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

Simon A. Haine

School of Mathematics and Physics, University of Queensland, Brisbane, Queensland, 4072, Australia

Stuart S. Szigeti

School 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

Quantum-enhanced metrology can be achieved by entangling a probe with an auxiliary system,passing the probe through an interferometer, and subsequently making measurements on both theprobe and auxiliary system. Conceptually, this corresponds to performing metrology with the pu-rification of a (mixed) probe state. We demonstrate via the quantum Fisher information how todesign mixed states whose purifications are an excellent metrological resource. In particular, wegive examples of mixed states with purifications that allow (near) Heisenberg-limited metrology andprovide examples of entangling Hamiltonians that can generate these states. Finally, we presentthe optimal measurement and parameter-estimation procedure required to realize these sensitivities(i.e., that saturate the quantum Cramér-Rao bound). Since pure states of comparable metrologicalusefulness are typically challenging to generate, it may prove easier to use this approach of entan-glement and measurement of an auxiliary system. An example where this may be the case is atominterferometry, where entanglement with optical systems is potentially easier to engineer than theatomic interactions required to produce nonclassical atomic states.

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 sensitiv-ity is the Heisenberg limit ∆φ = 1/N [1, 2]. Näıvely, thechoice of probe state is a solved problem; for instance,symmetric Dicke states [1, 3] and spin-cat states [4, 5]input into a Mach-Zehnder (MZ) interferometer yield

sensitivities of√

2/N and 1/N , respectively. However,in practice achieving quantum-enhanced sensitivities is asignificant challenge. This is due to both the deleteri-ous effect of losses [6] and the challenges associated withpreparing nonclassical states with an appreciable num-ber of particles [7–11]. For example, protocols for gen-erating a spin-cat state commonly require a large Kerrnonlinearity, which is either unavailable (e.g. in opti-cal systems [12]), difficult to engineer (e.g. in microwavecavities [13, 14]), or is incompatible with the efficientoperation of the metrological device (as in atom interfer-ometers [15–17]).

In this paper, we present an alternative route toquantum-enhanced metrology based on purifications ofmixed states. Physically, this involves entangling theprobe with an auxiliary system before the probe is af-fected by φ, making measurements on both the probe andauxiliary system, and subsequently using correlations be-tween the two measurement outcomes in order to reducethe uncertainty in the estimated parameter (see Fig. 1).

This approach is advantageous in cases where it is easierto entangle the probe system with another system, ratherthan directly create highly entangled states of the probesystem itself. An example of this is atom interferometry;although quantum squeezing can be produced in atomicsystems via atomic interactions [18–31], the technical re-quirements of high sensitivity, path separated atom inter-ferometers are better suited to enhancement via entan-glement 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 ex-cellent resource for quantum metrology. In Sec. III wespecialize to an N -boson probe state and Mach-Zehnder(MZ) interferometer, and show how to engineer purifica-tions that yield sensitivities at and near the Heisenberglimit. Finally, in Sec. IV we present optimal measure-ment schemes that allow these quantum-enhanced sensi-tivities to be achieved in practice.

II. QUANTUM FISHER INFORMATION FOR APURIFICATION

We can determine the best sensitivity possible for anygiven metrology scheme via the QFI, F , which placesan absolute lower bound on the sensitivity, ∆φ ≥ 1/

√F ,

called the quantum Cramér-Rao bound (QCRB) [42–45].This bound is independent of the choice of measurementand parameter estimation procedure, and depends only

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FIG. 1. The unitary ÛAB = exp(−iĤABt/~) entanglessystem A (probe) with system B (auxiliary) before system

A passes through a measurement device described by Ûφ =

exp(−iφĜA). If measurements are restricted to system A,then the QFI for an estimate of φ is FA = F [ĜA, ρ̂A], whereρ̂A = TrB {|ΨAB〉〈ΨAB |}. If measurements on both sys-tems are permitted, then the QFI is FAB = F [ĜA, |ΨAB〉] =4Var(ĜA)ρ̂A ≥ FA.

on the input state. Explicitly, if a state ρ̂A is input intoa metrological device described by the unitary operatorÛφ = exp(−iφĜA), then the QFI is

FA ≡ F [ĜA, ρ̂A] = 2∑i,j

(λi − λj)2

λi + λj|〈ei|ĜA|ej〉|2, (1)

where λi and |ei〉 are the eigenvalues and eigenvectors ofρ̂A, respectively. If ρ̂A is pure, then Eq. (1) reduces to

FA = 4Var(ĜA).A näıve consideration of the pure state QFI suggests

that engineering input states with a large variance inĜA is an excellent strategy for achieving a high precisionestimate of φ. However, there are many operations on ρ̂Athat increase Var(ĜA) at the expense of also decreasingthe purity γ = Tr

{ρ̂2A}

. Since the QFI is convex in thestate, any process that mixes the state typically decreasesthe QFI. Consequently, any improvement due to a largerVar(ĜA) is usually overwhelmed by reductions in the QFIdue to mixing.

In order to concretely demonstrate this point, we spe-cialize to an N -boson state input into a MZ interferome-ter. As discussed in [46], this system is conveniently de-

scribed by the SU(2) Lie algebra [Ĵi, Ĵj ] = i�ijkĴk, where�ijk is the Levi-Civita symbol, for i = x, y, z. A MZ in-

terferometer is characterized by ĜA = Ĵy, therefore, for

pure states, a large QFI requires a large Var(Ĵy).

Without loss of generality, we restrict ourselves to theclass of input states

ρ̂A =

∫ 2π0

dϕP(ϕ)|α(π2 , ϕ)〉〈α(π2 , ϕ)|. (2)

Here |α(θ, ϕ)〉 = exp(−iϕĴz) exp(−iθĴy)|j, j〉 are spin co-herent states, where |j,m〉 are Dicke states with totalangular momentum j = N/2 and Ĵz projection m. Wefocus on the following three states in class (2), which are

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

in order of increasing Var(Ĵy):

Case (I): P(ϕ) = δ(ϕ), (3a)

Case (II): P(ϕ) = 12π, (3b)

Case (III): P(ϕ) = 12

[δ(ϕ− π2

)+ δ

(ϕ+ π2

)]. (3c)

These states can be conveniently visualized by plottingthe Husimi-Q function [47, 48]

Q(θ, ϕ) = 2j + 14π

〈α(θ, ϕ)|ρ̂A|α(θ, ϕ)〉 , (4)

and the Ĵy projection of the state, P (Jy) = 〈Jy|ρ̂A|Jy〉,where Ĵy|Jy〉 = Jy|Jy〉 (see Fig. 2).

None of these states yield sensitivities that surpassthe standard quantum limit (SQL), ∆φ = 1/

√N . In

Case (I), ρ̂A is a pure spin coherent state, |α(π/2, 0)〉,with FA = 4Var(Ĵy) = N . In Case (II), ρ̂A is an in-coherent mixture of Dicke states (i.e. it contains no off-

diagonal terms in the |j,m〉 basis). Although 4Var(Ĵy) =N(N + 1)/2 is much larger than for Case (I), the QFIis only FA = N/2. Finally, Case (III) is an incoher-ent mixture of maximal and minimal Ĵy eigenstates with

4Var(Ĵy) = N2, which is the maximum possible value

in SU(2). However, since the state is mixed the QFI issignificantly less than this, with FA = N/2.

However, suppose the mixing in ρ̂A arises from en-tanglement with an auxiliary system B before systemA passes through the metrological device (see Fig. 1).Specifically, for an input pure state |ΨAB〉 of a compos-ite system A ⊗ B, where ρ̂A = TrB {|ΨAB〉〈ΨAB |}, theQFI is

FAB ≡ F [ĜA, |ΨAB〉]

= 4(〈ΨAB |Ĝ2A|ΨAB〉 − 〈ΨAB |ĜA|ΨAB〉2

)= 4

(TrA

[Ĝ2Aρ̂A

]− TrA

[ĜAρ̂A

]2)≡ 4Var(ĜA)ρ̂A . (5)

3

Consequently, for a purification of ρ̂A the QFI only de-pends on the variance in ĜA of ρ̂A [41, 49]. Our näıve

strategy of preparing a state with large Var(ĜA) irrespec-tive of its purity is now an excellent approach. Indeed,in this situation the states in Cases (I)-(III) are now alsoarranged in order of increasing QFI, with Case (II) andCase (III) attaining a QFI of N(N+1)/2 and N2, respec-tively. It is interesting to note that the QFI for Case (III)is the maximum allowable for N particles in SU(2) [50],and is usually obtained via the difficult to generate spin-cat state, which is a macroscopic superposition, ratherthan a classical mixture, of spin coherent states. Notealso that FAB is independent of any particular purifica-tion, and convexity implies that FAB ≥ FA. That is, inprinciple any purification of ρ̂A is capable of achievingsensitivities at least as good as, and usually much betterthan, ρ̂A itself.

Quantum metrology with purifications is not simplya mathematical ‘trick’; physically, a purification corre-sponds to entangling the probe system A with some aux-iliary system B, and permitting measurements on bothsystems [51]. Therefore, the practical utility of our pro-posal depends crucially on the existence of an entanglingHamiltonian that can prepare ρ̂A in a state with largeVar(ĜA)ρ̂A .

For the three cases described by Eq. (2) and Eqs. (3),a purification of ρ̂A can be written as

|ΨAB〉 =j∑

m=−jcm|j,m〉 ⊗ |Bm〉, (6)

with Case (I) corresponding to 〈Bm|Bn〉 = 1, Case (II)corresponding to 〈Bm|Bn〉 = δn,m, and Case (III) corre-sponding to 〈Bm|Bn〉 = 1(0) for |n −m| even (odd). Inthe following section, we present a simple scheme thatconverts a shot-noise limited spin coherent state [such asCase (I)] to the enhanced QFI purifications of Cases (II)and (III).

III. EXAMPLE ENTANGLING DYNAMICSLEADING TO INCREASED QFI

Consider again the N -boson probe state (system A)

input into a MZ interferometer (i.e. ĜA = Ĵy). The QFIfor a purification of ρ̂A can be written as

FAB = 4(〈Ĵ2y 〉 − 〈Ĵy〉2

)= F0 + F1 + F2, (7)

with

F0 = N2 (N + 2)− 2〈Ĵ2z 〉, (8a)

F1 = −〈i(Ĵ+ − Ĵ−)〉2, (8b)F2 = −〈Ĵ2+ + Ĵ2−〉, (8c)

where Ĵ± = Ĵx ± iĴy. Note that F0, F1, and F2 dependonly on the matrix elements of ρ̂A with |n−m| equal to

0, 1, and 2, in the Ĵz basis; writing FAB in this form isvery convenient for what follows.

Before the interferometer, we assume the probe is cou-pled to some auxiliary system B via the Hamiltonian

ĤAB = ~gĴzĤB . (9)

When system B is a photon field and ĤB is proportionalto the number of photons in the field, then ĤAB describesthe weak probing of the population difference of an en-semble of two-level atoms with far-detuned light [37, 52–61], or dispersive coupling between a microwave cavityand a superconducting qubit [62–64]. We will explore

this specific case shortly, however, for now we keep ĤBcompletely general. If the initial system state is a prod-uct state |ΨAB(0)〉 = |ΨA〉 ⊗ |ΨB〉, after some evolutiontime the state of the system will be given by Eq. (6) with

cm = 〈m|ΨA〉 and |Bm〉 = exp(−imgtĤB)|ΨB〉. Thereduced density operator of system A is then

ρ̂A =

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〉 = 〈ΨB |e−i(n−m)gtĤB |ΨB〉 . (11)When Cn−m = 1, the system remains separable and sys-tem A is a pure state, whereas if Cn−m = δn,m then ρ̂Ais an incoherent mixture of Dicke states.

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

F0 = N2 (N + 2)− 2〈Ĵ2z 〉, (12a)

F1 = −〈i(C1Ĵ+ − C∗1 Ĵ−)〉2, (12b)

F2 = −(C2〈Ĵ2+〉+ C∗2 〈Ĵ2−〉

), (12c)

where the above expectation values are calculated withrespect to |ΨA〉. The effect of the entanglement betweensystems A and B is entirely encoded in the coherences C1and C2; coherences greater than 2nd order do not affectthe QFI.

Let us consider the effect on the QFI of each term inEq. (7). F0 is independent of the entanglement betweensystems A and B, and will be of order N2/2 if |ΨA〉 has〈Ĵ2z 〉 ∼ N (e.g. the spin coherent state |α(π/2, ϕ)〉 has〈Ĵ2z 〉 = N/4). This suggests that a sufficient conditionfor Heisenberg scaling is F1 ∼ F2 ∼ 〈Ĵ2z 〉 ∼ N . In fact,since F1 ≤ 0, the maximum QFI state must necessarilyhave C1 = 0. In contrast, F2 can be positive or negative,in which case a state with C2 = 0 and another state withC2 = 1 and F2 ∼ +N2/2 might both be capable of (near)Heisenberg-limited metrology. We consider examples ofboth states below.

A. Case (II): Example dynamics yieldingFAB ' N2/2

To concretely illustrate the increased QFI a purifica-tion of ρ̂A can provide, we assume system B is a single

4

bosonic mode, described by annihilation operator b̂, and

take ĤB = b̂†b̂ such that

ĤAB = ~gĴz b̂†b̂ . (13)

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

Cn−m = exp[−|β|2

(1− e−i(n−m)gt

)]. (14)

|Cn−m|2 decays on a timescale gt ∼ [(n−m)|β|2]−1. Al-though the non-orthogonality of 〈β|βeiθ〉 ensures thatCn−m never actually reaches zero, it becomes very smallfor even modest values of |β|2.

If the initial condition of system A is |ΨA〉 = |α(θ, φ)〉,then FAB has the simple analytic form given by (see Ap-pendix A)

F0 = N(

1 + (N−1)2 sin2 θ), (15a)

F1 = −N2 sin2 θ sin2(|β|2 sin (gt) + φ

)e−4|β|

2 sin2(gt/2),

(15b)

F2 = N(1−N)2 sin2 θ cos

(|β|2 sin (2gt) + 2φ

)e−2|β|

2 sin2(gt).

(15c)

In contrast, calculating FA via Eq. (1) requires the diag-onalization of ρ̂A, which must be performed numerically.

We first demonstrate the effect of vanishing 1st and2nd order coherence on FAB by preparing system A inthe maximal Ĵx eigenstate, |α(π/2, 0)〉, with N = 100,and a Glauber coherent state for system B with aver-age particle number |β|2 = 500. The initial state forsystem A is precisely Case (I) [see Eq. (3a)], and hasa QFI of N . As shown in Fig. 3, under the evolutionof Eq. (13), ρ̂A tends towards an incoherent mixture ofDicke states [Case (II)], with the corresponding broaden-ing of the P (Jy) distribution.

Figure 4(a) shows that both coherences C1 and C2rapidly approach zero, which causes F1 and F2 to van-ish [see Fig. 4(b)]. Consequently, FAB approaches F0 =N(N + 1)/2, which allows a phase sensitivity of approxi-

mately√

2×Heisenberg limit [see Fig. 4(c)]. In contrast,the effect of the mixing causes the QFI of ρ̂A itself todecrease from N to FA = N/2, with FA ≤ N for all t.This remains true even if the if GA is rotated to lie inany arbitrary direction on the Bloch sphere.

The oscillations in F1 and F2 (and consequently FAand FAB) before the plateau are due to the complex ro-tation of C1 and C2, which causes rotations of ρ̂A aroundthe Jz axis before being overwhelmed by the overall de-cay in magnitude. Furthermore, although the purity ofthe state also decays, it never vanishes, thereby illustrat-ing that it is not the entanglement per se that is causingthe QFI enhancement for a purification of ρ̂A.

FIG. 3. (Color online) Time snapshots of the Husimi-Q func-tion and Ĵy projection illustrating the evolution of a maximal

Ĵx eigenstate under entangling Hamiltonian Eq. (13). Thesnapshots were chosen to correspond to times when the rota-tion around the Jz axis is such that 〈Ĵy〉 = 0, which roughlycorresponds to the local maxima of FAB in Fig. 4(c). (Pa-rameters: N = 100, |β|2 = 500).

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 Ĵx eigenstateunder entangling Hamiltonian Eq. (13). (a) Coherences |C1|2(blue solid line), |C2|2 (red dashed line) and purity γ (blackdot-dashed line). (b): Three components of FAB [see Eq. (7)]:F0 (black dot-dashed line), F1 (blue solid line), and F2 (reddashed line). (c) QFI for ρ̂A, FA (blue dashed line), and a pu-rification of ρ̂A, FAB (red solid line). For reference, we haveincluded N (black dotted line) and F0 = N(N+1)/2 ≈ N2/2(black dot-dashed line), which correspond to phase sensitivi-ties at the SQL and

√2×Heisenberg limit, respectively. (Pa-

rameters: N = 100, |β|2 = 500).

B. Case (III): Example dynamics yieldingFAB = 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 underthe evolution of Eq. (13) for times close to gt = π, when

the initial state of system A is the maximal Ĵy eigenstate|α(π/2, π/2)〉.

The QFI is initially zero, but the decay of C1 and C2rapidly increases to FAB = F0 = N(N + 1)/2 as in theprevious example. As gt→ π, the revival of |C2|2 causesFAB to briefly increase to N2 (see Fig. 6). This is theHeisenberg limit, which is the QFI of a (pure) spin-cat

5

FIG. 5. (Color online) Husimi-Q function and Ĵy projectionfor an initial state |ΨAB(0)〉 = |α(π/2, π/2)〉 ⊗ |β〉 under theevolution of Eq. (13) for different values of gt. The Q functionis symmetric about reflection of the Jy axis, resulting in partof the function being hidden from view on the reverse side ofthe sphere. (Parameters: N = 100, |β|2 = 500).

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 Ĵy eigenstatenear gt = π under entangling Hamiltonian Eq. (13). (a) |C1|2(blue solid line), |C2|2 (red dashed line), and γ (black dot-dashed line). (b): F0 (black dot-dashed line), F1 (blue solidline), and F2 (red dashed line). (c) FA (blue dashed line)and FAB (red solid line). For comparison, we have includedF0 = N(N+1)/2 ≈ N2/2 and the Heisenberg limit N2 (blackdot-dashed lines); FA ≤ N for all t. (Parameters: N = 100,|β|2 = 500).

state and the maximum QFI for SU(2) [50]. At gt =π, ρ̂A is identical to a classical mixture of |α(π/2, π/2)〉and |α(π/2,−π/2)〉, however, its Q-function is similar tothat of a spin-cat state, and purifications of it behaveas a spin-cat state for metrological purposes. For thesereasons, we call this state a pseudo-spin-cat state.

C. Example dynamics for particle-exchangeHamiltonian

In the previous two examples the Ĵz projection wasa conserved quantity, so any entanglement between sys-tems A and B can only degrade the coherence in the Ĵzbasis of system A (ultimately resulting in an enhancedQFI). The situation is more complicated when consider-

ing a Hamiltonian that does not conserve the Ĵz projec-tion, such as when a spin flip in system A is correlatedwith the creation or annihilation of a quantum in systemB. Here, we encounter scenarios where the interactioncan either create or destroy coherences in the Ĵz basis ofsystem A, and although a significant QFI enhancement isstill possible, it depends upon the initial state of systemB.

As a concrete illustration, consider the particle-exchange Hamiltonian

Ĥ± = ~g(Ĵ±b̂

† + Ĵ∓b̂), (16)

and assume that system A is initially prepared in themaximal Ĵz eigenstate |ΨA〉 = |α(0, 0)〉 = |j, j〉 (N.B. thishas Var(Ĵy) = N/4, and therefore a QFI of N). Then

Ĥ− and Ĥ+ physically correspond to Raman superradi-ance [38, 66] and quantum state transfer [34–37, 39, 40]processes, respectively. After some period of evolution,the combined state of systems A ⊗ B takes the form ofEq. (6).

First, consider the case when the initial state for sys-tem B is a large amplitude coherent state (i.e. |ΨB〉 =|β〉). Here the addition/removal of a quantum to/fromsystemB has a minimal effect on the state and the systemremains approximately separable, since |〈Bn|Bm〉|2 ≈ 1.It is therefore reasonable to make the undepleted pump

approximation b̂→ β, such that Ĥ± → ~gβĴx (assumingβ is real). Hence, the effect of the interaction is simply arotation around the Jx axis, which can create coherencein the Ĵz basis, and so FA = FAB ≤ N for all time.

In the opposite limit where the initial state of systemB is a Fock state with NB particles, |ΨB〉 = |NB〉, then

|Bm〉 = |NB ± (m− j)〉, (17)

and 〈Bm|Bn〉 = δn,m. This ensures that the first andsecond order coherences vanish, and F1 = F2 = 0 for alltime. That is, as illustrated in Fig. 7, the state movestowards the equator and ultimately evolves to an inco-herent Dicke mixture [i.e. Case (II)]. As described inSec. III A, and shown in Fig. 8, the QFI increases toa maximum of approximately FAB ≈ N2/2. Althoughsetting NB = 0 (i.e. a vacuum state) leads to a larger

variance in Ĵz, FAB still reaches approximately 70% ofN2/2.

We therefore see that for the Hamiltonian (16), a largeQFI enhancement is achieved provided the initial state|ΨB〉 has small number fluctuations. Compare this to theHamiltonian (13), where the choice |ΨB〉 = |NB〉 leads tono entanglement between systems A and B, while in con-trast an initial state with small phase fluctuations (andtherefore large number fluctuations), such as a coherentstate, causes rapid decoherence in ρ̂A.

6

FIG. 7. (Color online) Husimi-Q function and Ĵy projectionfor an initial state |ΨAB(0)〉 = |α(0, 0)〉 ⊗ |NB〉 under theevolution of Ĥ− 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 (blue dashed line) and FAB (redsolid line) for an initial state |ΨAB(0)〉 = |α(0, 0)〉 ⊗ |NB〉under the evolution of Ĥ−. We have indicated N and N2/2with black dotted lines for comparison. (Parameters: N =100, NB = 20).

IV. OPTIMAL MEASUREMENT SCHEMES

Although the QFI determines the optimum sensitivityfor a given initial state, it is silent on the question ofhow to achieve this optimum. It is therefore importantto identify a) which measurements to make on each sys-tem and b) a method of combining the outcomes of thesemeasurements - which we refer to as a measurement sig-nal (Ŝ) - that saturates the QCRB. We do this below forpurifications of the incoherent Dicke mixture [Case (II)]and the pseudo-spin-cat state [Case (III)].

A. Optimal measurements for incoherent Dickemixture [Case (II)]:

It is worthwhile briefly recounting the optimal estima-tion procedure for a symmetric Dicke state |j, 0〉 inputinto a MZ interferometer. A MZ interferometer rotatesĴz according to Ĵz(φ) = Û

†φĴzÛφ = cosφĴz − sinφĴx.

Since symmetric Dicke states satisfy 〈Ĵx〉 = 〈Ĵz〉 =〈Ĵ2z 〉 = 0 and 〈Ĵ2x〉 = N(N + 2)/8, it is clear that thefluctuations in Ĵz(φ) contain the phase information, and

therefore the quantity Ŝ = [Ĵz(φ)]2 oscillates between0 and N(N + 2)/8. It can be shown that at the op-

erating point φ → 0, Var(Ŝ) → 0, and the quantity(∆φ)2 → Var(Ŝ)/(∂φ〈Ŝ〉)2 = 1/FA, and therefore thesignal Ŝ saturates the QCRB [3, 67, 68].

For an incoherent Dicke mixture, we have 〈Ĵx〉 =〈Ĵy〉 = 〈Ĵz〉 = 0, and 〈Ĵ2x〉 = N(N+1)/8. Unfortunately,the non-zero variance in Ĵz (i.e. 〈Ĵ2z 〉 = N/4) impliesthat Var(Ŝ) � 0 for all φ, and the signal no longer sat-urates the QCRB. However, since the states |Bm〉 in thepurification Eq. (6) are orthonormal, a projective mea-surement of some system B operator diagonal in the |Bm〉basis projects system A into a Ĵz eigenstate (i.e. a Dickestate). That is, these measurement outcomes on system

B are correlated with Ĵz measurement outcomes on sys-tem A. Therefore, subtracting both measurements yieldsa quantity with very little quantum noise.

More precisely, if we can construct an operator ŜB onsystem B that is correlated with Ĵz measurements onsystem A (i.e. ŜB |ΨAB〉 = Ĵz|ΨAB〉), then we can con-struct the quantity Ŝ0 = Ĵz−ŜB which has the property〈Ŝ0〉 = 〈Ŝ20 〉 = 0. This motivates the signal choice

Ŝ =(Û†φŜ0Ûφ

)2= (cosφĴz − sinφĴx − ŜB)2. (18)

Using ŜB |ΨAB〉 = Ĵz|ΨAB〉 and the fact that non-Ĵz con-serving terms vanish due to the absence of off-diagonalterms in the Ĵz representation of ρ̂A, (e.g. expectation

values with an odd power of Ĵx vanish), we can show that

〈Ŝ〉 = 〈Ĵ2z 〉 (cosφ− 1)2

+ 〈Ĵ2x〉 sin2 φ, (19a)〈Ŝ2〉 = 〈Ĵ4z 〉(cosφ− 1)4 + 〈Ĵ4x〉 sin4φ

+ 〈Ĵ2z Ĵ2x + Ĵ2x Ĵ2z + 4ĴzĴxĴxĴz〉 sin2φ(cosφ− 1)2

+ 2i〈(ĴzĴxĴy − ĴyĴxĴz)〉 sin2φ cosφ(cosφ− 1)+ 〈Ĵ2y 〉 cos2φ sin2φ. (19b)

Note that the above expectation values can be taken withrespect to |ΨAB〉 or ρ̂A. The best sensitivity occurs atsmall displacements around φ = 0. Taking the limit asφ→ 0 and noting that 〈Ĵ2x〉 = 〈Ĵ2y 〉 gives

(∆φ)2 =Var(Ŝ)(∂φ〈Ŝ〉

)2∣∣∣∣∣φ=0

=1

4〈Ĵ2x〉=

1

4〈Ĵ2y 〉=

1

FAB.

(20)This demonstrates that the signal Eq. (18) is optimalsince it saturates the QCRB.

B. Optimal measurements for pseudo-spin-catstate [Case (III)]:

Pure spin-cat states have the maximum QFI possiblefor N particles in SU(2), are eigenstates of the parity

7

operator, and indeed parity measurements saturate theQCRB [69]. Pseudo-spin-cat states (case (III)) also havemaximal QFI, and since 〈Bn|Bm〉 = 1(0) for |n − m|even(odd), a projective measurement of system B yields

no information other than the parity of the Ĵz projec-tion. This suggests that a measurement of parity couldbe optimal.

In analogy with Case (II), our aim is to construct an

operator Ŝ0 where the correlations between systems Aand B lead to a reduction in Var(Ŝ0) and the systemmimics a pure spin-cat state. Introducing the quantity

Ŝ0 = Π̂AŜB ≡ Π̂AΠ̂B , (21)

where Π̂A(B) is the parity operator for system A(B),

defined by Π̂A|j,m〉 = (−1)m|j,m〉 and Π̂B |Bm〉 =(−1)m|Bm〉, we see that pseudo-spin-cat states satisfyŜ0|ΨAB〉 = |ΨAB〉, and therefore Var(Ŝ0) = 0. This mo-tivates the signal choice Ŝ = Û†φŜ0Ûφ.

To calculate the sensitivity, we need to compute 〈Ŝ〉and 〈Ŝ2〉. Trivially, 〈Ŝ2〉 = 1 for all states. For φ � 1,expanding Ûφ to second order in φ gives

〈Ŝ〉 ≈ 〈(1 + iφĴy − 12φ2Ĵ2y )Ŝ0(1− iφĴy − 12φ

2Ĵ2y )〉

= 1 + iφ(〈ĴyŜ0〉 − 〈Ŝ0Ĵy〉

)+ φ2

[〈ĴyŜ0Ĵy〉 −

1

2

(〈Ĵ2y Ŝ0〉+ 〈Ŝ0Ĵ2y 〉

)]+O(φ3), (22)

The relation 〈Bn|Bn±1〉 = 0 ensures that terms linear inĴy go to zero:

〈Ĵ+〉 =∑m,n

cmc∗n〈j, n|Ĵ+|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〉 = 1preserves terms such as 〈Ĵ2+〉. Noting that Ĵy flips theparity of any state in subsystem A but not subsystem B:

Π̂AĴy|ΨAB〉 = −ĴyΠ̂A|ΨAB〉, (24a)Π̂B Ĵy|ΨAB〉 = ĴyΠ̂B |ΨAB〉, (24b)

and using Ŝ0|ΨAB〉 = |ΨAB〉 gives

〈ĴyŜ0Ĵy〉 = −〈Ŝ0Ĵ2y 〉 = −〈Ĵ2y Ŝ0〉 = −〈Ĵ2y 〉. (25)

Therefore

〈Ŝ〉 = 1− 2φ2〈Ĵ2y 〉+O(φ3) . (26)

Since Ŝ20 = 1 implies that Ŝ2 = 1, we obtain

Var(Ŝ) = 4φ2〈Ĵ2y 〉+O(φ4), (27)

and consequently

(∆φ)2 =Var(Ŝ)

(∂φ〈Ŝ〉)2=

4φ2〈Ĵ2y 〉16φ2〈Ĵ2y 〉2

=1

4〈Ĵ2y 〉=

1

FAB.

(28)This demonstrates that the signal saturates the QCRBand is therefore optimal.

The optimal estimation schemes presented inSecs. IV A and IV B illustrate a somewhat counter-intuitive fact: although the optimal measurement ofsystem B for a pseudo-spin-cat state provides lessinformation about system A than for an incoherentDicke 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 ŜB . In general, the choiceof ŜB depends upon the specific purification of ρ̂A. Phys-ically, the initial state of system B and the entanglingHamiltonian matter. However, there is no guarantee thatŜB exists, and if it does there is no guarantee that ameasurement of this observable can be made in practice.Nevertheless, as we show below, it may be possible tomake a measurement of an observable that approximatesŜB , and can therefore give near-optimal sensitivities.

1. Case (II)

To begin, consider the situation in Sec. III A: the evo-lution of the state |α(π/2, 0)〉 ⊗ |β〉 under the Hamilto-nian (13). We require ŜB |ΨAB〉 = Ĵz|ΨAB〉. After someevolution time t:

|ΨAB〉 =j∑

m=−jcm|j,m〉|βe−imgt〉 . (29)

Clearly, the phase of the coherent state is correlated withthe Ĵz projection of system B. This can be extractedvia a homodyne measurement of the phase quadrature

ŶB = i(b̂ − b̂†) [70]. In fact, provided mgt � 1, phasequadrature measurements of |β exp(−imgt)〉 are linearlyproportional to the Ĵz projection:

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

where without loss of generality we have taken β to bereal and positive. Consequently, the scaled phase quadra-ture

ŜB =ŶB

2βgt(31)

8

satisfies

〈ΨAB |(Ĵz − ŜB

)|ΨAB〉 ≈ 0, (32a)

〈ΨAB |(Ĵz − ŜB

)2|ΨAB〉 ≈

1

(2βgt)2, (32b)

and so the fluctuations in (Ĵz − ŜB) become arbitrarilysmall (and ŜB becomes perfectly correlated with Ĵz) as(βgt)2 becomes large. This suggests that Eqs. (18) and(31) should be a good approximation to an optimal mea-surement signal.

More precisely, assume that

1

β� Ngt

2� 1. (33)

The first inequality ensures that 〈βe−ingt|βe−imgt〉 ≈δn,m and so ρ̂A is approximately an incoherent Dicke mix-ture, while the second inequality implies that we are inthe linearized regime where Eq. (30) and Eqs. (32) hold.Then the signal

Ŝapprox =

(Ĵz cosφ− Ĵx sinφ−

ŶB2gt

)2, (34)

yields the sensitivity

(∆φ)2 ≈ 1(∂φ〈Ŝ〉

)2{Var(Ŝ)− sin2(φ/2)β2 〈Ĵ4z 〉+ 2(2βgt)4+

4

(2βgt)2

[(cosφ− 1)2〈Ĵ2z 〉+ sin2 φ〈Ĵ2x〉

]},

(35)

where Var(Ŝ) and ∂φ〈Ŝ〉 are given by the expecta-tions (19) of the optimal signal Eq. (18). Figure (9) showsEq. (35) compared to an exact numeric calculation.

Condition (33) typically ensures that the term propor-

tional to 〈Ĵ4z 〉 is small in comparison to the term propor-tional to 1/(2βgt)2. We therefore see that our approx-

imate signal Ŝapprox gives a sensitivity worse than theQCRB, and furthermore at an operating point φ 6= 0.Nevertheless, ∆φ approaches the QCRB at φ = 0 asβ2 and (2βgt)2 approach infinity. Therefore, for a suf-ficiently large βgt, we can achieve near-optimal sensi-tivities close to φ = 0. This is illustrated in Fig. 9.When βgt = 10, we find that ∆φ is very close to theQCRB. In contrast, for βgt = 1, the imperfect corre-lations between Ĵz and Ŝb prevent the sensitivity fromreaching the QCRB; nevertheless, the sensitivity is stillbelow the SQL. Note that there is a slight deviation be-tween Eq. (35) and the numerical calculation of the sensi-tivity using the state (29). This is due to terms neglectedby our approximations; in particular, the nonlinear termsignored by linearizations such as Eq. (30) and those ne-glected terms that arise due to the small (but strictlynon-zero) off-diagonal elements of ρ̂A.

−0.4 −0.2 0 0.2 0.4

100

101

N∆φ

φ (rads)

FIG. 9. (Color online) ∆φ versus φ using Eq. (34) for a stateof the form Eq. (29), with N = 100, and gt = 10−2. Theblue dot-dashed line is with |β|2 = 106 (βgt = 10), andthe red solid line is for |β|2 = 104 (βgt = 1). The reddashed line shows the approximate expression for the sen-sitivity (Eq. (35)) for |β|2 = 104. For |β|2 = 106, the nu-merical calculation and Eq. (35) are identical. The upper andlower black dotted lines represent the standard quantum limit(1/√N), and

√2/N respectively. The divergence in ∆φ close

to φ = 0 in both cases is due to the imperfect correlationsbetween ŜB and Ĵz leading to non-zero variance in Ŝ. If thecorrelations were perfect and Var(Ŝ)|φ=0 = 0, ∆φ would reachexactly 1/

√FAB at φ = 0.

2. Case (III)

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

In order to find an operator that approximates ŜB =Π̂B , we introduce the amplitude quadrature operator

X̂B = (b̂+ b̂†), and notice that

〈βe−imπ|X̂B |βe−imπ〉 = 2β(−1)m

= 2β〈j,m|Π̂A|j,m〉 . (36)

That is, amplitude quadrature measurements of |βe−imπ〉are proportional to parity measurements on system B,which are directly correlated with parity measurementson Ĵz eigenstates. Indeed, the quantity

Ŝ0 = Π̂AX̂B2β

, (37)

has a variance Var(Ŝ0) = 1/(2β)2 that becomes vanish-ingly small as the amplitude of the coherent state is in-creased. We therefore expect the signal

Ŝapprox = Û†φ

(Π̂A

X̂B2β

)Ûφ (38)

will be a good approximation to the optimal measure-ment Ŝ.

9

−0.2 −0.1 0 0.1 0.20.8

1

1.2

1.4

1.6

φ (rads)

N∆φ

FIG. 10. (Color online) Phase sensitivity of the approximatesignal Eq. (38) for a state of the form Eq. (29) at gt = π (i.e.a pseudo-spin-cat state) with N = 20. The blue solid line andred dashed line are for |β|2 = 30 and |β|2 = 5, respectively.The black dotted line indicates the Heisenberg limit ∆φ =1/N (which is the QCRB). Note that the vertical axis is alinear scale.

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, whilefor |β|2 = 5 there is a slight degradation in the sensi-tivity due to imperfect correlations. In contrast to theapproximate optimal measurement scheme for Case (II),which requires a large amplitude coherent state, here thesignal (38) is almost optimal even for small amplitude co-herent states. This is because 〈βe−iπ|β〉 = exp(−2|β|2)is approximately zero even for modest values of β.

In situations where system A is an ensemble of atoms,and system B is an optical mode, it would be challengingto achieve the strong atom-light coupling regime requiredfor gt = π. On the other hand, the choice of an initialcoherent state for system B ensures that the sensitivityis reasonably insensitive to losses in system B. In partic-ular, since particle loss from a coherent state acts only toreduce the state’s amplitude, provided the coherent stateremains sufficiently large after losses in order to satisfythe 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)(see Sec. III C). The optimal measurement signal is sim-ply Eq. (18) with

ŜB =N

2±(Nb − b̂†b̂

). (39)

This choice of ŜB can be constructed by countingthe number of particles in system B, and it satisfiesŜB |ΨAB〉 = Ĵz|ΨAB〉 as required. As any entangling

Hamiltonian of the form

Ĥ± =∑k

Ak(Ĵ±b̂

† + Ĵ∓b̂)k

+∑j,k

Bj,kĴkz(b̂†b̂)j

(40)

will lead to a state of the form Eq. (6), with |Bm〉 givenby Eq. (17) (assuming an initial state |ΨA〉 = |α(0, 0)〉,|ΨB〉 = |NB〉), Eq. (39) also transfers to these systems.

V. DISCUSSION AND CONCLUSIONS

We have shown that purifications of mixed states rep-resent an excellent resource for quantum metrology. Inparticular, we showed that if probe system A and auxil-iary system B are entangled such that the 1st and 2nd or-der coherences of system A vanish, then near-Heisenberg-limited sensitivities can be achieved provided measure-ments on both systems A and B are allowed. Althoughwe focused on the situation where this entanglement isgenerated via a few specific Hamiltonians, our conclu-sions hold irrespective of the specific entanglement gen-eration scheme.

While preparing this paper, we also numerically exam-ined the effect of decoherence on the sensitivity of ourmetrological schemes. In particular, we found that theeffect of particle loss, spin flips, and phase diffusion onpurifications of the pseudo-spin-cat state from Fig. 2 wasidentical to that of a pure spin-cat state.

Although these purified states are no more or less ro-bust to decoherence than other nonclassical pure states,there are situations where they are easier to generate.The example we are most familiar with is atom interfer-ometry, where atom-light entanglement and informationrecycling is more compatible with the requirements ofhigh precision atom interferometry than the preparationof nonclassical atomic states via interatomic interactions[39]. However, controlled interactions are routinely en-gineered between atoms and light [37, 71–73], supercon-ducting circuits and microwaves [74, 75], light and me-chanical systems [76], and ions and light [77–79]. Giventhat high efficiency detection is available in all these sys-tems [80–85], the application of our proposal to a rangeof metrological platforms is plausible in the near term.

It is important to note that although the QFI ap-proaches the Heisenberg limit (FAB = N

2, in Case (III),for example), this is not the true Heisenberg limit, asN refers only to the number of particles in system A(which pass through the interferometer), rather than thetotal number of particles Nt in system A and system B.However, there are some situations where the number ofparticles in system A is by far the more valuable resource,which is why it makes sense to report the QFI in terms ofN rather than Nt. For example, consider the case of iner-tial sensing with atom interferometry, where system A isatoms, and system B is photons. The atoms are sensitiveto inertial phase shift, but it is difficult to arbitrarily in-crease the atomic flux. However, a gain can be achieved

10

by adding some number of photons to the system, whichare comparatively ‘cheep’ compared to atoms.

Finally, we note that not all quantum systems are cre-ated equal; certain quantum information protocols, suchas quantum error correction [86] and no-knowledge feed-back [87], are better suited to some platforms than oth-ers. Our proposal allows an experimenter to both per-form quantum-enhanced metrology and take advantageof any additional benefits a hybrid quantum system pro-vides.

VI. ACKNOWLEDGEMENTS

We would like to acknowledge useful discussions withCarlton Caves, Joel Corney, Jamie Fiess, Samuel Nolan,and Murray Olsen. This work was supported by Aus-tralian Research Council (ARC) Discovery Project No.DE130100575 and the ARC Centre of Excellence for En-gineered Quantum Systems (Project No. CE110001013).

Appendix A: Derivation of Eqs. (15)

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

|ΨAB(t)〉 = e−igtĴzN̂b |θ, ϕ〉 ⊗ |β〉, (A1)

where N̂b = b̂†b̂, system B is initially in a coherent state

|β〉, and system A is initially in a spin coherent state|θ, ϕ〉. Any spin coherent state can be defined by rotatingthe maximal Dicke state on the top pole of the Blochsphere an angle θ about the Jy axis and an angle ϕ aboutthe Jz axis:

|θ, ϕ〉 ≡ R̂(θ, ϕ)|j, j〉 = e−iϕĴze−iθĴy |j, j〉. (A2)

Recall that j = N/2, where N is the total number ofsystem A particles.

The QFI is

FAB = 4Var(eigtĴzN̂b Ĵye

−igtĴzN̂b), (A3)

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

By application of the Baker-Campbell-Hausdorff for-mula

eÂB̂e−Â = B̂ + [Â, B̂] +1

2!

[Â, [Â, B̂]

]+

1

3!

[Â,[Â, [Â, B̂]

]]+ · · · (A4)

it can be shown that

eigtĴzN̂b Ĵye−igtĴzN̂b = sin(gtN̂b)Ĵx + cos(gtN̂b)Ĵy. (A5)

Therefore, since the initial state is separable, we obtain

FAB = 〈sin2(gtN̂b)〉〈Ĵ2x〉+ 〈cos2(gtN̂b)〉〈Ĵ2y 〉

+ 〈sin(gtN̂b) cos(gtN̂b)〉〈ĴxĴy + ĴyĴx〉

−(〈sin(gtN̂b)〉〈Ĵx〉+ 〈cos(gtN̂b)〉〈Ĵy〉

)2. (A6)

The system A expectations are more easily computedby rotating the operators by R̂(θ, ϕ) and then taking ex-pectations with respect to the Dicke state |j, j〉. Specifi-cally, by virtue of

R̂†(θ, ϕ)ĴxR̂(θ, ϕ) = cos θ cosϕĴx + cos θ sinϕĴy

− sin θĴz, (A7a)R̂†(θ, ϕ)ĴyR̂(θ, ϕ) = − sinϕĴx + cosϕĴy, (A7b)

and the application of Ĵ± = Ĵx ± iĴy with

Ĵ±|j,m〉 =√j(j + 1)− (m± 1)|j,m± 1〉, (A8)

we obtain

〈Ĵx〉 = j sin θ cosϕ, (A9a)〈Ĵy〉 = j sin θ sinϕ, (A9b)

〈Ĵ2x〉 =j

2

(1 + (2j − 1) sin2 θ cos2 ϕ

), (A9c)

〈Ĵ2y 〉 =j

2

(1 + (2j − 1) sin2 θ sin2 ϕ

), (A9d)

〈ĴxĴy + ĴyĴx〉 = j(2j − 1) sin2 θ sinϕ cosϕ. (A9e)With some simplification this gives

FAB = 2j(

1 + sin2 θ[(2j − 1)〈sin2(gtN̂b + ϕ)〉

− 2j〈sin(gtN̂b + ϕ)〉2]). (A10)

Incidentally, by setting t = 0 we can see that the QFI fora spin coherent state input never exceeds the standardquantum limit:

F [Ĵy, |θ, ϕ〉] = 2j(1− sin2 θ sin2 ϕ

)≤ N. (A11)

In order to compute the system B expectations, notethat

〈sin(gtN̂b + ϕ)〉 = −i

2

(〈ei(gtN̂b+ϕ)〉 − 〈e−i(gtN̂b+ϕ)〉

)(A12a)

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

4.

(A12b)

Furthermore, for any m,

〈eim(gtN̂b+ϕ)〉 = eimϕ〈β|βeimgt〉= exp

[−|β|2

(1− eimgt

)+ 2mϕ

]. (A13)

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

FAB = F0 + F1 + F2, (A14)with the expressions for F0, F1, and F2 listed inEqs. (15).

11

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Quantum metrology with mixed states: When recovering lost information is better than never losing itAbstractI IntroductionII Quantum Fisher information for a purificationIII Example entangling dynamics leading to increased QFIA Case (II): Example dynamics yielding FAB N2/2B Case (III): Example dynamics yielding FAB = N2 C Example dynamics for particle-exchange Hamiltonian

IV Optimal measurement schemesA Optimal measurements for incoherent Dicke mixture [Case (II)]:B Optimal measurements for pseudo-spin-cat state [Case (III)]:C System B observables that approximate optimal measurements1 Case (II)2 Case (III)3 Particle-exchange Hamiltonian

V Discussion and ConclusionsVI AcknowledgementsA Derivation of Eqs. (15) References