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

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 the probe and auxiliary system. 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 an excellent metrological resource. In particular, we give examples of mixed states with purifications that allow (near) Heisenberg-limited metrology, and provide example entangling Hamiltonians that can generate these states. Finally, we present the optimal measurement and parameter-estimation procedure required to realize these sensitivities (i.e. that saturate the quantum Cram\'er-Rao bound). Since pure states of comparable metrological usefulness are typically challenging to generate, it may prove easier to use this approach of entanglement and measurement of an auxiliary system. An example where this may be the case is atom interferometry, where entanglement with optical systems is potentially easier to engineer than the atomic 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 a quantum probe at a higher precision than is possible with a classical probe of identical particle flux.Given a fixed number of particles, N , the ultimate limit to the sensitivity is the Heisenberg limit ∆φ = 1/N [1,2].Naïvely, the choice 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 a significant challenge.This is due to both the deleterious effect of losses [6] and the challenges associated with preparing nonclassical states with an appreciable number of particles [7][8][9][10][11].For example, protocols for generating a spin-cat state commonly require a large Kerr nonlinearity, which is either unavailable (e.g. in optical systems [12]), difficult to engineer (e.g. in microwave cavities [13,14]), or is incompatible with the efficient operation of the metrological device (as in atom interferometers [15][16][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 auxiliary system before the probe is affected by φ, making measurements on both the probe and auxiliary system, and subsequently using correlations between the two measurement outcomes in order to reduce the uncertainty in the estimated parameter (see Fig. 1).This approach is advantageous in cases where it is easier to entangle the probe system with another system, rather than directly create highly entangled states of the probe system itself.An example of this is atom interferometry; although quantum squeezing can be produced in atomic systems via atomic interactions [18][19][20][21][22][23][24][25][26][27][28][29][30][31], the technical requirements of high sensitivity, path separated atom interferometers are better suited to enhancement via entanglement with an optical system [32][33][34][35][36][37] and information recycling [38][39][40][41].
The structure of this paper is as follows.In Sec.II we introduce in detail the central idea of this paper: that purifications of mixed states can possess a large quantum Fisher information (QFI), and therefore represent an excellent resource for quantum metrology.In Sec.III we specialize to an N -boson probe state and Mach-Zehnder (MZ) interferometer, and show how to engineer purifications that yield sensitivities at and near the Heisenberg limit.Finally, in Sec.IV we present optimal measurement schemes that allow these quantum-enhanced sensitivities to be achieved in practice.

II. QUANTUM FISHER INFORMATION FOR A PURIFICATION
We can determine the best sensitivity possible for any given metrology scheme via the QFI, F, which places an absolute lower bound on the sensitivity, ∆φ ≥ 1/ √ F, called the quantum Cramér-Rao bound (QCRB) [42][43][44][45].This bound is independent of the choice of measurement and parameter estimation procedure, and depends only The unitary ÛAB = exp(−i ĤABt/ ) entangles system A (probe) with system B (auxiliary) before system A passes through a measurement device described by Ûφ = exp(−iφ ĜA).If measurements are restricted to system A, then the QFI for an estimate of φ is FA = F[ ĜA, ρA], where ρA = TrB {|ΨAB ΨAB|}.If measurements on both systems are permitted, then the QFI is on the input state.Explicitly, if a state ρA is input into a metrological device described by the unitary operator Ûφ = exp(−iφ ĜA ), then the QFI is where λ i and |e i are the eigenvalues and eigenvectors of ρA , respectively.If ρA is pure, then Eq. ( 1) reduces to A naïve consideration of the pure state QFI suggests that engineering input states with a large variance in ĜA is an excellent strategy for achieving a high precision estimate of φ.However, there are many operations on ρA that increase Var( ĜA ) at the expense of also decreasing the purity γ = Tr ρ2 A .Since the QFI is convex in the state, any process that mixes the state typically decreases the QFI.Consequently, any improvement due to a larger Var( ĜA ) is usually overwhelmed by reductions in the QFI due to mixing.
In order to concretely demonstrate this point, we specialize to an N -boson state input into a MZ interferometer.As discussed in [46], this system is conveniently described by the SU(2) Lie algebra [ Ĵi , Ĵj ] = i ijk Ĵk , where ijk is the Levi-Civita symbol, for i = x, y, z.A MZ interferometer 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 the class of input states Here |α(θ, ϕ) = exp(−iϕ Ĵz ) exp(−iθ Ĵy )|j, j are spin coherent states, where |j, m are Dicke states with total angular momentum j = N/2 and Ĵz projection m.We focus on the following three states in class (2), which are in order of increasing Var( Ĵy ): Case (I): Case (II): Case (III): These states can be conveniently visualized by plotting the Husimi-Q function [47,48] and the Ĵy projection of the state, P (J y ) = J y |ρ A |J y , where Ĵy |J y = J y |J y (see Fig. 2).None of these states yield sensitivities that surpass the standard quantum limit (SQL), ∆φ = 1/ √ N .In Case (I), ρA is a pure spin coherent state, |α(π/2, 0) , with F A = 4Var( Ĵy ) = N .In Case (II), ρA is an incoherent mixture of Dicke states (i.e. it contains no offdiagonal terms in the |j, m basis).Although 4Var( Ĵy ) = N (N + 1)/2 is much larger than for Case (I), the QFI is only F A = N/2.Finally, Case (III) is an incoherent mixture of maximal and minimal Ĵy eigenstates with 4Var( Ĵy ) = N 2 , which is the maximum possible value in SU (2).However, since the state is mixed the QFI is significantly less than this, with F A = N/2.
However, suppose the mixing in ρA arises from entanglement with an auxiliary system B before system A passes through the metrological device (see Fig. 1).Specifically, for an input pure state |Ψ AB of a composite system A ⊗ B, where ρA = Tr B {|Ψ AB Ψ AB |}, the QFI is Consequently, for a purification of ρA the QFI only depends on the variance in ĜA of ρA [41,49].Our naïve strategy of preparing a state with large Var( ĜA ) irrespective of its purity is now an excellent approach.Indeed, in this situation the states in Cases (I)-(III) are now also arranged in order of increasing QFI, with Case (II) and Case (III) attaining a QFI of N (N +1)/2 and N 2 , respectively.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 spincat state, which is a macroscopic superposition, rather than a classical mixture, of spin coherent states.Note also that F AB is independent of any particular purification, and convexity implies that F AB ≥ F A .That is, in principle any purification of 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 a mathematical 'trick'; physically, a purification corresponds to entangling 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 the existence of an entangling Hamiltonian that can prepare ρA in a state with large Var( ĜA ) ρA .
For the three cases described by Eq. ( 2) and Eqs.(3), a purification of ρA can be written as with Case (I) corresponding to B m |B n = 1, Case (II) corresponding to B m |B n = δ n,m , and Case (III) corresponding to B m |B n = 1(0) for |n − m| even (odd).In the following section, we present a simple scheme that converts a shot-noise limited spin coherent state [such as Case (I)] to the enhanced QFI purifications of Cases (II) and (III).

III. EXAMPLE ENTANGLING DYNAMICS LEADING TO INCREASED QFI
Consider again the N -boson probe state (system A) input into a MZ interferometer (i.e.ĜA = Ĵy ).The QFI for a purification of ρA can be written as with where Ĵ± = Ĵx ± i Ĵy .Note that F 0 , F 1 , and F 2 depend only on the matrix elements of ρA with |n − m| equal to 0, 1, and 2, in the Ĵz basis; writing F AB in this form is very convenient for what follows.
Before the interferometer, we assume the probe is coupled to some auxiliary system B via the Hamiltonian ĤAB = g Ĵz ĤB .
When system B is a photon field and ĤB is proportional to the number of photons in the field, then ĤAB describes the weak probing of the population difference of an ensemble of two-level atoms with far-detuned light [37,[52][53][54][55][56][57][58][59][60][61], or dispersive coupling between a microwave cavity and a superconducting qubit [62][63][64].We will explore this specific case shortly, however, for now we keep ĤB completely general.If the initial system state is a product state |Ψ AB (0) = |Ψ A ⊗ |Ψ B , after some evolution time the state of the system will be given by Eq. ( 6) with The reduced density operator of system A is then where the coherence of system A is determined via When C n−m = 1, the system remains separable and system A is a pure state, whereas if C n−m = δ n,m then ρA is an incoherent mixture of Dicke states.Using Eq. ( 10), F 0 , F 1 , and F 2 can be written as where the above expectation values are calculated with respect to |Ψ A .The effect of the entanglement between systems A and B is entirely encoded in the coherences C 1 and C 2 ; coherences greater than 2nd order do not affect the QFI.
Let us consider the effect on the QFI of each term in Eq. ( 7).F 0 is independent of the entanglement between systems A and B, and will be of order ).This suggests that a sufficient condition for Heisenberg scaling is In fact, since F 1 ≤ 0, the maximum QFI state must necessarily have C 1 = 0.In contrast, F 2 can be positive or negative, in which case a state with C 2 = 0 and another state with C 2 = 1 and F 2 ∼ +N 2 /2 might both be capable of (near) Heisenberg-limited metrology.We consider examples of both states below.

A. Case (II): Example dynamics yielding
To concretely illustrate the increased QFI a purification of ρA can provide, we assume system B is a single If the initial state of system B is a Glauber coherent state |β [65], then the coherences described by Eq. ( 11) simplify to Although the non-orthogonality of β|βe iθ ensures that C n−m never actually reaches zero, it becomes very small for even modest values of |β| 2 .
If the initial condition of system A is |Ψ A = |α(θ, φ) , then F AB has the simple analytic form given by (see Appendix A) In contrast, calculating F A via Eq.( 1) requires the diagonalization of ρA , which must be performed numerically.
We first demonstrate the effect of vanishing 1st and 2nd order coherence on F AB by preparing system A in the maximal Ĵx eigenstate, |α(π/2, 0) , with N = 100, and a Glauber coherent state for system B with average particle number |β| 2 = 500.The initial state for system A is precisely Case (I) [see Eq. (3a)], and has a QFI of N .As shown in Fig. 3, under the evolution of Eq. ( 13), ρA tends towards an incoherent mixture of Dicke states [Case (II)], with the corresponding broadening of the P (J y ) distribution.
Figure 4(a) shows that both coherences C 1 and C 2 rapidly approach zero, which causes F 1 and F 2 to vanish [see Fig. 4(b)].Consequently, F AB approaches F 0 = N (N + 1)/2, which allows a phase sensitivity of approximately √ 2×Heisenberg limit [see Fig. 4(c)].In contrast, the effect of the mixing causes the QFI of ρA itself to decrease from N to F A = N/2, with F A ≤ N for all t.This remains true even if the if G A is rotated to lie in any arbitrary direction on the Bloch sphere.
The oscillations in F 1 and F 2 (and consequently F A and F AB ) before the plateau are due to the complex rotation of C 1 and C 2 , which causes rotations of ρA around the J z axis before being overwhelmed by the overall decay in magnitude.Furthermore, although the purity of the state also decays, it never vanishes, thereby illustrating that it is not the entanglement per se that is causing the QFI enhancement for a purification of ρA .At gt = π, there is a revival in |C n | 2 for n even, but not for n odd. Figure 5 shows the Husimi-Q function under the 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 C 1 and C 2 rapidly increases to F AB = F 0 = N (N + 1)/2 as in the previous example.As gt → π, the revival of |C 2 | 2 causes F AB to briefly increase to N 2 (see Fig. 6).This is the Heisenberg limit, which is the QFI of a (pure) spin-cat state and the maximum QFI for SU(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, its Q-function is similar to that of a spin-cat state, and purifications of it behave 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 Ĵz projection was a conserved quantity, so any entanglement between systems A and B can only degrade the coherence in the Ĵz basis of system A (ultimately resulting in an enhanced QFI).The situation is more complicated when consider-ing a Hamiltonian that does not conserve the Ĵz projection, such as when a spin flip in system A is correlated with the creation or annihilation of a quantum in system B. Here, we encounter scenarios where the interaction can either create or destroy coherences in the Ĵz basis of system A, and although a significant QFI enhancement is still possible, it depends upon the initial state of system B.
First, consider the case when the initial state for system B is a large amplitude coherent state (i.e.|Ψ B = |β ).Here the addition/removal of a quantum to/from system B has a minimal effect on the state and the system remains approximately separable, since It is therefore reasonable to make the undepleted pump approximation b → β, such that Ĥ± → gβ Ĵx (assuming β is real).Hence, the effect of the interaction is simply a rotation around the J x axis, which can create coherence in the Ĵz basis, and so F A = F AB ≤ N for all time.
In the opposite limit where the initial state of system B is a Fock state with N B particles, |Ψ B = |N B , then and B m |B n = δ n,m .This ensures that the first and second order coherences vanish, and F 1 = F 2 = 0 for all time.That is, as illustrated in Fig. 7, the state moves towards the equator and ultimately evolves to an incoherent Dicke mixture [i.e.Case (II)].As described in Sec.III A, and shown in Fig. 8, the QFI increases to a maximum of approximately F AB ≈ N 2 /2.Although setting N B = 0 (i.e. a vacuum state) leads to a larger variance in Ĵz , F AB still reaches approximately 70% of N 2 /2.
We therefore see that for the Hamiltonian ( 16), a large QFI enhancement is achieved provided the initial state |Ψ B has small number fluctuations.Compare this to the Hamiltonian (13), where the choice |Ψ B = |N B leads to no entanglement between systems A and B, while in contrast an initial state with small phase fluctuations (and therefore large number fluctuations), such as a coherent state, causes rapid decoherence in ρA .

IV. OPTIMAL MEASUREMENT SCHEMES
Although the QFI determines the optimum sensitivity for a given initial state, it is silent on the question of how to achieve this optimum.It is therefore important to identify a) which measurements to make on each system and b) a method of combining the outcomes of these measurements -which we refer to as a measurement signal ( Ŝ) -that saturates the QCRB.We do this below for purifications of the incoherent Dicke mixture [Case (II)] and the pseudo-spin-cat state [Case (III)].
For an incoherent Dicke mixture, we have Ĵx = Ĵy = Ĵz = 0, and Ĵ2 x = N (N + 1)/8.Unfortunately, the non-zero variance in Ĵz (i.e.Ĵ2 z = N/4) implies that Var( Ŝ) 0 for all φ, and the signal no longer saturates the QCRB.However, since the states |B m in the purification Eq. ( 6) are orthonormal, a projective measurement of some system B operator diagonal in the |B m basis projects system A into a Ĵz eigenstate (i.e. a Dicke state).That is, these measurement outcomes on system B are correlated with Ĵz measurement outcomes on system A. Therefore, subtracting both measurements yields a quantity with very little quantum noise.
More precisely, if we can construct an operator ŜB on system B that is correlated with Ĵz measurements on system A (i.e.ŜB |Ψ AB = Ĵz |Ψ AB ), then we can construct the quantity Ŝ0 = Ĵz − ŜB which has the property Ŝ0 = Ŝ2 0 = 0.This motivates the signal choice Using ŜB |Ψ AB = Ĵz |Ψ AB and the fact that non-Ĵz conserving terms vanish due to the absence of off-diagonal terms in the Ĵz representation of ρA , (e.g.expectation values with an odd power of Ĵx vanish), we can show that Note that the above expectation values can be taken with respect to |Ψ AB or ρA .The best sensitivity occurs at small displacements around φ = 0. Taking the limit as φ → 0 and noting that Ĵ2 (20) This demonstrates that the signal Eq. ( 18) is optimal since it saturates the QCRB.

B. Optimal measurements for pseudo-spin-cat state [Case (III)]:
Pure spin-cat states have the maximum QFI possible for N 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, and since B n |B m = 1(0) for |n − m| even(odd), a projective measurement of system B yields no information other than the parity of the Ĵz projection.This suggests that a measurement of parity could be optimal.
In analogy with Case (II), our aim is to construct an operator Ŝ0 where the correlations between systems A and B lead to a reduction in Var( Ŝ0 ) and the system mimics a pure spin-cat state.Introducing the quantity where ΠA(B) is the parity operator for system A(B), defined by ΠA |j, m = (−1) m |j, m and ΠB |B m = (−1) m |B m , we see that pseudo-spin-cat states satisfy Ŝ0 |Ψ AB = |Ψ AB , and therefore Var( Ŝ0 ) = 0.This motivates the signal choice Ŝ = Û † φ Ŝ0 Ûφ .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 The relation B n |B n±1 = 0 ensures that terms linear in Ĵy go to zero: However, unlike Case (II), the condition B n |B n±2 = 1 preserves terms such as Ĵ2 + .Noting that Ĵy flips the parity of any state in subsystem A but not subsystem B: and using Ŝ0 Therefore Since Ŝ2 0 = 1 implies that Ŝ2 = 1, we obtain and consequently (28) This demonstrates that the signal saturates the QCRB and is therefore optimal.
The optimal estimation schemes presented in Secs.IV A and IV B illustrate a somewhat counterintuitive fact: although the optimal measurement of system B for a pseudo-spin-cat state provides less information about system A than for an incoherent Dicke mixture, the pseudo-spin-cat state yields the better (in fact best) sensitivity.

C. System B observables that approximate optimal measurements
We now turn to the explicit construction of physical observables that approximate ŜB .In general, the choice of ŜB depends upon the specific purification of ρA .Physically, the initial state of system B and the entangling Hamiltonian matter.However, there is no guarantee that ŜB exists, and if it does there is no guarantee that a measurement of this observable can be made in practice.Nevertheless, as we show below, it may be possible to make a measurement of an observable that approximates ŜB , and can therefore give near-optimal sensitivities.

Case (II)
To begin, consider the situation in Sec.III A: the evolution of the state |α(π/2, 0) ⊗ |β under the Hamiltonian (13).We require ŜB |Ψ AB = Ĵz |Ψ AB .After some evolution time t: Clearly, the phase of the coherent state is correlated with the Ĵz projection of system B.This can be extracted via a homodyne measurement of the phase quadrature ŶB = i( b − b † ) [70].In fact, provided mgt 1, phase quadrature measurements of |β exp(−imgt) are linearly proportional to the Ĵz projection: (30) where without loss of generality we have taken β to be real and positive.Consequently, the scaled phase quadrature satisfies and so the fluctuations in ( Ĵz − ŜB ) become arbitrarily small (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 measurement signal.More precisely, assume that The first inequality ensures that βe −ingt |βe −imgt ≈ δ n,m and so ρA is approximately an incoherent Dicke mixture, while the second inequality implies that we are in the linearized regime where Eq. ( 30) and Eqs.(32) hold.
Then the signal yields the sensitivity where Var( Ŝ) and ∂ φ Ŝ are given by the expectations (19) of the optimal signal Eq.(18). Figure (9) shows Eq. ( 35) compared to an exact numeric calculation.Condition (33) typically ensures that the term proportional to Ĵ4 z is small in comparison to the term proportional to 1/(2βgt) 2 .We therefore see that our approximate signal Ŝapprox gives a sensitivity worse than the QCRB, and furthermore at an operating point φ = 0. Nevertheless, ∆φ approaches the QCRB at φ = 0 as β 2 and (2βgt) 2 approach infinity.Therefore, for a sufficiently large βgt, we can achieve near-optimal 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 between Ĵz and Ŝb prevent the sensitivity from reaching the QCRB; nevertheless, the sensitivity is still below the SQL.Note that there is a slight deviation between Eq. ( 35) and the numerical calculation of the sensitivity using the state (29).This is due to terms neglected by our approximations; in particular, the nonlinear terms ignored by linearizations such as Eq. ( 30) and those neglected terms that arise due to the small (but strictly non-zero) off-diagonal elements of ρA .34) for a state of the form Eq. ( 29), with N = 100, and gt = 10 −2 .The blue dot-dashed line is with |β| 2 = 10 6 (βgt = 10), and the red solid line is for |β| 2 = 10 4 (βgt = 1).The red dashed line shows the approximate expression for the sensitivity (Eq.( 35)) for |β| 2 = 10 4 .For |β| 2 = 10 6 , the numerical calculation and Eq. ( 35) are identical.The upper and lower 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 correlations between ŜB and Ĵz leading to non-zero variance in Ŝ.If the correlations were perfect and Var( Ŝ)| φ=0 = 0, ∆φ would reach exactly 1/ √ FAB at φ = 0.

Case (III)
Now, consider the situation in Sec.III B: the evolution of the state |α(0, 0) ⊗ |β under the Hamiltonian (13) that at gt = π approximately results in a pseudo-spincat state.
In order to find an operator that approximates ŜB = ΠB , we introduce the amplitude quadrature operator XB = ( b + b † ), and notice that That is, amplitude quadrature measurements of |βe −imπ are proportional to parity measurements on system B, which are directly correlated with parity measurements on Ĵz eigenstates.Indeed, the quantity has a variance Var( Ŝ0 ) = 1/(2β) 2 that becomes vanishingly small as the amplitude of the coherent state is increased.We therefore expect the signal will be a good approximation to the optimal measurement Ŝ.  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 and red 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 a linear scale.
Figure 10 shows the sensitivity for a state of the form Eq. ( 29) at gt = π with N = 20.When |β| 2 = 30, the sensitivity is very close to the Heisenberg limit, while for |β| 2 = 5 there is a slight degradation in the sensitivity due to imperfect correlations.In contrast to the approximate optimal measurement scheme for Case (II), which requires a large amplitude coherent state, here the signal (38) is almost optimal even for small amplitude coherent 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 challenging to achieve the strong atom-light coupling regime required for gt = π.On the other hand, the choice of an initial coherent state for system B ensures that the sensitivity is reasonably insensitive to losses in system B. In particular, since particle loss from a coherent state acts only to reduce the state's amplitude, provided the coherent state remains sufficiently large after losses in order to satisfy the requirements for near-optimal measurements, near-Heisenberg-limited sensitivities should be obtainable.

Particle-exchange Hamiltonian
Finally, for completeness we include the optimal measurement scheme for the state attained after evolving the product state |α(0, 0) ⊗|N B under the Hamiltonian (16) (see Sec. III C).The optimal measurement signal is simply Eq. ( 18) with This choice of ŜB can be constructed by counting the number of particles in system B, and it satisfies ŜB |Ψ AB = Ĵz |Ψ AB as required.As any entangling Hamiltonian of the form will lead to a state of the form Eq. ( 6), with |B m given by Eq. ( 17) (assuming an initial state |Ψ A = |α(0, 0) , |Ψ B = |N B ), Eq. ( 39) also transfers to these systems.
V. DISCUSSION AND CONCLUSIONS We have shown that purifications of mixed states represent an excellent resource for quantum metrology.In particular, we showed that if probe system A and auxiliary system B are entangled such that the 1st and 2nd order coherences of system A vanish, then near-Heisenberglimited sensitivities can be achieved provided measurements on both systems A and B are allowed.Although we focused on the situation where this entanglement is generated via a few specific Hamiltonians, our conclusions hold irrespective of the specific entanglement generation scheme.
While preparing this paper, we also numerically examined the effect of decoherence on the sensitivity of our metrological schemes.In particular, we found that the effect of particle loss, spin flips, and phase diffusion on purifications of the pseudo-spin-cat state from Fig. 2 was identical to that of a pure spin-cat state.
Although these purified states are no more or less robust 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 interferometry, where atom-light entanglement and information recycling is more compatible with the requirements of high precision atom interferometry than the preparation of nonclassical atomic states via interatomic interactions [39].However, controlled interactions are routinely engineered between atoms and light [37,[71][72][73], superconducting circuits and microwaves [74,75], light and mechanical systems [76], and ions and light [77][78][79].Given that high efficiency detection is available in all these systems [80][81][82][83][84][85], the application of our proposal to a range of metrological platforms is plausible in the near term.
It is important to note that although the QFI approaches the Heisenberg limit (F AB = N 2 , in Case (III), for example), this is not the true Heisenberg limit, as N refers only to the number of particles in system A (which pass through the interferometer), rather than the total number of particles N t in system A and system B.However, there are some situations where the number of particles in system A is by far the more valuable resource, which is why it makes sense to report the QFI in terms of N rather than N t .For example, consider the case of inertial sensing with atom interferometry, where system A is atoms, and system B is photons.The atoms are sensitive to inertial phase shift, but it is difficult to arbitrarily increase the atomic flux.However, a gain can be achieved by adding some number of photons to the system, which are comparatively 'cheep' compared to atoms.
Finally, we note that not all quantum systems are created equal; certain quantum information protocols, such as quantum error correction [86] and no-knowledge feedback [87], are better suited to some platforms than others.Our proposal allows an experimenter to both perform quantum-enhanced metrology and take advantage of any additional benefits a hybrid quantum system provides.
bosonic mode, described by annihilation operator b, and take ĤB = b †b such that ĤAB = g Ĵz b † b .

FIG. 3 .FIG. 4 .
FIG. 3. (Color online) Time snapshots of the Husimi-Q function and Ĵy projection illustrating the evolution of a maximal Ĵx eigenstate under entangling Hamiltonian Eq. (13).The snapshots were chosen to correspond to times when the rotation around the Jz axis is such that Ĵy = 0, which roughly corresponds to the local maxima of FAB in Fig.4(c).(Parameters: N = 100, |β| 2 = 500).

FIG. 10
FIG.10.(Color online) Phase sensitivity of the approximate signal 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 and red 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 a linear scale.