absence of a low-reﬂectivity target [1, 2]. Ac-

cording to the frequency range of the probe signal used

for QI, photon loss is the dominant limiting factor for

the performance in optical wave range or thermal noise

is the dominant one in microwave range. In a laboratory,

a low-reﬂectivity beam splitter plays the role of the tar-

get and thermal noise is intentionally injected into the

low-reﬂectivity beam splitter. Thermal noise is replaced

by thermal state which is produced by scattering coher-

ent state light with a rotating glass disk or blocking one

mode of a two-mode squeezed vacuum (TMSV) state.

We consider a scenario where the signal mode of an in-

put state interacts with a target having a low reﬂectivity

in a strong thermal-noise environment, and then the re-

ﬂected signal mode is measured in a receiver. Given an

idler mode of the input state, it is best to measure the

idler mode with the reﬂected signal mode.

The performance of QI can be evaluated by quantum

Chernoﬀ bound (QCB) [3–5], which is the upper limit of

the lower bound on a target-detection error probability.

Under a weak thermal-noise environment, in the begin-

ning, an entangled state takes quantum advantage over

a separable state under single-photon level [1]. The idea

was extended to Gaussian states under a strong thermal-

noise environment [2, 6], where a TMSV state outper-

forms a coherent state. As an input state, it is feasible

to prepare Gaussian states, such as coherent, thermal,

∗suyong2@add.re.kr

†kiow639@gmail.com

squeezed, and TMSV states. Based on QCB, coherent

state presents the best performance in single-mode Gaus-

sian states. In two-mode Gaussian states, TMSV state is

a nearly optimal state [7, 8] for symmetric discrimination

and an optimal state [9] for asymmetric one. A classi-

cally correlated thermal (CCT) state, which is produced

by impinging a thermal state into a beam splitter, can-

not outperform the TMSV state and the coherent state

in QCB.

Although QCB is not directly related to any physi-

cal observable, it can be achieved with signal-to-noise

ratio (SNR) with speciﬁc measurement schemes. QCB

and SNR are independently derived from the detection

error probability. The QCB is represented by the expo-

nential of the decay constant [3, 10], exp[−Mγ], as the

upper limit of the lower bound, and the SNR is repre-

sented by the exponent of the detection error probability,

exp[−SNR(M)], where Mis the number of modes. As

examples, coherent state asymptotically approaches its

QCB with homodyne detection [2, 11], and CCT state

can do that with photon number diﬀerence measurement

[12]. However, TMSV state asymptotically can attain its

QCB not by the SNR that only considers joint local mea-

surement but by collective measurement, e.g., sum fre-

quency generation with feedforward [13], which requires

a quantum memory. Except the homodyne detection, the

two other measurement schemes were not implemented in

a laboratory.

Since the ﬁrst QI experiment [14], there were several

experiments implemented in optical range [15–19] and

microwave range [20–23]. Theoretically, there are fea-

sible proposals with optical parametric ampliﬁer (OPA)

and phase-conjugate (PC) receivers [24], photon number

diﬀerence measurement [12, 25, 26], homodyne (or het-

erodyne) detection [27–29], and on-oﬀ detection scheme

[30, 31] that is conditionally to prepare a signal mode by

detecting the idler mode with an on-oﬀ detector (or on-oﬀ

detector arrays). Note that the OPA and PC receivers

arXiv:2210.01471v4 [quant-ph] 2 Nov 2023

can asymptotically approach a half of the exponent of

the QCB for TMSV state. However, all the measure-

ments schemes require nonlinear interactions or interfer-

ence with additional modes, except the on-oﬀ detection

and the photon number diﬀerence measurement. Thus,

it is worthwhile to ﬁnd the detection error probability

bound using direct photon measurement, without any

other additional modes.

In this paper, we consider the most feasible measure-

ments for Gaussian-state quantum-illumination, using

on-oﬀ detection and photon number resolving (PNR)

detection. Note that we have already considered ho-

modyne (or heterodyne) detection in our previous work

[12]. Physically, the on-oﬀ detection can be implemented

by an ideal avalanche photodiode (APD) as well as an

ideal superconducting nanowire single-photon detector

(SNSPD), where both detectors are commercially avail-

able. The PNR detection can be implemented by an ideal

transition-edge-sensor (TES) that is extensively investi-

gated [32] as well as SNSPD arrays [33]. CCD or EM-

CCD is also implemented to discriminate photon num-

bers, even with low eﬃciency. Theoretically, it is valu-

able to ﬁnd the best theoretical bounds using the on-oﬀ

and PNR detection, without additional modes or com-

bining the signal-and-idler mode. By directly measuring

the idler and the reﬂected signal, we evaluate the perfor-

mance bound with signal-to-noise ratio (SNR). The per-

formance is shown by coincidence counting events, and

then it is enhanced by Fisher information approach in-

cluding all possible counting events.

II. SINGLE- AND TWO-MODE GAUSSIAN

STATES

A Gaussian state is described with covariance matrix

and ﬁrst-order moments [34]. The covariance matrix

is described with σjk =⟨ˆ

Rjˆ

Rk+ˆ

Rkˆ

Rj⟩ − 2⟨ˆ

Rj⟩⟨ ˆ

Rk⟩,

where ˆ

Rk=ˆ

Xk(ˆ

Pk) and ˆak=1

√2(ˆ

Xk+iˆ

Pk). The

ﬁrst-order moments are described with displacement,

µT=√2(Re(α),Im(α)). ˆakis the annihilation op-

erator in mode kthat can be described with the posi-

tion and momentum operators, ˆ

Xkand ˆ

Pk. According

to the relation among the annihilation operator, the po-

sition and momentum operators, the coeﬃcients in the

covariance matrix and the ﬁrst-order moment are de-

termined, such as 2 in σjk and √2 in µT. After in-

teracting a signal mode with a target in thermal noise

environment, the covariance matrix is transformed into

σ(κ) = XσinXT+Y, where X= diag(√κ, √κ, 1,1), Y =

diag(1−κ+2NB,1−κ+2NB,0,0),and σin is the input co-

variance matrix. The ﬁrst-order moment is transformed

into µ(κ) = Xµin, where µin is the input ﬁrst-order mo-

ment. κis a target reﬂectivity while 1 −κis the target

transmittivity.

Previously, most of the QI works compared the perfor-

mances of TMSV state and coherent state, while there

are few works that compared the performances of TMSV

state and CCT state [14, 20, 21]. Recently one com-

pared the performances of coherent state, CCT state, and

TMSV state [12, 22]. Here we consider coherent state,

CCT state, TMSV state, and displaced squeezed (DS)

state.

A single-mode pure Gaussian state is represented by

a DS state ˆ

D(α)|ξ⟩, where α=|α|eiϕ and ξ=reiφ.α

is a displacement parameter, ξis a squeezing parame-

ter, and ris the amplitude of the squeezing parameter.

After interacting the input signal with a target having

reﬂectivity κin thermal noise environment, the reﬂected

output state is given by

σDS(κ) = A1−A2cos φ−A2sin φ

−A2sin φ A1+A2cos φ,(1)

µ=√2κ|α|cos ϕ

sin ϕ,

where A1= 1 + 2NB+ 2κNsq,A2= 2κpNsq(Nsq + 1),

Nsq = sinh2r, and NBis the mean photon number of

thermal noise observed at a detector. When the target

is absent, the covariance matrix is σDS(0) and the ﬁrst-

order moment is µ= 0.

We also consider two-mode squeezed vacuum (TMSV)

states and classically correlated thermal (CCT) states

as representatives of two-mode Gaussian states with no

ﬁrst-order moment. The former is a representative of

continuous variable entangled states, and the latter is

a kind of classically correlated states. After the signal

mode of the TMSV state interacts with a target, the

output is given by

σTMSV(κ) = 





B0C0

0B0−C

C0 1 + 2NS0

0−C0 1 + 2NS





,(2)

where B= 1 + 2NB+ 2κNS, and C= 2pκNS(NS+ 1).

NSis the mean photon number of the signal mode. When

the target is absent, the covariance matrix is σTMSV(0).

After the signal mode of the CCT state interacts with

the target, the output state is given by

σCCT(κ) = 





B0D0

0B0D

D0 1 + 2NI0

0D0 1 + 2NI





,(3)

where D= 2√κNSNIand NIis the mean photon num-

ber of the idler mode. When the target is absent, the

covariance matrix is σCCT(0). The pre-interaction co-

variance matrices of the TMSV and CCT states are ob-

tained by writing down Eqs. (2) and (3) at κ= 1 and

NB= 0.

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BoundforGaussian-stateQuantumilluminationusingdirectphotonmeasurementSu-YongLee,∗DongHwanKim,†YonggiJo,TaekJeong,DukY.Kim,andZaeillKimAdvancedDefenseScience&TechnologyResearchInstitute,AgencyforDefenseDevelopment,Daejeon34186,RepublicofKorea(Dated:November6,2023)Itisimportanttofindfeasiblemeasuremen...

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Bound for Gaussian-state Quantum illumination using direct photon measurement Su-Yong LeeDong Hwan KimYonggi Jo Taek Jeong Duk Y. Kim and Zaeill Kim Advanced Defense Science Technology Research Institute

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