PHYSICAL REVIEW A 87, 043833 (2013)

Phase estimation at the quantum Cramer-Rao bound via parity detection

Kaushik P. Seshadreesan,1 Sejong Kim,2,* Jonathan P. Dowling,1,3 and Hwang Lee1

1Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge,Louisiana 70803, USA

2Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, USA3Beijing Computational Science Research Center, Beijing, 100084, China

(Received 29 June 2012; revised manuscript received 18 January 2013; published 25 April 2013)

We show that, in two-path optical interferometry, the measurement of photon number parity achieves maximalphase sensitivity at the quantum Cramer-Rao bound for a certain type of path-symmetric states. All pure states oflight proposed for sub–shot-noise phase sensitivity thus far in the literature are of the form of these path-symmetricstates. Thus, parity measurement presents an optimal detection strategy for all of them. Our scheme is applicablefor local phase estimation.

DOI: 10.1103/PhysRevA.87.043833 PACS number(s): 42.50.St, 42.50.Dv, 42.50.Ex, 42.50.Lc

I. INTRODUCTION

Interferometry is a vital component of various precisionmeasurement, sensing, and imaging techniques. It worksbased on mapping the quantity of interest onto the unknownphase of a system and estimating the latter, for example, therelative phase between the two modes or “arms” of an opticalinterferometer.

Optical interferometry, often described in the Mach-Zehnder configuration (see Fig. 1), in general differs inthe strategies of probe-state preparation and detection. Theconventional choice is to use a coherent light source andintensity difference detection. Assuming linear interferometry,i.e., when the unitary phase evolution operator is of the form

Uφ = e−iφ(na−nb)/2 (1)

(note that the generator of the unitary operator is a linearoperator), where na and nb are the number operators associatedwith the modes, the phase sensitivity of the conventionalMach-Zehnder interferometer (MZI) is bounded by the shotnoise limit δφ = 1/

√n (for n photons in the coherent state

on average), whereas the use of states with nonclassicalphoton correlations opens up the possibility of enhanced phasesensitivities that can reach up to the so-called Heisenberg limit(HL) δφ = 1/n in the absence of photon losses [1–3].

The theory of quantum phase estimation aids in identifyingthe different probe states and detection observables that arecapable of such enhanced phase sensitivities [4]. The phasesensitivity δφ of a quantum state in an MZI with a givendetection observable is bounded by the Cramer-Rao bound(CRB), which, for an unbiased observable, is attained inthe limit of a large number of measurements. The CRB isfurther bounded by the so-called quantum Cramer-Rao bound(QCRB), which depends on the quantum state alone and iscalculated by assuming the most optimal detection allowedby quantum mechanics, namely, the symmetric logarithmicderivative operation (SLDO) [5]. A detection observable isconsidered optimal for a state if the CRB achieves the QCRB.

Uys and Meystre considered photon number counting anda Bayesian update protocol in place of intensity-difference

*ksejong@math.lsu.edu

detection and derived probe states whose CRB reaches theHL [6]. Lee et al. included photon losses in the scheme andworked out the optimal states and their phase sensitivity [7].Dorner et al. found optimal states in the presence of photonlosses for an SLDO-based detection strategy—in other words,states with the best QCRB under certain lossy conditions [8].Meanwhile, Pezzı and Smerzi revived Caves’ original schemeof mixing the coherent state with the squeezed-vacuum stateat the input of the MZI [1], but now with photon numbercounting. They showed that the CRB of the scheme, whenthe two states are mixed in equal proportions (n/2 photonson average in each state), reaches the HL independently ofthe actual value of phase [9]. Later, Hofmann derived a classof pure states that achieve their QCRB with photon numbercounting independently of the actual value of the phase—called path-symmetric states [10].

Apart from number counting and the SLDO, a detectionstrategy based on the measurement of parity of photon numberin one of the output modes of the MZI, described by theoperator � = (−1)n, where n is the number operator for themode, has been widely studied [11]. Parity detection wasfirst proposed by Bollinger et al. for enhanced frequencymeasurement with an entangled state of trapped ions [12].Gerry and Campos later applied it to optical interferometrywith the NOON state (|N,0〉 + e−iNφ|0,N〉)/√2 for achievingphase sensitivity at the HL [13]. Parity detection achievessub–shot-noise-limit phase sensitivities with various inputs[14]. Parity was recently shown to reach the QCRB fortwo-mode squeezed-vacuum light input and coherent mixedwith squeezed-vacuum light input [15,16].

A theoretical question of interest is whether parity detectionis capable of achieving phase sensitivity at the QCRB forall two-mode quantum states. In this paper, we attempt toanswer this question. We study parity detection for the classof path-symmetric pure states, for which number countingis known to achieve the QCRB. The path-symmetric statespossess a certain kind of coherence, i.e., phase relation betweenprobability amplitudes, as given in Eq. (15). We show thatparity detection attains the QCRB of all path-symmetric stateswhose coherences further satisfy the condition of Eq. (31). Allsingle-mode inputs to the MZI result in path-symmetric statesthat satisfy Eq. (31) after passing through the first 50:50 beamsplitter. The NOON state and the states that result inside the

043833-11050-2947/2013/87(4)/043833(6) ©2013 American Physical Society

SESHADREESAN, KIM, DOWLING, AND LEE PHYSICAL REVIEW A 87, 043833 (2013)

ai

bi

af

bf

|ψ1 ψ2 ψ3 ψ4

BS BS

φ

FIG. 1. (Color online) An MZI with a two-mode input |ψ1〉,which, after the 50:50 beam-splitter and phase-shifter transformationsUBS = exp(−i π

2 Jy), Uφ = exp(−iφJz) and U†BS = exp(i π

2 Jy) (in thatorder), is denoted |ψ2〉, |ψ3〉 and |ψ4〉, at the respective stages.

MZI from inputs such as the twin-Fock state [3,13,17], theYuen state [18], the two-mode squeezed-vacuum state [15],the coherent state mixed with squeezed-vacuum state [9,16,19], and the pair-coherent state [20], which are capable ofHL phase sensitivity, are all path-symmetric states satisfyingEq. (31). Consequently, it suffices to measure the parity ofphoton number in one of the output modes alone, in place ofnumber counting in both the modes, in order to achieve theQCRB of those states.

The paper is organized as follows. In Sec. II, we presentthe mathematical framework required to identify optimalstates and observables for phase estimation in Mach-Zehnderinterferometry. In Sec. III, we revisit Hofmann’s treatmentof photon-number-counting-based detection strategies [10]and rederive the so-called path-symmetric states. In Sec. IV,we present phase estimation based on photon number paritymeasurement for Hofmann’s path-symmetric states. We thendiscuss some general pros and cons of parity detection inSec. V. Finally, we conclude with a summary in Sec. VI.

II. MATHEMATICAL FRAMEWORK

Consider a typical MZI with an unknown phase φ, as shownin Fig. 1. The quantum state, and its dynamics in such a device(i.e., its propagation through the various components of theMZI), can be described in the Schwinger representation [21].A two-mode N -photon state in this representation resides in thej = N/2 subspace of the angular momentum Hilbert space,with the angular momentum operators Jx , Jy , and Jz given interms of the mode operators a, a†, b, and b† as

Jx = 1

2(a†b + b†a), Jy = 1

2i(a†b − b†a),

(2)

Jz = 1

2(a†a − b†b).

The unitary phase evolution operator of Eq. (1) becomesUφ = exp(−iφJz), and the 50:50 beam-splitter transformationcan be chosen to be UBS = exp(−i π

2 Jy). Using the SU(2) al-gebra of the angular momentum operators, namely, [Jq ,Jr ] =iJsεqrs , where {q,r,s} ∈ {x,y,z} and ε is the antisymmetrictensor, and the Baker-Hausdorff lemma [22], the overallMZI transformation, UMZI = U

†BSUφUBS, can be shown to be

UMZI = exp(−iφJx).

Our goal is to estimate the unknown phase φ. The error inthe estimate based on the measurement of an observable O

can be written as

δφ = |�O/(∂〈O〉/∂φ)|, (3)

where 〈O〉 is the expectation value of O, and �O theuncertainty in the measurement. Throughout this paper, weconsider observables O acting on state |ψ3〉 in Fig. 1. Givenan observable A acting on the state |ψ4〉 at the output ofthe MZI, the corresponding observable O is related to A

via the transformation of the second beam splitter as O =exp(i π

2 Jy)Aexp(−i π2 Jy).

Based on the Heisenberg equation of motion for operators,the expectation value of observable O with respect to state|ψ3〉 satisfies

∂

∂φ〈O〉 = −i〈[O,Jz]〉. (4)

(Note that the commutator involves Jz, which is the generatorof phase evolution between the two beam splitters) Accordingto the uncertainty principle [23], we have

�O�Jz � 1

2|〈[O,Jz]〉| = 1

2

∣∣∣∣ ∂

∂φ〈O〉

∣∣∣∣ . (5)

Consequently, the error in the estimate δφ obeys

δφ = |�O/(∂〈O〉/∂φ)| � 1

2�Jz

. (6)

For pure quantum states |ψ3〉, the right-hand side of the aboveinequality is a tight bound and is identically equal to theQCRB. The equivalent (necessary and sufficient) conditionon an observable O and state |ψ3〉 for achieving the bound isexactly the same as the condition for equality in Eq. (5), givenby

O|ψ3〉 = iλJz|ψ3〉, (7)

for any nonzero λ ∈ R, where O = O − 〈O〉I , Jz = Jz −〈Jz〉I , and I is the (2j + 1) × (2j + 1) identity operator [23].(Note that O and Jz are also Hermitian operators.)

First, we discuss the implications of the condition in Eq. (7)for number-counting-based detection strategies, as done byHofmann [10].

III. PHOTON NUMBER COUNTING REVISITED

From Eq. (2), we know that the photon number differenceoperator na − nb is the Jz operator in the Schwinger represen-tation (up to a factor of 1/2). More generally, observables basedon photon number counting, in the Schwinger representation,are Hermitian operators whose eigenkets are the same as thoseof the Jz operator. In other words, they are diagonal in theJz eigenbasis. Consequently, if an observable A acting onstate |ψ4〉 in Fig. 1 is based on number counting, then thecorresponding observable O (and O) acting on |ψ3〉 mustbe diagonal in the eigenbasis of the Jx operator. This isbecause the eigenkets of the Jz operator transform into theeigenkets of the Jx operator under the action of the secondbeam splitter [24]:

exp

(iπ

2Jy

)Jzexp

(− i

π

2Jy

)= Jx . (8)

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For such a number-counting-based observable O acting onthe state |ψ3〉, multiplying both sides of Eq. (7) by an eigenbraof Jx , 〈mx |, we get

om〈mx |ψ3〉 = iλ〈mx |Jz|ψ3〉, (9)

where om is the eigenvalue of O satisfying

O|mx〉 = om|mx〉. (10)

Rearranging the terms of Eq. (9), we obtain

om

λ= i

〈mx |Jz|ψ3〉〈mx |ψ3〉 . (11)

Since om and λ are purely real numbers, 〈mx |Jz|ψ3〉/〈mx |ψ3〉has to be purely imaginary in order for the state |ψ3〉 to satisfyEq. (11).

On the right-hand side of Eq. (11), by introducing the iden-tity operator I = ∑

|m〉 |m〉〈m|, where {|m〉} is the eigenbasis

of the Jx operator, we can rewrite 〈mx |Jz|ψ3〉/〈mx |ψ3〉 as

〈mx |Jz|ψ3〉〈mx |ψ3〉 =

∑|m〉

〈mx |Jz|m〉〈m|ψ3〉〈mx |ψ3〉 . (12)

Owing to the cyclic property of the commutation relationbetween the angular momentum operators, the matrix elementsof the Jz operator in the Jx basis are purely imaginary (justlike the matrix elements of the Jy operator in the Jz basis arepurely imaginary). This, coupled to the fact that Jz and Jz areHermitian operators, implies that all the off-diagonal entries ofJz are purely imaginary numbers, and all its diagonal entriesidentically equal to −〈Jz〉. Therefore, Eq. (12) reduces to

〈mx |Jz|ψ3〉〈mx |ψ3〉 =

∑|m〉�=|mx 〉

〈mx |Jz|m〉 〈m|ψ3〉〈mx |ψ3〉 − 〈Jz〉I , (13)

where 〈mx |Jz|m〉, |m〉 �= |mx〉 are all purely imaginary num-bers. One can see that states with (a) purely real coefficientsin the Jx basis (up to a global phase), i.e.,

〈mx |ψ3〉 = 〈mx |ψ3〉∗e−i2χ , ∀ mx ∈ {−j, . . . , + j}, (14)

and (b) 〈Jz〉 = 0 result in a purely imaginary quantity on theright-hand side of Eq. (13). [Note that the condition in Eq. (14)implies 〈m|ψ3〉/〈mx |ψ3〉 ∈ R ∀ {m, mx} ∈ {−j, . . . , + j}.]Such states are therefore capable of reaching their maximalphase sensitivities at the QCRB with number-counting-baseddetection strategies. Hofmann [10] identified the conditionin Eq. (14), and 〈Jz〉 = 0, with a symmetry property inthe Heisenberg picture, namely, J = {Jx, Jy, Jz} ≡ J′ ={Jx, Jy, − Jz}. He called such states path-symmetric states,since the operation Jz → −Jz corresponds to an exchange ofpaths (modes) in the Schwinger representation. (It is, however,an unphysical exchange of paths, since Jx and Jy remainunchanged.)

The above conditions of path symmetry, when transformedinto the eigenbasis of the Jz operator, yield

〈mz|ψ3〉 = 〈−mz|ψ3〉∗e−i2χ , ∀ mz ∈ {−j, . . . , + j} (15)

[10]. It is easy to verify that states |ψ3〉 that obeyEq. (15) implicitly satisfy 〈Jz〉 = 0. Also, if |ψ2〉 =∑+j

mz=−j cm|mz〉, and hence |ψ3〉 = ∑+j

mz=−j cme−imzφ|m〉,

then 〈mz|ψ3〉/〈−mz|ψ3〉∗ = e−i2χ obviously implies〈mz|ψ2〉/〈−mz|ψ2〉∗ also equals e−i2χ . In other words, thecondition of Eq. (15) is satisfied independently of the valueof the unknown phase φ. Thus, the path-symmetric states ofEq. (15) are capable of reaching their QCRB with numbercounting independently of the actual value of phase φ.

Next we consider a detection strategy based on photon num-ber parity measurement, in particular, for the path-symmetricstates of Eq. (15).

IV. PARITY DETECTION

The parity operator for the mode bf of Fig. 1 is givenby � = (−1)nb . In the Schwinger representation, it can bewritten as � = (−1)j−Jz . The observable O acting on state|ψ3〉 corresponding to parity measurement on |ψ4〉, which wewill call Q, is given by

Q = exp

(iπ

2Jy

)(−1)j−Jz exp

(− i

π

2Jy

)

= (−1)j exp

(iπ

2Jy

)exp(−iπJz)exp

(− i

π

2Jy

)= (−1)j exp(−iπJx). (16)

When expanded in the eigenbasis of the Jz operator, {|mz〉},the Q operator takes the form [25]

Q = (−1)2j∑|mz〉

|mz〉〈−mz|. (17)

Based on Eq. (7), in order to achieve the QCRB, the parityobservable Q acting on a path-symmetric state |ψ3〉 mustsatisfy

Q|ψ3〉 = iλJz|ψ3〉 (18)

for some nonzero λ ∈ R, where Q = Q − 〈Q〉I , and I is the(2j + 1) × (2j + 1) identity operator. (Note that 〈Jz〉 has beenmade 0 since |ψ3〉 is assumed to be path symmetric.) Multiply-ing throughout by the identity operator I = ∑

|mz〉 |mz〉〈mz|,we can rewrite Eq. (18) as∑

|mz〉|mz〉〈mz|(Q − iλJz)|ψ3〉 = 0

⇒∑|mz〉

(〈mz|Q|ψ3〉 − 〈Q〉〈mz|ψ3〉

− iλmz〈mz|ψ3〉)|mz〉 = 0. (19)

Dividing Eq. (19) throughout by 〈mz|ψ3〉 (assuming them tobe nonzero without loss of generality), we obtain

∑|mz〉

( 〈mz|Q|ψ3〉〈mz|ψ3〉 − 〈Q〉 − iλmz

)|mz〉 = 0. (20)

However, since∑+j

mz=−j mz = 0, independently of the value

of the real number λ,∑+j

mz=−j iλmz = 0. Therefore, Eq. (20)further reduces to∑

|mz〉

( 〈mz|Q|ψ3〉〈mz|ψ3〉 − 〈Q〉

)|mz〉 = 0. (21)

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SESHADREESAN, KIM, DOWLING, AND LEE PHYSICAL REVIEW A 87, 043833 (2013)

Using 〈mz|ψ3〉 = cme−imzφ , and the form in Eq. (17) for theQ operator, Eq. (21) can be rewritten as∑

|mz〉(S − 〈Q〉)|mz〉 = 0, (22)

where S is a function of mz, given by

S = (−1)2j c−m

cm

ei2mzφ. (23)

Let cm = rmeiθm , where rm and θm are real. Then, based on thepath-symmetry condition of Eq. (15), it is easy to show thatc−m

cm= e−2i(θm+χ ), and therefore, we rewrite

S = (−1)2j ei2(mzφ−χ−θm) ≡ S ′ + iS ′′, (24)

where S ′ and S ′′ are the real and imaginary parts of S,respectively. (Note that |S|2 = 1 for all mz.)

Meanwhile, the expectation of the Q operator with state|ψ3〉, 〈Q〉, can be written as

〈Q〉 = (−1)2j

j∑m=−j

c−mc∗mei2mzφ. (25)

We now provide a sufficient condition. Let us sup-pose that S ′ = ±1 for all mz ∈ {−j, . . . , + j} for someφ. Then S ′′ = 0 since S ′2 + S ′′2 = |(−1)2j ei2(mzφ−χ−θm)|2 =1. In Eq. (23), this implies S|cm|2 = (−1)2j c−mc∗

mei2mzφ =S ′|cm|2 = ±|cm|2, which subsequently, in Eq. (25), implies〈Q〉 = ±∑j

m=−j |cm|2 = ±1. Thus, if S ′ = ±1 for all mz ∈{−j, . . . , + j} for some φ, then in Eq. (22), S − 〈Q〉 isidentically equal to 0 for all mz. Hence, Eq. (22) is satisfied, andthe QCRB achieved. Therefore, S ′ = ±1 ∀ mz ∈ {−j, . . . , +j} for some φ is a sufficient condition on a path-symmetricstate of Eq. (15) to reach the QCRB with photon number paritymeasurement.

We now prove that S ′ = ±1 ∀ mz ∈ {−j, . . . , + j} forsome φ is also a necessary condition. Let us assume thatparity detection achieves phase sensitivity at the QCRBat some phase φ for a path-symmetric state |ψ3〉; i.e., inEq. (22),

S ′ + iS ′′ − 〈Q〉 = 0 (26)

for all mz at that value of φ. Since Q is a Hermitian operator,we have

S ′ = 〈Q〉, S ′′ = 0. (27)

Further, since |S|2 = 1, the above equation implies

S ′ = 〈Q〉 = ±1 (28)

at that value of phase φ. This proves that S ′ = ±1 ∀ mz ∈{−j, . . . , + j} for some φ is also a necessary condition ona path-symmetric state |ψ3〉 to reach its QCRB with paritydetection. Hence, we conclude that the phase sensitivity of a2j -photon state,

|ψ3〉 =j∑

mz=−j

rmei(θm−mzφ)|mz〉, (29)

which is path symmetric according to Eq. (15), reaches theQCRB with parity detection if, and only if, there exists a real

number φ such that ∀ mz ∈ {−j, . . . , + j},Re((−1)2j ei2(mzφ−χ−θm)) = ±1. (30)

Such a real number φ, obviously, is not expected to exist forany arbitrary path-symmetric state. It exists if, and only if,the path-symmetric state |ψ3〉 [of the form given in Eq. (29)]satisfies

θm = mzβ − χ − nm

π

2, n−m = −nm,

(31)∀ mz ∈ {−j, . . . , + j},

where nm ∈ Z, β ∈ R, and its value is given by φ = φ0 + β,where

φ0 ={

0 or π if 2j is an odd integer,

0 or π2 or π or 3π

2 if 2j is an even integer.

(32)

The linear dependence of θm on mz can be understood as aconsequence of linear interferometry [Eq. (1)], wherein theacquired phase is always proportional to the difference inphoton numbers of the two modes. Proof for the necessityand sufficiency of the condition is provided in the Appendix.The result extends to states with indefinite photon numberstoo. If the different 2j -photon components of a state satisfyEq. (15), and also Eq. (31) for the same value of β, then parityattains the QCRB of the state at φ = (2l + 1)π/2 + β, l ∈ Z,where the values of φ0 in Eq. (32) common to both evenand odd photon numbers have been identified. Interestingly,all two-mode states of definite and indefinite photon numberthus far considered for sub–shot-noise interferometry, withoutexception, are path symmetric, satisfying Eq. (31) after passingthrough the first 50:50 beam splitter [26].

V. DISCUSSION

Since parity detection achieves maximal phase sensitivitiesat particular values of phase φ, its applicability, in general,is restricted to estimating “local” phases.1 Local parameterestimation is concerned with detecting small changes inparameters that are more or less known, as opposed to the“global” one, wherein a complete lack of knowledge about theparameter is initially assumed [27]. It is assumed that we havecomplete control over the initial phase of the interferometer,which is tuned to an optimal bias phase or “sweet spot,”φ = φ0 + β, where β, as discussed, can be determined a prioribased on the quantum state input to the interferometer, andφ0 is given by Eq. (32). At this optimal operating point, ourscheme can detect very tiny changes in phase with sensitivityat the QCRB of the quantum state used. Potential applicationsinclude quantum sensing and imaging.

After all, number counting reveals all possible informationabout the output state and leads to phase estimation at the

1The NOON state is an exception since it satisfies Eq. (30)independently of the value of φ. Parity for the NOON state alone,is hence optimal at all values of phase φ.

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QCRB for all values of phase (global phase estimation). Paritydetection, on the other hand, contains only partial informationabout the state (not the number, but the parity only) and leadsto phase estimation at the QCRB for some particular valuesof phase (local phase estimation). The important advantage ofparity detection over number counting is that there is no needfor any pre or post data processing. Number-counting-basedstrategies typically work via the construction of the likelihoodfunction ahead of every detection event based on the condi-tional probability distribution of phase conditioned on the pre-vious detection outcome [3,9]. In general, after a sequence ofdetection events, the error in the phase estimate is determinedby the variance of the likelihood function, σ 2 = 1/(MFC),where M is the number of measurements and FC is the classicalFisher information. The classical Fisher information is givenby [6]

FC =∑

i

1

P (i|φ)

(dP (i|φ)

dφ

)2

, (33)

where i represents the outcome of the measurement and P (i|φ)is the probability of the measurement resulting in the ithoutcome conditioned on a specific value of phase φ. Forparity measurement described by the operator � = (−1)n,there are only two outcomes, + for even and − for odd, suchthat

P (+|φ) + P (−|φ) = 1,(34)

(+1)P (+|φ) + (−1)P (−|φ) = 〈�〉.Also, �2 = 1, and therefore,

��2 = 1 − 〈�〉2 = 4P (+|φ)P (−|φ),(35)

dP (+|φ)

dφ= 1

2

d〈�〉dφ

= −dP (−|φ)

dφ.

From Eqs. (33)–(35) we can see that the classical Fisherinformation for parity measurement is

FC =⎛⎝��2

/ ∣∣∣∣∣d〈�〉dφ

∣∣∣∣∣2⎞⎠

−1

=

⎛⎜⎝1 − 〈�〉2∣∣∣ d〈�〉

dφ

∣∣∣2

⎞⎟⎠

−1

. (36)

Hence, the error in the estimate based on parity measurement,or, in other words, the phase sensitivity of parity measurement,can be directly determined from the average signal 〈�〉without the need for any pre or post data processing asis required with photon counting. The major drawback ofparity detection, however, is that its performance is highlysusceptible to photon losses. Thus, it becomes very crucialto maintain lossless conditions in order to apply paritydetection.

One way to measure parity is obviously to performnumber counting at the output and to infer the parity fromit. High-precision number-resolving detection of photons atthe single-photon level has been demonstrated experimentallywith superconducting transition-edge sensors [28]. However,inferring parity via number counting would necessitate postprocessing of measurement data, negating the advantage of

parity detection. Hence, efforts are being made to devisetechniques that measure parity directly. Gerry and co-workerssuggested the use of optical nonlinearities [13]. Plick et al.showed that homodyne quantum state tomography can be usedto construct the expected parity signal, at least in the caseof Gaussian states, since the expectation value of the parityoperator is proportional to the value of the Wigner function ofthe state at the origin in phase space for such states [29]. Thedirect implementation of parity measurement continues to bean area of ongoing research.

VI. SUMMARY

In conclusion, we have shown that the detection strategybased on the measurement of the parity of photon numberachieves maximal phase sensitivity at the quantum CRB forall pure states that are path symmetric satisfying Eq. (31).All two-mode inputs of definite and indefinite photon numberthus far considered for sub–shot-noise interferometry resultin such path symmetric states after passing through the first50:50 beam splitter. Therefore, parity detection could beused with all of them to achieve optimal phase sensitivities.The advantage of parity measurement over photon countingis that the value of the unknown phase can be estimatedwithout the need for any post data processing of measurementdata. However, phase estimation at the QCRB with paritydetection is only possible in the vicinity of particular valuesof phase (local phase estimation), and only under losslessconditions.

ACKNOWLEDGMENTS

The authors wish to thank P. M. Anisimov for stimulatingdiscussions. This work was supported by the National ScienceFoundation.

APPENDIX

We now prove that Eq. (31) is a necessary and sufficientcondition on a path-symmetric state |ψ3〉 for parity detectionto achieve the QCRB of the state. First, it is easy to verifythat states of the form in Eq. (31) implicitly satisfy the pathsymmetry condition of Eq. (15). The necessity of the conditionin Eq. (31) can be proved as follows. Say Eq. (30) is satisfiedat some value of phase φ = β for a path-symmetric state|ψ3〉. Then cos 2(mβ − χ − θm) = ±1 ∀ m ∈ {−j, . . . , +j}. This is equivalent to sin 2(mβ − χ − θm) = 0 ∀ m ∈{−j, . . . , + j}, which implies 2(mβ − χ − θm) = nmπ ∀ m ∈{−j, . . . , + j}, where nm in general are different integers.Rearranging the terms, we obtain θm = mβ − χ − nm

π2 ∀ m ∈

{−j, . . . , + j}. However, the state being path symmetric,the θm values must satisfy Eq. (15), which necessitates thatθm = mβ − χ − nm

π2 , n−m = −nm, ∀ m ∈ {−j, . . . , + j}.

This is precisely the condition of Eq. (31). We next proveits sufficiency. Substituting Eq. (31) into Eq. (30), weobtain sin 2(m(φ − β) + nm

π2 ) = 0. Expanding the sine as

sin 2m(φ − β) cos nmπ + cos 2m(φ − β) sin nmπ , the aboveequation reduces to sin 2m(φ − β) = 0, whose solution isφ = φ0 + β for φ0 as given in Eq. (32).

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