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Sub-shot-noise quantum opticalinterferometry: a comparison ofentangled state performance within aunified measurement schemeYang Gao a & Hwang Lee aa Hearne Institute for Theoretical Physics , Department of Physicsand Astronomy, Louisiana State University , Baton Rouge, LA, USAPublished online: 02 Dec 2010.

To cite this article: Yang Gao & Hwang Lee (2008) Sub-shot-noise quantum optical interferometry:a comparison of entangled state performance within a unified measurement scheme, Journal ofModern Optics, 55:19-20, 3319-3327, DOI: 10.1080/09500340802428298

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Journal of Modern OpticsVol. 55, Nos. 19–20, 10–20 November 2008, 3319–3327

Sub-shot-noise quantum optical interferometry: a comparison of

entangled state performance within a unified measurement scheme

Yang Gao and Hwang Lee*

Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, LouisianaState University, Baton Rouge, LA, USA

(Received 27 March 2008; final version received 21 August 2008)

Phase measurement using a lossless Mach–Zehnder interferometer with certainentangled N-photon states can lead to a phase sensitivity of the order of 1/N, theHeisenberg limit. However, previously considered output measurement schemesare different for different input states to achieve this limit. We show that it ispossible to achieve this limit just by the parity measurement for all the commonlyproposed entangled states. Based on the parity measurement scheme, thereductions of the phase sensitivity in the presence of photon loss are examinedfor the various input states.

Keywords: quantum entanglement; Heisenberg limited interferometry; NOONstate; correlated Fock states; parity measurement

1. Introduction

The notion of quantum entanglement holds great promise for certain computational andcommunication tasks. It is also at the heart of metrology and precision measurementsin extending their capabilities beyond the so-called standard quantum limit [1–4]. Forexample, the phase sensitivity of a usual two-port interferometer has a shot-noise limit(SL) that scales as 1/N1/2, where N is the number of the photons entering the input port.However, a properly correlated Fock-state input for the Mach–Zehnder interferometercan lead to an improved phase sensitivity that scales as 1/N, i.e. the Heisenberg limit (HL)[5–8]. In the subsequent development, the dual Fock-state [9] and the so-called intelligentstate [10,11] were proposed to reach a sub-shot-noise sensitivity as well. Recently, muchattention has been paid to the so-called NOON state to reach the exact HL ininterferometry as well as super-resolution imaging [12–16].

The utilization of those quantum correlated input states are accompanied by variousoutput measurement schemes. In some cases the conventional measurement scheme ofphoton-number difference is used, whereas a certain probability distribution [17–21], aspecific adaptive measurement [22–24], and the parity measurement are used for other cases.

Gerry first showed the use of the parity measurement for the ‘maximally entangledstate’ – the NOON state – of light to reach the exact HL [25,26], following the earliersuggestion of the HL spectroscopy with N two-level atoms [27]. Campos et al. later

*Corresponding author. Email: hwlee@phys.lsu.edu

ISSN 0950–0340 print/ISSN 1362–3044 online

� 2008 Taylor & Francis

DOI: 10.1080/09500340802428298

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suggested that the parity measurement scheme can also be used for the dual Fock stateinputs by comparing the quantum state inside the interferometer with the NOON state[28]. As was discussed in [28], since the parity is not a continuous function of photonnumber, we need to have photodetectors with single-photon resolution. In other words,the number-resolving photodetectors are required to perform the parity measurement[29,30]. Though it is a very challenging problem, there have been some recent works on thetechniques of high-resolution photon-number detection in literatures, such as the atomic-vapor-based high-efficiency photon detectors [31], the time-multiplexing method [32], andthe noise-free transition-edge sensors [33].

In this paper we show that the parity measurement can be used as a detection schemefor sub-shot-noise interferometry with the correlated Fock states first proposed by Yurkeet al. [5], as well as with the intelligent states first suggested by Hillery and Mlodinow [10].Extension of its use for all these input states then promote the parity measurement toa kind of universal detection scheme for quantum interferometry. Then, based on sucha universal detection scheme comparisons of performance of various quantum states canbe made in a common ground. As an example, we present a comparison of the phasesensitivity reduction for various quantum states of light in the presence of photon loss.

2. Mach–Zehnder interferometer

In order to describe the notations, we briefly review the group theoretical formalism of theMach–Zehnder interferometer. The key point of such a formalism is that any passivelossless four-port optical system can be described by the SU(2) group [5]. First, we usethe mode annihilation operators ain(out) and bin(out), which satisfy boson commutationrelations, to represent the two light beams entering (leaving) the beam splitter (BS),respectively. Then the action of BS takes the form

aout

bout

� �¼

eið�þ�Þ=2 cos�

2e�ið���Þ=2 sin

�

2

�eið���Þ=2 sin�

2e�ið�þ�Þ=2 cos

�

2

0BBB@

1CCCA ain

bin

� �: ð1Þ

Here �, �, and � denote the Euler angles parameterizing SU(2), and they are related to thecomplex transmission and reflection coefficients. Through the Schwinger representation ofangular momentum we can construct the operators for the angular momentum and for theoccupation number from the mode operators a and b,

J ¼

Jx

Jy

Jz

0B@

1CA ¼ 1

2

aby þ bay

iðaby � bayÞ

aay � bby

0B@

1CA, ð2Þ

and N¼ ayaþ byb. The commutation relations [a, b]¼ [a, by]¼ 0 and [a, ay]¼ [b, by]¼ 1 leadto the relation J� J¼ iJ. The Casimir invariant has the form J2 ¼ J2x þ J2yþJ2z ¼ ðN=2ÞðN=2þ 1Þ that commutes with Ji and N. Next, it was shown that the operationof the BS is equivalent to [5]

Jout ¼ expði�JzÞ expði�JyÞ expði�JzÞJin expð�i�JzÞ expð�i�JyÞ expð�i�JzÞ, ð3Þ

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in the Heisenberg picture, and to

jouti ¼ expð�i�JzÞ expð�i�JyÞ expð�i�JzÞ jini, ð4Þ

in the Schrodinger picture. If we use the symbols j and m to indicate the eigenvalues of N/2

and Jz, then the theory of angular momentum tells that the representation Hilbert space

is spanned by the complete orthonormal basis j j, mi with m2 [�j, j], which can also be

labeled by the Fock states of the two modes, j j, mi¼ j jþmia j j�mib. In terms of this

language, we may make the geometrical interpretation of the elements of a Mach–Zehnder

interferometer. For example, the effect of a 50/50 BS, which leads to a ��/2 rotation

around the x axis (given by the unitary transformation exp[�i(�/2)Jx]), is equivalent to the

transformation

aout

bout

� �¼

1

21=21 �i

�i 1

� �ain

bin

� �: ð5Þ

Similarly, the relative phase shift ’ acquired between the two arms of the Mach–Zehnder

interferometer can be expressed by aout¼ ain, bout¼ exp(i’)bin, or by the unitary

transformation exp(�i’Jz) equivalently.The Mach–Zehnder interferometer can be illustrated schematically in Figure 1, where

the two light beams a and b first enter the BSþ, and then acquire a relative phase shift ’,and finally pass through the BS�. The photons leaving the BS� are counted by detectors

Da and Db. Therefore, in the language of the group theory, the input states of BSþ and the

output states of BS� is connected by a simple unitary transformation U¼ exp[i(�/2)Jx]exp(�i’Jz) exp[�i(�/2)Jx]¼ exp(�i’Jy).

1

The information on the phase shift ’ is inferred from the photon statistics of the output

beams. There are many statistical methods to extract such information. The most common

one is to use the difference between the number of photons in the two output modes,

Nd ¼ ayoutaout � byoutbout, or equivalently, Jz,out¼Nd/2. The minimum detectable phase shift

then can be estimated by [34]

�’ ¼�Jz,out

j@hJz,outi=@’j, ð6Þ

where �Jz,out ¼ ðhJ2z,outi � hJz,outi

2Þ1=2. The expectation value of Jz,out and J2z,out are calcula-

ted by hJz,outi¼ hinjJz,outjini¼ hinjUyJz,inUjini, hJ

2z,outi ¼ hinjJ

2z,outjini ¼ hinjU

yJ2z,inUjini,

and UyJnz,inU ¼ ð� sin’Jx,in þ cos ’Jz,inÞn.

ain

ϕ

bin

aout

bout

Figure 1. Schematic of the Mach–Zehnder interferometer. The angle ’ denotes the relative phasedifference between the arms. Note that the Yurke, dual-Fock, and intelligent states are inserted tothe left of the first beam splitter, and NOON to the right.

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3. Parity measurement

Now the application of the group formalism to analyze the phase sensitivity of the

ideal Mach–Zehnder interferometer is straightforward. Let us first consider the

correlated photon-number states [5,6,8]. In particular, the so-called Yurke state

has the form jini ¼ [j j, 0i þ j j, 1i]/21/2, which is one of the earliest proposals of utilizing

the correlated photon-number states [5]. A simple calculation for the Yurke-state input

gives

�’ ¼jð jþ 1Þ � 1½ � sin2 ’þ cos2 ’

� �1=2j½ jð jþ 1Þ�1=2 cos ’þ sin’j

, ð7Þ

which has its minimum value �’min� 1/[j(jþ 1)]1/2 when sin ’� 0. Hence, when the Yurke

state is fed into the input ports of an interferometer, the minimum of �’ has the order of

2/N limit since j¼N/2. We should bear in mind that the minimum phase sensitivity is

achieved only at particular values of ’� 0. For other values of ’ the phase sensitivity

is decreased. However, one can always control the phase shift by a feedback loop which

keeps ’ at any particular value.On the other hand, the parity measurement, represented by the observable

P¼ (�1)byb¼ exp[i�(j� Jz)] has an advantage when the simple photon number counting

method ceases to be appropriate to infer the phase shift and provides a wider applicability

than Jz. The parity measurement scheme was first introduced by Bollinger et al. for

spectroscopy with trapped ions of maximally entangled form [27]. Gerry and Campos

adopted such a measurement scheme to the optical interferometry with the NOON

state [26]. The NOON state can be formally written as |NOONi ¼ [| j, jiþ | j,�ji]/21/2. Note

that the NOON state is not the input state of MZI, but the state after the first beam splitter

BSþ. Hence, the output state is described as |outi¼ exp[i(�/2)Jx]exp(�i’Jz) jNOONi.The expectation value for the parity operator is then given by hPi ¼ iN hNOONj �

exp(i’Jz)exp(i�Jy)exp(�i’Jz) jNOONi¼ iN[exp(iN’)þ (�1)Nexp(�iN’)]/2, so that we

have

hPi ¼

�iNþ1 sinN’, N odd,

iN cosN’, N even,ð8Þ

where the identity exp[�i(�/2)Jx]exp(�i�Jz)exp[i(�/2)Jx]¼ exp(i�Jy) is applied. Since

P2¼ 1, Equation (8) then immediately leads to the result �’¼ 1/N, exactly.Now, let us consider the dual Fock-state as the input state, j j, 0i¼ j jia j jib. Here, if we

still use Jz,out as our observable, we have hJz,outi¼ h j, 0j � sin ’Jxþ cos ’Jz j j, 0i ¼ 0.

The expectation value of the difference of the output photon number is now independent

of the phase shift. Therefore, in this case the measurement of Jz,out contains no

information about the phase shift. A method of reconstruction of the probability

distribution has been proposed to avoid this phase independence and to reach the

Heisenberg limit [9,19,20]. More recently, Campos et al. suggested the use of the parity

measurement for the dual Fock-state inputs [28].

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The expectation value of P can be derived from hPouti¼ hinjexp(i’Jy) Pin exp(�i’Jy)jini and hP2

outi ¼ hinjini ¼ 1. For the dual Fock-state, we have hPoutid-Fock ¼

hj, 0j expði’JyÞð�1Þj�Jz expð�i’JyÞjj, 0i ¼ ð�1Þ

j d j0,0ð2’Þ, where d j

m,n denotes the rotationmatrix element: expð�i’JyÞjj, ni ¼

Pjm¼�j d

jm,nð’Þjj,mi, and

d jm,nð’Þ ¼ ð�1Þ

m�n2�mð j�mÞ!ð jþmÞ!

ð j� nÞ!ð jþ nÞ!

� �1=2

� Pðm�n,mþnÞj�m ðcos’Þð1� cos ’Þðm�nÞ=2ð1þ cos ’ÞðmþnÞ=2,

where Pð�,�Þn ðxÞ represents the Jacobi polynomial. Thus, the phase sensitivity is obtainedas �’d-Fock ¼ f1� ½d j

0,0ð2’Þ�2g1=2=j@d j

0,0ð2’Þ=@’j for the dual Fock-state, and in the limit of

’! 0, we have �’d-Fock! 1/[2j(jþ 1)]1/2� 21/2/N.

If we use the parity measurement scheme for the Yurke-state input, we obtain

hPoutiYurke ¼ hinjexpði’JyÞð�1Þ

j�Jz expð�i’JyÞjini

¼Xjm¼�j

ð�1Þ j�m

2d jm,0 þ d j

m,1

� �d jm,0 þ d j

m,1

� �

¼ð�1Þj

2d j0,0 þ d j

0,1 � d j1,0 � d j

1,1

h ið2’Þ, ð9Þ

where we have used the following properties of the matrix element [35] in the last lineof (9):

d jm,n ¼ d j

m,n ¼ ð�1Þm�nd j

n,m ¼ d j�n,�m,Xj

m¼�j

d jk,mð’1Þ d

jm,nð’2Þ ¼ d j

k,nð’1 þ ’2Þ:ð10Þ

Again, using �’¼ {1� [hPoutiYurke]2}1/2/j@hPouti

Yurke/@’j, we have �’Yurke! 1/[j(jþ 1)]1/2�2/

N, in the limit of ’! 0. This shows that, for the Yurke state, the paritymeasurement schemeleads to the same phase sensitivity as the Jz,out measurement scheme. The dual-Fock statethen performs better than the Yurke-state by a factor of 21/2 within the parity measurementscheme.

We can also use a parity observable for the intelligent state entering the first beamsplitter BSþ in Figure 1. The intelligent state is defined as the solution of the equation(Jyþ i�Jz) j j, m0, �i ¼� j j, m0, �i, where �

2¼ (�Jy)

2/(�Jz)2 and m0 is an integer belonging

to [�j, j] [10]. The eigenvalue corresponding to | ji, m0, � is �¼ im0(�2� 1)1/2 and the

eigenvector j j,m0, �i ¼Pj

k¼�j Ckj j, ki, where an explicit form of the expansion coefficientCk is given in [11]. The expectation value of the parity operator is then obtainedas hPouti

Int ¼ ð�1Þ jPj

k,n¼�j CkCnð�1Þ

kd jk,nð2’Þ. It follows that from the explicit form of

Ck’s the phase sensitivity scales better with a larger � and a smaller jm0j. As �!1, thephase sensitivity becomes

�’Int !1

½2ð j2 �m20 þ jÞ�1=2

�21=2

N: ð11Þ

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On the other hand, as �! 1, we have �’Int! 1/(2j)1/2� 1/N1/2, which is the standard shot-noise limit. So the minimum value of �’ is only accessible for m0¼ 0. This limitingbehavior is the same as the phase sensitivity with Jz measurement at ’¼ 0 [11]. We note

that, within the parity measurement scheme, of all states considered here only the NOONstate reaches exactly the HL [36].

Now that we have seen we can adopt the parity measurement as a universal detectionscheme for all the commonly used entangled states, we will use it as a common ground tocompare the effect of photon loss on phase sensitivity, thus we can put each input state onthe same footing.

4. Photon loss

The effect of photon loss has been recently studied for the NOON states. Gilbert andcoworkers applied a model for loss as a series of beam splitters in the propagation

paths [37]. Rubin and Kaushik applied a single beam splitter model for loss on thedetection operator [38]. Whereas the two approaches are equivalent, we adopt theone given in [37] by putting the the effect of photon loss in the following form [39]:aout ¼ exp½ð�i�a!=c� Ka=2ÞLa�ain þ iK1=2

a

Ð La

0 dz exp½ð�i�a!=c� Ka=2ÞðLa � zÞ�dðzÞ, where�i is the index of refraction for arm i of the interferometer, Ki is the absorptioncoefficient, and Li is the path length. The annihilation operator d(z) is the modes

into which photons are scattered. A similar expression for the mode b is obtained byreplacing a with b.

The observable used for the output detection schemes in both [37,38], namely,A¼ jN, 0ih0, Nj þ j0, NihN, 0j, is equivalent to the parity measurement for the NOON state[12]. In addition, if we now only consider the measurement performed in the N-photonsubspace of the output state, we can ignore the scattering term of the abovetransformation.

Following [37], we assume that the losses are present only in one of the two arms of theinterferometer and set exp(�KaLa)¼ 1 and exp(�KbLb) �. The associated operation ofthe lossy Mach–Zehnder interferometer then can be expressed as

aout

bout

� �¼

1

2

1þ �ei’ �ið1� �ei’Þ

ið1� �ei’Þ 1þ �ei’

!ain

bin

� �, ð12Þ

which is non-unitary unless �¼ 1. In the angular momentum representation, thistransformation can be rephrased as L(’)¼ exp[iJx(�/2)]�exp[�iJx(�/2)]exp[iJx(�/2)]exp(�iJz’)exp[�iJx(�/2)]¼ exp[iJx(�/2)]�exp[�iJx(�/2)]exp(�iJy’), where � is a matrix

representing the effect of path absorption. Then we get

LyPNL ¼ expðiJy’Þ exp iJxp2

� �L exp �iJx

p2

� �PN exp iJx

p2

� �� L exp �iJx

p2

� �expð�iJy’Þ

¼ �N expðiJy’ÞPN expð�iJy’Þ Y1: ð13Þ

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with PN ¼ P�Pm¼j

m¼�j j j,mih j,mj denoting the N-photon projected parity operator. That

is to say, one needs to detect all the N photons, even though that probability decreases

exponentially with N.Similarly, we find

LyP2NL ¼ LyL ¼ expðiJy’Þ exp iJx

p2

� �L2 exp �iJx

p2

� �expð�iJy’Þ

¼ exp iJxp2

� �L2 exp �iJx

p2

� � Y2, ð14Þ

where the commutability of Y2 and exp(�iJy’) is applied, which can be simply proved

in the spinor representation. Now, for a general input state, jini ¼Pj

m¼�j cmj j,mi,

0.7 0.75 0.8 0.85 0.9 0.95 1

λ

0

0.5

1

1.5

2

δϕm

in

δϕm

in

(a) (c)(b)

(f) (e) (d)

N=6

0.7 0.75 0.8 0.85 0.9 0.95 1

λ

0

0.25

0.5

0.75

1

1.25

1.5

1.75

(a) (b) (c)

(f) (e) (d)

N=4

Figure 2. The minimum phase sensitivity, �’min, for the various entangled states as a function of �,the transmission coefficient. The upper and lower figures are for N¼ 4 and N¼ 6, respectively.The dotted line (a) represents that of the uncorrelated input state [37]. The solid lines represent(b) the NOON state, (c) the dual Fock state, (d) the intelligent (�¼ 10) state, (e) the Yurke, and(f) the intelligent (�¼ 1) state, respectively.

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we obtain hPNiout ¼ hY1iin ¼ ð�1Þj�2j

Pm,n c

mcnð�1Þ

md jmnð2’Þ, and hP2

Niout ¼ hY2iin ¼

ð1=2ÞP

m,n cmcn½Qmn þQnm�ð�Þ. Here, the polynomial Qmn(�) is defined as the matrix

element h j, mjY2j j, ni such that

Qmnð�Þ ¼i�m�n

ð j� nþ 1Þjþn

ð2jÞ!½ð jþ nÞ!�1=2

½ð j�mÞ!ð jþmÞ!ð j� nÞ!�1=2�

x2 � 1

4

� �j1þ x

1� x

� �jþn�m

� Pð�2j�1,m�nÞjþn 1�

8x

ðxþ 1Þ2

� �, ð15Þ

where x �2 and pq�( pþ q)/�( p).We now compare the phase sensitivity for different entangled states in the presence

of photon loss. The plots depicted in Figure 2 show the reduced phase sensitivity due

to the photon loss, in this case as a function of � (the transmission coefficient). All the

commonly proposed entangled states are compared to the lossy-environment shot-noise

limit. Among the entangled states, the best possible phase sensitivity can be achieved

by the NOON state, and it gets worse in the following order: the dual Fock state, the

�¼ 10 intelligent state, the Yurke state, and then the �¼ 1 intelligent state. Within the

restricted parity measurement scheme the NOON states show the best performance for

phase detection and can still beat the shot-noise limit if the transmittance of the

interferometer is not too small and the photon number is not too large. We see that

beating the shot-noise limit (dotted line, represented by the uncorrelated input state)

requires less attenuation as the number of photons increases. For example, the lowest

solid line (representing the NOON states) requires 75% transmission for N¼ 4 and

80% for N¼ 6.

5. Summary

To summarize, we showed that the utilization of the parity measurement in sub-shot-noise

interferometry is applicable to a wide range of quantum entangled input states, so far

known entangled states of light. Comparison of the performance of the various quantum

states then can be made within such a unified output measurement scheme. Furthermore,

it may lead to a great reduction of the efforts in precise quantum state preparation as well

as in various optimization strategies involving quantum state engineering for the sub-shot-

noise interferometry [40].

Acknowledgements

The authors wish to thank J.P. Dowling for stimulating discussions, and would like to acknowledgesupport from US AFRL, ARO, IARPA, and DARPA.

Note

1. For the sake of simplicity, BS� is chosen to be conjugate of BSþ such that, if all the photonsenter the upper port, they will also come out at the upper port in the absence of the relativephase shift.

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