PHYSICAL REVIEW A 92, 032317 (2015)

Quantum metrology with mixed states: When recovering lost information is betterthan never losing it

Simon A. HaineSchool of Mathematics and Physics, University of Queensland, Brisbane, Queensland, 4072, Australia

Stuart S. SzigetiSchool of Mathematics and Physics, University of Queensland, Brisbane, Queensland, 4072, Australia

and ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane, Queensland 4072, Australia(Received 10 July 2015; published 17 September 2015)

Quantum-enhanced metrology can be achieved by entangling a probe with an auxiliary system, passingthe probe through an interferometer, and subsequently making measurements on both the probe and auxiliarysystem. Conceptually, this corresponds to performing metrology with the purification of a (mixed) probe state.We demonstrate via the quantum Fisher information how to design mixed states whose purifications are anexcellent metrological resource. In particular, we give examples of mixed states with purifications that allow(near) Heisenberg-limited metrology and provide examples of entangling Hamiltonians that can generate thesestates. Finally, we present the optimal measurement and parameter-estimation procedure required to realize thesesensitivities (i.e., that saturate the quantum Cramer-Rao bound). Since pure states of comparable metrologicalusefulness are typically challenging to generate, it may prove easier to use this approach of entanglement andmeasurement of an auxiliary system. An example where this may be the case is atom interferometry, whereentanglement with optical systems is potentially easier to engineer than the atomic interactions required toproduce nonclassical atomic states.

DOI: 10.1103/PhysRevA.92.032317 PACS number(s): 03.67.−a, 42.50.Dv, 42.50.St, 03.75.Dg

I. INTRODUCTION

There is currently great interest in quantum metrology,the science of estimating a classical parameter φ with aquantum probe at a higher precision than is possible witha classical probe of identical particle flux. Given a fixednumber of particles, N , the ultimate limit to the sensitivity isthe Heisenberg limit �φ = 1/N [1,2]. Naıvely, the choice ofprobe state is a solved problem; for instance, symmetric Dickestates [1,3] and spin-cat states [4,5] input into a Mach-Zehnder(MZ) interferometer yield sensitivities of

√2/N and 1/N , re-

spectively. However, in practice achieving quantum-enhancedsensitivities is a significant challenge. This is due to both thedeleterious effect of losses [6] and the challenges associatedwith preparing nonclassical states with an appreciable numberof particles [7–11]. For example, protocols for generating aspin-cat state commonly require a large Kerr nonlinearity,which either is unavailable (e.g., in optical systems [12]), isdifficult to engineer (e.g., in microwave cavities [13,14]), ordetrimentally affects the efficient operation of the metrologicaldevice (e.g., atom interferometers where the interfering modeshave large space-time separation [15–17]).

In this paper, we present an alternative route to quantum-enhanced metrology based on purifications of mixed states.Physically, this involves entangling the probe with an aux-iliary system before the probe is affected by φ, makingmeasurements on both the probe and auxiliary system, andsubsequently using correlations between the two measurementoutcomes in order to reduce the uncertainty in the estimatedparameter (see Fig. 1). This approach is advantageous incases where it is easier to entangle the probe system withanother system, rather than directly create highly entangledstates of the probe system itself. An example of this isatom interferometry; although quantum squeezing can be

produced in atomic systems via atomic interactions [18–31],the technical requirements of high-sensitivity, path-separatedatom interferometers are better suited to enhancement viaentanglement with an optical system [32–37] and informationrecycling [38–41].

The structure of this paper is as follows. In Sec. IIwe introduce in detail the central idea of this paper: thatpurifications of mixed states can possess a large quantumFisher information (QFI) and therefore represent an excellentresource for quantum metrology. In Sec. III we specialize to anN -boson probe state and Mach-Zehnder (MZ) interferometer,and show how to engineer purifications that yield sensitivitiesat and near the Heisenberg limit. Finally, in Sec. IV we presentoptimal measurement schemes that allow these quantum-enhanced sensitivities to be achieved in practice.

II. QUANTUM FISHER INFORMATIONFOR A PURIFICATION

We can determine the best sensitivity possible for anygiven metrology scheme via the QFI, F , which places anabsolute lower bound on the sensitivity, �φ � 1/

√F , called

the quantum Cramer-Rao bound (QCRB) [42–45]. This boundis independent of the choice of measurement and parameterestimation procedure, and depends only on the input state.Explicitly, if a state ρA is input into a metrological devicedescribed by the unitary operator Uφ = exp(−iφGA), then theQFI is

FA ≡ F[GA,ρA] = 2∑i,j

(λi − λj )2

λi + λj

|〈ei |GA|ej 〉|2, (1)

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SIMON A. HAINE AND STUART S. SZIGETI PHYSICAL REVIEW A 92, 032317 (2015)

FIG. 1. The unitary UAB = exp(−iHABt/�) entangles systemA (probe) with system B (auxiliary) before system A passesthrough a measurement device described by Uφ = exp(−iφGA).If measurements are restricted to system A, then the QFI for anestimate of φ is FA = F[GA,ρA], where ρA = TrB{|�AB〉〈�AB |}.If measurements on both systems are permitted, then the QFI isFAB = F[GA,|�AB〉] = 4Var(GA)ρA

� FA.

where λi and |ei〉 are the eigenvalues and eigenvectors ofρA, respectively. If ρA is pure, then Eq. (1) reduces toFA = 4Var(GA).

A naıve consideration of the pure state QFI suggests thatengineering input states with a large variance in GA is anexcellent strategy for achieving a high-precision estimate ofφ. However, there are many operations on ρA that increaseVar(GA) at the expense of a decrease to the purity γ = Tr{ρ2

A}.Since the QFI is convex in the state, any process that mixesthe state typically decreases the QFI. Consequently, anyimprovement due to a larger Var(GA) is usually overwhelmedby reductions in the QFI due to mixing.

In order to concretely demonstrate this point, we focus on anN -boson state input into a MZ interferometer. As discussed inRef. [46], this system is conveniently described by the SU(2)Lie algebra [Ji ,Jj ] = iεijkJk , where εijk is the Levi-Civitasymbol, for i = x,y,z. A MZ interferometer is characterizedby GA = Jy ; therefore, for pure states, a large QFI requires alarge Var(Jy).

Without loss of generality, we restrict ourselves to the classof input states

ρA =∫ 2π

0dϕ P(ϕ)

∣∣∣∣α(

π

2,ϕ

)⟩⟨α

(π

2,ϕ

)∣∣∣∣. (2)

Here |α(θ,ϕ)〉 = exp(−iϕJz) exp(−iθ Jy)|j,j 〉 are spin coher-ent states, where |j,m〉 are Dicke states with total angularmomentum j = N/2 and Jz projection m. We focus on thefollowing three states in class (2), which are in order ofincreasing Var(Jy):

Case I: P(ϕ) = δ(ϕ), (3a)

Case II: P(ϕ) = 1

2π, (3b)

Case III: P(ϕ) = 1

2

[δ

(ϕ − π

2

)+ δ

(ϕ + π

2

)]. (3c)

These states can be conveniently visualized by plotting theHusimi-Q function [47,48]

Q(θ,ϕ) = 2j + 1

4π〈α(θ,ϕ)|ρA|α(θ,ϕ)〉 , (4)

and the Jy projection of the state, P (Jy) = 〈Jy |ρA|Jy〉, whereJy |Jy〉 = Jy |Jy〉 (see Fig. 2).

None of these states yield sensitivities that surpass thestandard quantum limit (SQL), �φ = 1/

√N . In case I, ρA is

a pure spin coherent state, |α(π/2,0)〉, with FA = 4Var(Jy) =

FIG. 2. (Color online) Husimi-Q function for case I (a), case II(b), and case III (c). The projection in the Jy basis, P (Jy) is shownfor case I (d), case II (e), and case III (f). N = 20 for all frames.

N . In case II, ρA is an incoherent mixture of Dicke states (i.e.,it contains no off-diagonal terms in the |j,m〉 basis). Although4Var(Jy) = N (N + 1)/2 is much larger than for case I, the QFIis only FA = N/2. Finally, case III is an incoherent mixtureof maximal and minimal Jy eigenstates with 4Var(Jy) = N2,which is the maximum possible value in SU(2). However, sincethe state is mixed the QFI is significantly less than this, withFA = N/2.

However, suppose the mixing in ρA arises from entan-glement with an auxiliary system B before system A passesthrough the metrological device (see Fig. 1). Specifically, foran input pure state |�AB〉 of a composite system A ⊗ B, whereρA = TrB{|�AB〉〈�AB |}, the QFI is

FAB ≡ F[GA,|�AB〉]= 4

(〈�AB |G2A|�AB〉 − 〈�AB |GA|�AB〉2

)= 4

(TrA[G2

AρA] − TrA[GAρA]2)

≡ 4Var(GA)ρA. (5)

Consequently, for a purification of ρA the QFI only dependson the variance in GA of ρA [41,49]. Our naıve strategy ofpreparing a state with large Var(GA) irrespective of its purityis now an excellent approach. Indeed, in this situation the statesin cases I–III are now also arranged in order of increasing QFI,with cases II and III attaining a QFI of N (N + 1)/2 and N2,respectively. It is interesting to note that the QFI for caseIII is the maximum allowable for N particles in SU(2) [50]and is usually obtained via the difficult to generate spin-catstate, which is a macroscopic superposition, rather than aclassical mixture, of spin coherent states. Note also that FAB

is independent of any particular purification, and convexityimplies that FAB � FA. That is, in principle any purificationof ρA is capable of achieving sensitivities at least as good as,and usually much better than, ρA itself.

Quantum metrology with purifications is not simply amathematical “trick”; physically, a purification corresponds toentangling the probe system A with some auxiliary system B,and permitting measurements on both systems [51]. Therefore,the practical utility of our proposal depends crucially on theexistence of an entangling Hamiltonian that can prepare ρA ina state with large Var(GA)ρA

.

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QUANTUM METROLOGY WITH MIXED STATES: WHEN . . . PHYSICAL REVIEW A 92, 032317 (2015)

For the three cases described by Eqs. (2) and (3), apurification of ρA can be written as

|�AB〉 =j∑

m=−j

cm|j,m〉 ⊗ |Bm〉, (6)

with case I corresponding to 〈Bm|Bn〉 = 1, case II corre-sponding to 〈Bm|Bn〉 = δn,m, and case III corresponding to〈Bm|Bn〉 = 1(0) for |n − m| even (odd). In the followingsection, we present a simple scheme that converts a shot-noiselimited spin coherent state [such as case I] to the enhancedQFI purifications of cases II and III.

III. EXAMPLE ENTANGLING DYNAMICS LEADINGTO INCREASED QFI

Consider again the N -boson probe state (system A) inputinto a MZ interferometer (i.e., GA = Jy). The QFI for apurification of ρA can be written as

FAB = 4(⟨J 2

y

⟩ − 〈Jy〉2) = F0 + F1 + F2, (7)

with

F0 = N

2(N + 2) − 2

⟨J 2

z

⟩, (8a)

F1 = −〈i(J+ − J−)〉2, (8b)

F2 = −〈J 2+ + J 2

−〉, (8c)

where J± = Jx ± iJy . Note that F0,F1, and F2 depend onlyon the matrix elements of ρA with |n − m| equal to 0,1, and2, in the Jz basis; writing FAB in this form is very convenientfor what follows.

Before the interferometer, we assume the probe is coupledto some auxiliary system B via the Hamiltonian

HAB = �gJzHB. (9)

When system B is a photon field and HB is proportional to thenumber of photons in the field, then HAB describes the weakprobing of the population difference of an ensemble of two-level atoms with far-detuned light [37,52–61], or dispersivecoupling between a microwave cavity and a superconductingqubit [62–64]. We will explore this specific case shortly;however, for now we keep HB completely general. If the initialsystem state is a product state |�AB(0)〉 = |�A〉 ⊗ |�B〉, aftersome evolution time the state of the system will be given byEq. (6) with cm = 〈m|�A〉 and |Bm〉 = exp(−imgtHB)|�B〉.The reduced density operator of system A is then

ρA =j∑

n,m=−j

cnc∗mCn−m|j,n〉〈j,m|, (10)

where the coherence of system A is determined via

Cn−m = 〈Bm|Bn〉 = 〈�B |e−i(n−m)gtHB |�B〉 . (11)

When Cn−m = 1, the system remains separable and system A

is a pure state, whereas if Cn−m = δn,m then ρA is an incoherentmixture of Dicke states.

Using Eq. (10), F0,F1, and F2 can be written as

F0 = N

2(N + 2) − 2

⟨J 2

z

⟩, (12a)

F1 = −〈i(C1J+ − C∗1 J−)〉2, (12b)

F2 = −(C2〈J 2+〉 + C∗

2 〈J 2−〉), (12c)

where the above expectation values are calculated with respectto |�A〉. The effect of the entanglement between systems A andB is entirely encoded in the coherences C1 and C2; coherencesgreater than second order do not affect the QFI.

Let us consider the effect on the QFI of each term in Eq. (7).F0 is independent of the entanglement between systems A andB, and will be of order N2/2 if |�A〉 has 〈J 2

z 〉 ∼ N (e.g., thespin coherent state |α(π/2,ϕ)〉 has 〈J 2

z 〉 = N/4). This suggeststhat a sufficient condition for Heisenberg scaling isF1 ∼ F2 ∼〈J 2

z 〉 ∼ N . In fact, since F1 � 0, the maximum QFI state mustnecessarily have C1 = 0. In contrast, F2 can be positive ornegative, in which case a state with C2 = 0 and another statewith C2 = 1 and F2 ∼ +N2/2 might both be capable of (near)Heisenberg-limited metrology. We consider examples of bothstates below.

A. Case II: Example dynamics yielding FAB � N2/2

To concretely illustrate the increased QFI a purification ofρA can provide, we assume system B is a single bosonic mode,described by annihilation operator b, and take HB = b†b suchthat

HAB = �gJzb†b. (13)

If the initial state of system B is a Glauber coherent state|β〉 [65], then the coherences described by Eq. (11) simplifyto

Cn−m = exp[−|β|2(1 − e−i(n−m)gt )]. (14)

|Cn−m|2 decays on a time scale gt ∼ [(n − m)|β|2]−1. Al-though the nonorthogonality of 〈β|βeiθ 〉 ensures that Cn−m

never actually reaches zero, it becomes very small for evenmodest values of |β|2.

If the initial condition of system A is |�A〉 = |α(θ,φ)〉, thenFAB has the simple analytic form given by (see Appendix)

F0 = N

(1 + (N − 1)

2sin2 θ

), (15a)

F1 = −N2 sin2 θ sin2(|β|2 sin(gt) + φ)e−4|β|2 sin2(gt/2), (15b)

F2 = N (1 − N )

2sin2 θ cos(|β|2 sin(2gt) + 2φ)e−2|β|2 sin2(gt).

(15c)

In contrast, calculating FA via Eq. (1) requires the diagonal-ization of ρA, which must be performed numerically.

We first demonstrate the effect of vanishing first- andsecond-order coherence on FAB by preparing system A inthe maximal Jx eigenstate, |α(π/2,0)〉, with N = 100, anda Glauber coherent state for system B with average particlenumber |β|2 = 500. The initial state for system A is preciselycase I [see Eq. (3a)] and has a QFI of N . As shown in Fig. 3,

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SIMON A. HAINE AND STUART S. SZIGETI PHYSICAL REVIEW A 92, 032317 (2015)

FIG. 3. (Color online) Time snapshots of the Husimi-Q functionand Jy projection illustrating the evolution of a maximal Jx eigenstateunder entangling Hamiltonian Eq. (13). The snapshots were chosento correspond to times when the rotation around the Jz axis is suchthat 〈Jy〉 = 0, which roughly corresponds to the local maxima of FAB

in Fig. 4(c). (Parameters: N = 100,|β|2 = 500).

under the evolution of Eq. (13), ρA tends towards an incoherentmixture of Dicke states [case II], with the correspondingbroadening of the P (Jy) distribution.

Figure 4(a) shows that both coherences C1 and C2 rapidlyapproach zero, which causes F1 and F2 to vanish [seeFig. 4(b)]. Consequently, FAB approaches F0 = N (N + 1)/2,which allows a phase sensitivity of approximately

√2×

Heisenberg limit [see Fig. 4(c)]. In contrast, the effect of themixing causes the QFI of ρA itself to decrease from N toFA = N/2, with FA � N for all t . This remains true evenif GA is rotated to lie in an arbitrary direction on the Blochsphere.

The oscillations in F1 and F2 (and consequently FA andFAB) before the plateau are due to the complex rotationof C1 and C2, which causes rotations of ρA around theJz axis before being overwhelmed by the overall decay inmagnitude. Furthermore, although the purity of the state also

0

0.5

1

−10000

−5000

0

5000

0 0.05 0.1 0.1510

0

102

104

QF

I

gt

(a)

(b)

(c)

FIG. 4. (Color online) Evolution of a maximal Jx eigenstateunder entangling Hamiltonian Eq. (13). (a) Coherences |C1|2 (solidline), |C2|2 (dashed line), and purity γ (dot-dashed line). (b) Threecomponents of FAB [see Eq. (7)]: F0 (dot-dashed line), F1 (solidline), and F2 (dashed line). (c) QFI for ρA,FA (dashed line) and apurification of ρA,FAB (solid line). For reference, we have included N

(dotted line) andF0 = N (N + 1)/2 ≈ N2/2 (dot-dashed line), whichcorrespond to phase sensitivities at the SQL and

√2× Heisenberg

limit, respectively. (Parameters: N = 100,|β|2 = 500.)

FIG. 5. (Color online) Husimi-Q function and Jy projection foran initial state |�AB (0)〉 = |α(π/2,π/2)〉 ⊗ |β〉 under the evolutionof Eq. (13) for different values of gt . The Q function is symmetricabout reflection of the Jy axis, resulting in part of the function beinghidden from view on the reverse side of the sphere. (Parameters:N = 100,|β|2 = 500.)

decays, it never vanishes, thereby illustrating that it is not theentanglement per se that is causing the QFI enhancement fora purification of ρA.

B. Case III: Example dynamics yielding FAB = N2

At gt = π , there is a revival in |Cn|2 for n even, but notfor n odd. Figure 5 shows the Husimi-Q function under theevolution of Eq. (13) for times close to gt = π , when the initialstate of system A is the maximal Jy eigenstate |α(π/2,π/2)〉.

The QFI is initially zero, but the decay of C1 and C2 rapidlyincreases to FAB = F0 = N (N + 1)/2 as in the previousexample. As gt → π , the revival of |C2|2 causes FAB to brieflyincrease to N2 (see Fig. 6). This is the Heisenberg limit, whichis the QFI of a (pure) spin-cat state and the maximum QFI forSU(2) [50]. At gt = π,ρA is identical to a classical mixture of|α(π/2,π/2)〉 and |α(π/2, − π/2)〉; however, itsQ function issimilar to that of a spin-cat state, and purifications of it behave

0

0.5

1

−5000

0

5000

0.97 0.98 0.99 1 1.01 1.02 1.030

5000

10000

QF

I

gt/π

(a)

(b)

(c)

FIG. 6. (Color online) Evolution of a maximal Jy eigenstate neargt = π under entangling Hamiltonian Eq. (13). (a) |C1|2 (solid line),|C2|2 (dashed line), and γ (dot-dashed line). (b) F0 (dot-dashed line),F1 (solid line), and F2 (dashed line). (c) FA (dashed line) and FAB

(solid line). For comparison, we have included F0 = N (N + 1)/2 ≈N 2/2 and the Heisenberg limit N2 (dot-dashed lines); FA � N forall t . (Parameters: N = 100,|β|2 = 500.)

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QUANTUM METROLOGY WITH MIXED STATES: WHEN . . . PHYSICAL REVIEW A 92, 032317 (2015)

as a spin-cat state for metrological purposes. For these reasons,we call this state a pseudo-spin-cat state.

C. Example dynamics for particle-exchange Hamiltonian

In the previous two examples the Jz projection was aconserved quantity, so any entanglement between systems A

and B can only degrade the coherence in the Jz basis of systemA (ultimately resulting in an enhanced QFI). The situation ismore complicated when considering a Hamiltonian that doesnot conserve the Jz projection, such as when a spin flip insystem A is correlated with the creation or annihilation ofa quantum in system B. Here, we encounter scenarios wherethe interaction can either create or destroy coherences in the Jz

basis of system A, and although a significant QFI enhancementis still possible, it depends upon the initial state of system B.

As a concrete illustration, consider the particle-exchangeHamiltonian

H± = �g(J±b† + J∓b), (16)

and assume that system A is initially prepared in the max-imal Jz eigenstate |�A〉 = |α(0,0)〉 = |j,j 〉 (note this hasVar(Jy) = N/4, and therefore a QFI of N ). Then H− and H+physically correspond to Raman superradiance [38,66] andquantum state transfer [34–37,39,40] processes, respectively.After some period of evolution, the combined state of systemsA ⊗ B takes the form of Eq. (6).

First, consider the case when the initial state for systemB is a large amplitude coherent state (i.e., |�B〉 = |β〉). Herethe addition or removal of a quantum to or from system B

has a minimal effect on the state and the system remainsapproximately separable, since |〈Bn|Bm〉|2 ≈ 1. It is thereforereasonable to make the undepleted pump approximation b →β, such that H± → �gβJx (assuming β is real). Hence, theeffect of the interaction is simply a rotation around the Jx

axis, which can create coherence in the Jz basis, and soFA = FAB � N for all time.

In the opposite limit where the initial state of system B is aFock state with NB particles, |�B〉 = |NB〉, then

|Bm〉 = |NB ± (m − j )〉, (17)

and 〈Bm|Bn〉 = δn,m. This ensures that the first- and second-order coherences vanish, and F1 = F2 = 0 for all time. Thatis, as illustrated in Fig. 7, the state moves towards the equator

FIG. 7. (Color online) Husimi-Q function and Jy projection foran initial state |�AB (0)〉 = |α(0,0)〉 ⊗ |NB〉 under the evolution ofH− for different values of gt . (Parameters: N = 100,NB = 20).

0 0.05 0.1 0.1510

1

102

103

104

QF

I

gt

FIG. 8. (Color online) FA (dashed line) and FAB (solid line) foran initial state |�AB (0)〉 = |α(0,0)〉 ⊗ |NB〉 under the evolution ofH−. We have indicated N and N2/2 with black dotted lines forcomparison. (Parameters: N = 100,NB = 20.)

and ultimately evolves to an incoherent Dicke mixture [i.e.,case II]. As described in Sec. III A, and shown in Fig. 8, theQFI increases to a maximum of approximately FAB ≈ N2/2.Although setting NB = 0 (i.e., a vacuum state) leads to a largervariance in Jz,FAB still reaches approximately 70% of N2/2.

We therefore see that for the Hamiltonian (16), a largeQFI enhancement is achieved provided the initial state|�B〉 has small number fluctuations. Compare this to theHamiltonian (13), where the choice |�B〉 = |NB〉 leads to noentanglement between systems A and B, while in contrast aninitial state with small phase fluctuations (and therefore largenumber fluctuations), such as a coherent state, causes rapiddecoherence in ρA.

IV. OPTIMAL MEASUREMENT SCHEMES

Although the QFI determines the optimum sensitivity for agiven initial state, it is silent on the question of how to achievethis optimum. It is therefore important to identify (a) whichmeasurements to make on each system and (b) a method ofcombining the outcomes of these measurements—which werefer to as a measurement signal (S)—that saturates the QCRB.We do this below for purifications of the incoherent Dickemixture (case II) and the pseudo-spin-cat state (case III).

A. Optimal measurements for incoherentDicke mixture (case II)

It is worthwhile briefly recounting the optimal estimationprocedure for a symmetric Dicke state |j,0〉 input into aMZ interferometer. A MZ interferometer rotates Jz accord-ing to Jz(φ) = U

†φJzUφ = cos φJz − sin φJx . Since symmet-

ric Dicke states satisfy 〈Jx〉 = 〈Jz〉 = 〈J 2z 〉 = 0 and 〈J 2

x 〉 =N (N + 2)/8, it is clear that the fluctuations in Jz(φ) containthe phase information, and therefore the quantity S = [Jz(φ)]2

oscillates between 0 and N (N + 2)/8. It can be shown thatat the operating point φ → 0,Var(S) → 0, and the quantity(�φ)2 → Var(S)/(∂φ〈S〉)2 = 1/FA, and therefore the signalS saturates the QCRB [3,67,68].

For an incoherent Dicke mixture, we have 〈Jx〉 = 〈Jy〉 =〈Jz〉 = 0, and 〈J 2

x 〉 = N (N + 1)/8. Unfortunately, the nonzerovariance in Jz (i.e., 〈J 2

z 〉 = N/4) implies that Var(S) � 0 forall φ, and the signal no longer saturates the QCRB. However,since the states |Bm〉 in the purification Eq. (6) are orthonormal,

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SIMON A. HAINE AND STUART S. SZIGETI PHYSICAL REVIEW A 92, 032317 (2015)

a projective measurement of some system B operator diagonalin the |Bm〉 basis projects system A into a Jz eigenstate (i.e., aDicke state). That is, these measurement outcomes on systemB are correlated with Jz measurement outcomes on system A.Therefore, subtracting both measurements yields a quantitywith very little quantum noise.

More precisely, if we can construct an operator SB onsystem B that is correlated with Jz measurements on system A

(i.e., SB |�AB〉 = Jz|�AB〉), then we can construct the quantityS0 = Jz − SB which has the property 〈S0〉 = 〈S2

0 〉 = 0. Thismotivates the signal choice

S = (U †φS0Uφ)2 = (cos φJz − sin φJx − SB)2. (18)

Using SB |�AB〉 = Jz|�AB〉 and the fact that non-Jz-conserving terms vanish due to the absence of off-diagonalterms in the Jz representation of ρA (e.g., expectation valueswith an odd power of Jx vanish), we can show that

〈S〉 = ⟨J 2

z

⟩(cos φ − 1)2 + ⟨

J 2x

⟩sin2 φ, (19a)

〈S2〉 = ⟨J 4

z

⟩(cos φ − 1)4 + ⟨

J 4x

⟩sin4φ

+ ⟨J 2

z J 2x + J 2

x J 2z + 4JzJx Jx Jz

⟩sin2φ(cos φ − 1)2

+ 2i〈(JzJx Jy − Jy Jx Jz)〉 sin2φ cos φ(cos φ − 1)

+ ⟨J 2

y

⟩cos2φ sin2φ. (19b)

Note that the above expectation values can be taken withrespect to |�AB〉 or ρA. The best sensitivity occurs at smalldisplacements around φ = 0. Taking the limit as φ → 0 andnoting that 〈J 2

x 〉 = 〈J 2y 〉 gives

(�φ)2 = Var(S)

(∂φ〈S〉)2

∣∣∣∣φ=0

= 1

4⟨J 2

x

⟩ = 1

4⟨J 2

y

⟩ = 1

FAB

. (20)

This demonstrates that the signal Eq. (18) is optimal since itsaturates the QCRB.

B. Optimal measurements for pseudo-spin-cat state (case III)

Pure spin-cat states have the maximum QFI possible forN particles in SU(2), are eigenstates of the parity operator,and indeed parity measurements saturate the QCRB [69].Pseudo-spin-cat states (case III) also have maximal QFI, andsince 〈Bn|Bm〉 = 1(0) for |n − m| even (odd), a projectivemeasurement of system B yields no information other than theparity of the Jz projection. This suggests that a measurementof parity could be optimal.

In analogy with case II, our aim is to construct an operatorS0 where the correlations between systems A and B lead toa reduction in Var(S0) and the system mimics a pure spin-catstate. Introducing the quantity

S0 = �ASB ≡ �A�B, (21)

where �A(B) is the parity operator for system A(B), definedby �A|j,m〉 = (−1)m|j,m〉 and �B |Bm〉 = (−1)m|Bm〉, wesee that pseudo-spin-cat states satisfy S0|�AB〉 = |�AB〉, andtherefore Var(S0) = 0. This motivates the signal choice S =U

†φS0Uφ .

To calculate the sensitivity, we need to compute 〈S〉 and〈S2〉. Trivially, 〈S2〉 = 1 for all states. For φ 1, expanding

Uφ to second order in φ gives

〈S〉 ≈ ⟨(1 + iφJy − 1

2φ2J 2y

)S0

(1 − iφJy − 1

2φ2J 2y

)⟩= 1 + iφ(〈Jy S0〉 − 〈S0Jy〉)

+φ2[〈Jy S0Jy〉 − 12

(⟨J 2

y S0⟩ + ⟨

S0J2y

⟩)] + O(φ3), (22)

The relation 〈Bn|Bn±1〉 = 0 ensures that terms linear in Jy goto zero:

〈J+〉 =∑m,n

cmc∗n〈j,n|J+|j,m〉〈Bn|Bm〉

∝∑n,m

cmc∗nδn,m+1〈Bn|Bm〉

=∑m

cmc∗m+1〈Bm+1|Bm〉 = 0 . (23)

However, unlike case II, the condition 〈Bn|Bn±2〉 = 1 pre-serves terms such as 〈J 2

+〉. Noting that Jy flips the parity ofany state in subsystem A but not subsystem B:

�AJy |�AB〉 = −Jy�A|�AB〉, (24a)

�BJy |�AB〉 = Jy�B |�AB〉, (24b)

and using S0|�AB〉 = |�AB〉 gives

〈Jy S0Jy〉 = −⟨S0J

2y

⟩ = −⟨J 2

y S0⟩ = −⟨

J 2y

⟩. (25)

Therefore

〈S〉 = 1 − 2φ2⟨J 2

y

⟩ + O(φ3) . (26)

Since S20 = 1 implies that S2 = 1, we obtain

Var(S) = 4φ2⟨J 2

y

⟩ + O(φ4), (27)

and consequently

(�φ)2 = Var(S)

(∂φ〈S〉)2= 4φ2

⟨J 2

y

⟩16φ2

⟨J 2

y

⟩2 = 1

4⟨J 2

y

⟩ = 1

FAB

. (28)

This demonstrates that the signal saturates the QCRB and istherefore optimal.

The optimal estimation schemes presented in Secs. IV Aand IV B illustrate a somewhat counterintuitive fact: Althoughthe optimal measurement of system B for a pseudo-spin-catstate provides less information about system A than for anincoherent Dicke mixture, the pseudo-spin-cat state yields thebetter (in fact best) sensitivity.

C. System B observables that approximate optimalmeasurements

We now turn to the explicit construction of physicalobservables that approximate SB . In general, the choice of SB

depends upon the specific purification of ρA. Physically, theinitial state of system B and the entangling Hamiltonian matter.However, there is no guarantee that SB exists, and if it doesthere is no guarantee that a measurement of this observablecan be made in practice. Nevertheless, as we show below, itmay be possible to make a measurement of an observablethat approximates SB and can therefore give near-optimalsensitivities.

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QUANTUM METROLOGY WITH MIXED STATES: WHEN . . . PHYSICAL REVIEW A 92, 032317 (2015)

1. Case II

To begin, consider the situation in Sec. III A: the evolutionof the state |α(π/2,0)〉 ⊗ |β〉 under the Hamiltonian (13). Werequire SB |�AB〉 = Jz|�AB〉. After some evolution time t ,

|�AB〉 =j∑

m=−j

cm|j,m〉|βe−imgt 〉 . (29)

Clearly, the phase of the coherent state is correlated with the Jz

projection of system B. This can be extracted via a homodynemeasurement of the phase quadrature YB = i(b − b†) [70]. Infact, provided mgt 1, phase quadrature measurements of|β exp(−imgt)〉 are linearly proportional to the Jz projection:

〈βe−imgt |YB |βe−imgt 〉 = 2β sin (mgt) ≈ 2βmgt, (30)

where without loss of generality we have taken β to be realand positive. Consequently, the scaled phase quadrature

SB = YB

2βgt(31)

satisfies

〈�AB |(Jz − SB)|�AB〉 ≈ 0, (32a)

〈�AB |(Jz − SB)2|�AB〉 ≈ 1

(2βgt)2, (32b)

and so the fluctuations in (Jz − SB) become arbitrarily small(and SB becomes perfectly correlated with Jz) as (βgt)2

becomes large. This suggests that Eqs. (18) and (31) shouldbe a good approximation to an optimal measurement signal.

More precisely, assume that

1

β Ngt

2 1. (33)

The first inequality ensures that 〈βe−ingt |βe−imgt 〉 ≈ δn,m andso ρA is approximately an incoherent Dicke mixture, while thesecond inequality implies that we are in the linearized regimewhere Eqs. (30) and (32) hold. Then the signal

Sapprox =(

Jz cos φ − Jx sin φ − YB

2gt

)2

(34)

yields the sensitivity

(�φ)2 ≈ 1

(∂φ〈S〉)2

{Var(S) − sin2(φ/2)

β2

⟨J 4

z

⟩ + 2

(2βgt)4

+ 4

(2βgt)2

[(cos φ − 1)2⟨J 2

z

⟩ + sin2 φ⟨J 2

x

⟩]}, (35)

where Var(S) and ∂φ〈S〉 are given by the expectations (19) ofthe optimal signal Eq. (18). Figure 9 shows Eq. (35) comparedto an exact numeric calculation.

Condition (33) typically ensures that the term proportionalto 〈J 4

z 〉 is small in comparison to the term proportional to1/(2βgt)2. We therefore see that our approximate signal Sapprox

gives a sensitivity worse than the QCRB, and furthermore atan operating point φ �= 0. Nevertheless, �φ approaches theQCRB at φ = 0 as β2 and (2βgt)2 approach infinity. Therefore,for a sufficiently large βgt , we can achieve near-optimal

−0.4 −0.2 0 0.2 0.4

100

101

NΔ

φ

φ (rad)

FIG. 9. (Color online) �φ vs φ using Eq. (34) for a state of theform Eq. (29), with N = 100, and gt = 10−2. The dot-dashed lineis with |β|2 = 106 (βgt = 10), and the solid line is for |β|2 = 104

(βgt = 1). The dashed line shows the approximate expression for thesensitivity [Eq. (35) ] for |β|2 = 104. For |β|2 = 106, the numericalcalculation and Eq. (35) are identical. The upper and lower blackdotted lines represent the standard quantum limit (1/

√N ) and

√2/N ,

respectively. The divergence in �φ close to φ = 0 in both cases is dueto the imperfect correlations between SB and Jz leading to nonzerovariance in S. If the correlations were perfect and Var(S)|φ=0 = 0,�φ

would reach exactly 1/√FAB at φ = 0.

sensitivities close to φ = 0. This is illustrated in Fig. 9.When βgt = 10, we find that �φ is very close to the QCRB.In contrast, for βgt = 1, the imperfect correlations betweenJz and Sb prevent the sensitivity from reaching the QCRB;nevertheless, the sensitivity is still below the SQL. Note thatthere is a slight deviation between Eq. (35) and the numericalcalculation of the sensitivity using the state (29). This is dueto terms neglected by our approximations, in particular, thenonlinear terms ignored by linearizations such as Eq. (30) andthose neglected terms that arise due to the small (but strictlynonzero) off-diagonal elements of ρA.

2. Case III

Now, consider the situation in Sec. III B: the evolution ofthe state |α(0,0)〉 ⊗ |β〉 under the Hamiltonian (13) that atgt = π approximately results in a pseudo-spin-cat state.

In order to find an operator that approximates SB = �B , weintroduce the amplitude quadrature operator XB = (b + b†),and notice that

〈βe−imπ |XB |βe−imπ 〉 = 2β(−1)m

= 2β〈j,m|�A|j,m〉. (36)

That is, amplitude quadrature measurements of |βe−imπ 〉 areproportional to parity measurements on system B, which aredirectly correlated with parity measurements on Jz eigenstates.Indeed, the quantity

S0 = �A

XB

2β, (37)

has a variance Var(S0) = 1/(2β)2 that becomes vanishinglysmall as the amplitude of the coherent state is increased. Wetherefore expect the signal

Sapprox = U†φ

(�A

XB

2β

)Uφ (38)

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SIMON A. HAINE AND STUART S. SZIGETI PHYSICAL REVIEW A 92, 032317 (2015)

−0.2 −0.1 0 0.1 0.20.8

1

1.2

1.4

1.6

φ (rad)

NΔ

φ

FIG. 10. (Color online) Phase sensitivity of the approximate sig-nal Eq. (38) for a state of the form Eq. (29) at gt = π (i.e., apseudo-spin-cat state) with N = 20. The solid line and dashed lineare for |β|2 = 30 and |β|2 = 5, respectively. The black dotted lineindicates the Heisenberg limit �φ = 1/N (which is the QCRB).Note that the vertical axis is a linear scale.

will be a good approximation to the optimalmeasurement S.

Figure 10 shows the sensitivity for a state of the formEq. (29) at gt = π with N = 20. When |β|2 = 30, thesensitivity is very close to the Heisenberg limit, while for|β|2 = 5 there is a slight degradation in the sensitivity dueto imperfect correlations. In contrast to the approximateoptimal measurement scheme for case II, which requires alarge amplitude coherent state, here the signal (38) is almostoptimal even for small amplitude coherent states. This isbecause 〈βe−iπ |β〉 = exp(−2|β|2) is approximately zero evenfor modest values of β.

In situations where system A is an ensemble of atoms, andsystem B is an optical mode, it would be challenging to achievethe strong atom-light coupling regime required for gt = π . Onthe other hand, the choice of an initial coherent state for systemB ensures that the sensitivity is reasonably insensitive to lossesin system B. In particular, since particle loss from a coherentstate acts only to reduce the state’s amplitude, provided thecoherent state remains sufficiently large after losses in orderto satisfy the requirements for near-optimal measurements,near-Heisenberg-limited sensitivities should be obtainable.

3. Particle-exchange Hamiltonian

Finally, for completeness we include the optimal mea-surement scheme for the state attained after evolving theproduct state |α(0,0)〉 ⊗ |NB〉 under the Hamiltonian (16) (seeSec. III C). The optimal measurement signal is simply Eq. (18)with

SB = N

2± (Nb − b†b). (39)

This choice of SB can be constructed by counting the numberof particles in system B, and it satisfies SB |�AB〉 = Jz|�AB〉as required. As any entangling Hamiltonian of the form

H± =∑

k

Ak(J±b† + J∓b)k +∑j,k

Bj,kJkz (b†b)j (40)

will lead to a state of the form Eq. (6), with |Bm〉 given byEq. (17) (assuming an initial state |�A〉 = |α(0,0)〉,|�B〉 =|NB〉), Eq. (39) also transfers to these systems.

V. DISCUSSION AND CONCLUSIONS

We have shown that purifications of mixed states representan excellent resource for quantum metrology. In particular,we showed that if probe system A and auxiliary system B

are entangled such that the first- and second-order coherencesof system A vanish, then near-Heisenberg-limited sensitivitiescan be achieved provided measurements on both systems A andB are allowed. Although we focused on the situation wherethis entanglement is generated via a few specific Hamiltonians,our conclusions hold irrespective of the specific entanglementgeneration scheme.

It is important to note that when the QFI approaches theHeisenberg limit FAB = N2 (in case III, for example), this isnot the true Heisenberg limit, since N is only the number ofparticles in system A (which pass through the interferometer),rather than the total number of particles Nt in system A andsystem B. However, we have chosen to report the QFI in termsof N instead of Nt , since we envisage our protocol to be usefulin situations where the number of particles in system A is byfar the more valuable resource. For example, consider the caseof inertial sensing with atom interferometry, where system A isatoms and system B is photons. Here the atoms are sensitive toan inertial phase shift, but it is difficult to arbitrarily increasethe atomic flux. Using the techniques discussed above, animproved inertial measurement can be achieved at the cost ofintroducing some number of auxiliary photons, which is cheapcompared to increasing the total number of atoms.

While preparing this paper, we also numerically examinedthe effect of decoherence on the sensitivity of our metrologicalschemes. In particular, we found that the effect of particleloss, spin flips, and phase diffusion on purifications of thepseudo-spin-cat state from Fig. 2 was identical to that of apure spin-cat state.

Although these purified states are no more or less robustto decoherence than other nonclassical pure states, there aresituations where they are easier to generate. The examplewe are most familiar with is atom interferometry, whereatom-light entanglement and information recycling is morecompatible with the requirements of high-precision atominterferometry than the preparation of nonclassical atomicstates via interatomic interactions [39]. However, controlledinteractions are routinely engineered between atoms and light[37,71–73], superconducting circuits and microwaves [74,75],light and mechanical systems [76], and ions and light [77–79].Given that high-efficiency detection is available in all thesesystems [80–85], the application of our proposal to a range ofmetrological platforms is plausible in the near term.

Finally, we note that not all quantum systems are createdequal; certain quantum information protocols, such as quantumerror correction [86] and no-knowledge feedback [87], arebetter suited to some platforms than others. Our proposalallows an experimenter to both perform quantum-enhancedmetrology and take advantage of any additional benefits ahybrid quantum system provides.

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QUANTUM METROLOGY WITH MIXED STATES: WHEN . . . PHYSICAL REVIEW A 92, 032317 (2015)

ACKNOWLEDGMENTS

We acknowledge useful discussions with Carlton Caves,Joel Corney, Jamie Fiess, Samuel Nolan, and Murray Olsen.This work was supported by Australian Research Council(ARC) Discovery Project No. DE130100575 and the ARCCentre of Excellence for Engineered Quantum Systems(Project No. CE110001013).

APPENDIX: DERIVATION OF EQS. (15)

Here we derive the QFI FAB for a MZ interferometer withthe following entangled input:

|�AB(t)〉 = e−igtJzNb |θ,ϕ〉 ⊗ |β〉, (A1)

where Nb = b†b, system B is initially in a coherent state |β〉,and system A is initially in a spin coherent state |θ,ϕ〉. Anyspin coherent state can be defined by rotating the maximalDicke state on the top pole of the Bloch sphere an angle θ

about the Jy axis and an angle ϕ about the Jz axis:

|θ,ϕ〉 ≡ R(θ,ϕ)|j,j 〉 = e−iϕJz e−iθ Jy |j,j 〉. (A2)

Recall that j = N/2, where N is the total number of systemA particles.

The QFI is

FAB = 4Var(eigtJzNb Jye−igtJzNb ), (A3)

where the expectations in the variance are taken with respectto the initial separable state, |θ,ϕ〉 ⊗ |β〉.

By application of the Baker-Campbell-Hausdorff formula

eABe−A = B + [A,B] + 1

2![A,[A,B]]

+ 1

3![A,[A,[A,B]]] + · · · (A4)

it can be shown that

eigtJzNb Jye−igtJzNb = sin(gtNb)Jx + cos(gtNb)Jy . (A5)

Therefore, since the initial state is separable, we obtain

FAB = 〈sin2(gtNb)〉⟨J 2x

⟩ + 〈cos2(gtNb)〉⟨J 2y

⟩+〈sin(gtNb) cos(gtNb)〉〈Jx Jy + Jy Jx〉− (〈sin(gtNb)〉〈Jx〉 + 〈cos(gtNb)〉〈Jy〉)2. (A6)

The system A expectations are more easily computedby rotating the operators by R(θ,ϕ) and then taking expec-tations with respect to the Dicke state |j,j 〉. Specifically,

by virtue of

R†(θ,ϕ)JxR(θ,ϕ) = cos θ cos ϕJx + cos θ sin ϕJy − sin θJz,

(A7a)

R†(θ,ϕ)JyR(θ,ϕ) = − sin ϕJx + cos ϕJy, (A7b)

and the application of J± = Jx ± iJy with

J±|j,m〉 =√

j (j + 1) − (m ± 1)|j,m ± 1〉, (A8)

we obtain

〈Jx〉 = j sin θ cos ϕ, (A9a)

〈Jy〉 = j sin θ sin ϕ, (A9b)

〈J 2x 〉 = j

2(1 + (2j − 1) sin2 θ cos2 ϕ), (A9c)

〈J 2y 〉 = j

2(1 + (2j − 1) sin2 θ sin2 ϕ), (A9d)

〈Jx Jy + Jy Jx〉 = j (2j − 1) sin2 θ sin ϕ cos ϕ. (A9e)

With some simplification this gives

FAB = 2j (1 + sin2 θ [(2j − 1)〈sin2(gtNb + ϕ)〉− 2j 〈sin(gtNb + ϕ)〉2]). (A10)

Incidentally, by setting t = 0 we can see that the QFI for aspin coherent state input never exceeds the standard quantumlimit:

F[Jy,|θ,ϕ〉] = 2j(1 − sin2 θ sin2 ϕ

)� N. (A11)

In order to compute the system B expectations, note that

〈sin(gtNb + ϕ)〉 = − i

2(〈ei(gtNb+ϕ)〉 − 〈e−i(gtNb+ϕ)〉)

(A12a)

〈sin2(gtNb + ϕ)〉 = 2 − 〈e2i(gtNb+ϕ)〉 − 〈e−2i(gtNb+ϕ)〉4

.

(A12b)

Furthermore, for any m,

〈eim(gtNb+ϕ)〉 = eimϕ〈β|βeimgt 〉= exp[−|β|2(1 − eimgt ) + 2mϕ]. (A13)

Substituting Eqs. (A12) and (A13) into Eq. (A10) gives

FAB = F0 + F1 + F2, (A14)

with the expressions for F0,F1, and F2 listed in Eqs. (15).

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