Heisenberg scaling in Gaussian quantum metrology
We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires non-classical, but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to monitor only finitely many modes, for which we identify a lower bound.
pacs:06.20.-f, 42.50.-p, 04.62.+v, 03.65.Ta
Quantum metrology exploits distinctive quantum features, such as entanglement, to enhance the estimation precision of parameters governing the dynamical evolution of the probe systems beyond that achievable by classical means. This enhancement is manifested in the form of a scaling gap in precision with respect to the available resources (the number of probe systems or the average input energy) between the corresponding optimal quantum and classical strategies, and depends on the specific encoding of the parameter in the Hamiltonian describing the evolution. In the case where the parameter of interest is a multiplicative factor of a local Hamiltonian acting on probes, the optimal quantum strategy provides a quadratic scaling gap in , known as the Heisenberg limit, over the best classical strategy (1); (2). This quadratic improvement is also present for a class of quasilocal Hamiltonians (3); (4). For Hamiltonians involving highly non-local interactions, super-Heisenberg scaling is also possible (5); (6); (7). Nonetheless, the paradigmatic example for a scaling gap is the estimation of a phase acquired in one arm of a Mach-Zehnder interferometer (1). There, the two input modes are subject to a particular Gaussian transformation, i.e., a combination of beam splitters and a phase shifters.
Here we are interested in determining the ultimate precision limits for a more general type of parameter estimation task, namely, where the parameter of interest is encoded nontrivially in an arbitrary Gaussian transformation of not two but possibly infinitely many modes of a quantum field. This problem is of broad interest since such transformations, often cast in the form of Bogoliubov transformations, feature in a large variety of physical systems throughout quantum optics (8) and condensed matter physics (9). In exchange for allowing for a broader class of Gaussian transformations, we restrict our approach to a regime of small parameters to gain analytical insights. Besides applications such as estimating (weak) single-mode or multimode squeezing in optical or superconducting (see, e.g., (11); (10)) systems, perturbative Bogoliubov transformations of this kind are of particular interest for the description of quantum effects in curved spacetime, such as the Unruh effect (12) and the dynamical Casimir effect (13), or analogous realizations thereof (14); (15); (16); (17); (18). A paradigm that highlights the challenges encountered in the context of estimating relativistic quantum effects (19); (20); (21); (22); (23) is the estimation of the acceleration of a nonuniformly moving cavity (24). While the parameter in question is small, the Gaussian transformation couples all pairs of modes in nontrivial ways (25); (26). Due to the resulting notoriously cumbersome perturbative calculations, the only known bounds on precision involve Gaussian input states. Consequently, the ultimate limits on how precisely one can determine small accelerations, as well as whether distinctive quantum features provide an improvement, are not known.
Here we show that, within such a perturbative approach, there exists a quadratic scaling gap between the optimal quantum and classical strategies for a fixed average input energy. We construct an optimal estimation strategy utilizing separable input states of fixed particle number and boson counting measurements, where we use the quantum Fisher information (QFI) as a figure of merit for the estimation precision. We provide simple formulas for the leading-order contributions to the QFI for arbitrary pure states, both when all modes can be controlled and when only part of the spectrum is accessible, and show that, within the considered regime, Heisenberg scaling is the ultimate precision limit. Moreover, we identify the family of states that exhibit Heisenberg scaling and show that, while some of these states may be entangled, the crucial feature is their nonclassicality rather than their correlations. Furthermore, we lower bound the loss in precision due to tracing out inaccessible modes and provide criteria for minimizing such losses.
In its most general form, our metrological protocol can be described as follows: We wish to estimate as precisely as possible a single real parameter that is encoded in a unitary transformation acting on the Fock space of an arbitrary number of bosonic modes. Our probe system is a set of (in principle infinitely many) noninteracting harmonic oscillators with creation and annihilation operators and , respectively, where can be a multilabel distinguishing frequencies, polarizations, or other degrees of freedom. The operators satisfy the usual commutation relations and and the ground state is annihilated by all , i.e., . Arbitrary pure states can be decomposed into superpositions of Fock states, e.g., , with fixed numbers of excitations. These Fock states form a basis of the total Hilbert space and can be obtained from the vacuum by applying the appropriate creation operators, i.e., .
For a given input state and a suitable measurement whose outcomes are used to estimate , the precision of the estimation of , quantified by the variance of the corresponding (unbiased) estimator, is lower bounded by the inverse of the QFI (27) via the (quantum) Cramr-Rao inequality (28); (29) , where is the number of repetitions. Intuitively, the QFI quantifies how well small changes of the parameter in question may be inferred from measurements of the final state after the dynamical evolution of the probes. One may express as
where is the Uhlmann fidelity between the states and . If one simply recovers . The latter expression applies for the case where all modes can be controlled and measured. Expanding in powers of , one obtains
where . When some of the modes are not accessible and are traced out, Eq. (2) provides an upper bound on the QFI. To allow for more specific statements about the QFI, additional information about the initial states or transformations encoding is required.
Iii QFI in the perturbative regime
iii.1 Parameter estimation for pure states
We now consider cases where is very small and close to zero. For this case the transformation is amenable to a perturbative approach and has a Taylor-Maclaurin expansion of the form
where is a unitary that encodes the free time evolution of the noninteracting bosonic modes, while for the represent (nonunitary) perturbations. Consequently, the final state can be written as
Hence, given that all the bosonic modes are accessible, the QFI can be straightforwardly computed for any initial (pure) state from just its linear perturbations.
As all our results utilize the perturbative approach, a few important remarks concerning the applicability of the perturbative approach are in order. In the latter we assume that all higher-order terms in are small and are therefore neglected. To ensure such reasoning is justified, we regard our approach as valid so long as the perturbation to the state remains small, such that . This implies that and all optimizations will be performed with this constraint in mind.
iii.2 Tracing losses
Thus far we have assumed that all bosonic modes are accessible and can be controlled. This is an unrealistic assumption in practice, as only a finite subset of the modes can be addressed simultaneously. Moreover, as the estimation strategy is to be optimized at a fixed investment of energy, it appears unwise to initially populate modes in the complementary subset , which cannot be measured. We therefore assume that the preparation of nontrivial initial states is also limited to . Hence, the input states that we consider are of the form . The unitary , on the other hand, acts on all modes such that the final state is , but only the reduced state is accessible for the estimation of (see Fig. 1). Thus, Eq. (5) provides an upper bound, i.e., for the precision with which can be estimated. However, a precise expression for the losses incurred by tracing can be established as we will show now. Let us expand the reduced state in powers of ,
where the leading order is and the corrections are given by
As the modes are noninteracting, the free evolution is a local operation that leaves the vacuum invariant and we may write . It then easily follows that is a pure state. In a similar way we may expand as
for which a detailed proof is given in the Appendix. We pause briefly to highlight the simplicity of Eq. (10): One only requires a single matrix element in the second-order expansion of in order to compute the QFI, to leading order in , for any input state .
where is given as in Eq. (5) and the sum runs over all basis states of the Fock space of that are orthogonal to the vacuum state . Note that Eq. (11) suggests that in order to compute the QFI to leading-order in one requires only leading order corrections to the initial state, whereas Eq. (10) above implied that second-order corrections are needed. However, as is shown in detail in the Appendix, the requirement that the transformation is unitary implies that the second-order correction in Eq. (10) can be expressed in terms of the linear order corrections to the state.
iii.3 Parameter estimation from Bogoliubov transformations
For the remainder of this paper, we will work with Eq. (11) to investigate how tracing losses can be minimized and optimal scaling can be achieved. For this task, we will specialize our discussion to Bogoliubov transformations (30), that is, unitary transformations on the Fock space that can be viewed as linear transformations of the mode operators, i.e.,
where unitarity places restrictions on the complex coefficients and . Another way to view such transformations is as unitaries that are generated by quadratic combinations of the mode operators. In the perturbative regime that we consider here, the Bogoliubov coefficients are expanded as
where encodes the phases that are accumulated due to the free time evolution . The coefficients and are responsible for the shifts of single excitations between modes and or the creation and annihilation of pairs of particles in these modes, respectively. Any initial state of the Fock space may simply be written in terms of linear combinations of products of creation operators acting on the vacuum. The operators transform according to Eq. (12), whereas the vacuum transforms as
where and must be a Hilbert-Schmidt operator to guarantee unitarity (see, e.g., Ref. ((31), pp. 93 and 101)). We now determine the tracing losses and the scaling of with the average energy of the initial state in the above scenario.
The linear order operation can be (uniquely) separated into (all) terms that leave the particle content of or invariant and a term that correlates both sets via the creation or exchange of excitations, i.e.,
From Eq. (11) it is obvious that does not contribute to . The other two terms generally provide nonzero contributions. However, to linear order, the operation populates with pairs of particles, whereas may either create one particle each in and or shift one excitation from to . As a result, for the same fixed state , which contains at most two excitations, and cannot both be nonzero and hence contribute separately to the tracing loss. The latter term can be avoided by choosing such that superpositions of states that differ only by one excitation are excluded. The former term, on the other hand, cannot be avoided as it is independent of the state . It can, however, straightforwardly be computed using the vacuum state of Eq. (15), for which . For this case we find (see the Appendix)
Thus, by choosing our initial state such that superpositions of states that differ only by one excitation are excluded, the inequality in Eq. (17) becomes a strict equality and the QFI can be precisely determined to linear order in a straightforward way using Eq. (11). We will restrict our discussion to such states from now on.
We now show that the optimally achievable precision in this setup is Heisenberg scaling and how it can be realized even with nonentangled states. To determine the optimal states, the exact values of the Bogoliubov coefficients and have to be known. This makes the procedure highly task specific. Nonetheless, it is possible to identify families of states that scale optimally with the input energy under some minimal assumptions about the Bogoliubov coefficients. First, consider a single-mode Fock state with excitations. Using Eqs. (12)-(15), this state transforms to
As , one immediately obtains the QFI
Notice that, here, the coefficient , which leaves the occupation number of the mode unchanged, does not contribute to the QFI, but it may do so for superpositions of different particle numbers. However, when , which corresponds to single-mode squeezing transformations, terms that shift the population of the mode by two excitations yield QFI that scales quadratically with , i.e., Heisenberg scaling. On the other hand, classical states, which in the present context are all coherent states, yield QFI that scales linearly with the average particle number (24).
As an example for the application of Eq. (19), consider a scenario in quantum optics (32) where only a single mode labeled is subject to weak single-mode squeezing, while all other modes are left invariant. The corresponding Bogoliubov transformation is represented by the coefficients and . The optimal state for the estimation of is hence already and the QFI is .
When the diagonal linear coefficients vanish, as in the transformation of field modes of nonuniformly accelerating rigid cavities (25); (26), excitations in a single mode are not sufficient to obtain Heisenberg scaling and one requires initial states with at least two occupied modes. For instance, when the modes and can be controlled and , the QFI for the state is (see the Appendix)
The factor allows the QFI to scale optimally with the average number of excitations for the estimation of beam-splitting () and two-mode squeezing () terms. Indeed, inspection of the first-order expansion of reveals that Heisenberg scaling is the ultimate achievable precision scaling within our approach (see the Appendix). For the explicit example of a pure two-mode squeezing transformation, which may be realized in a variety of physical systems, e.g., in superconducting circuitry (33), the Bogoliubov coefficients are given by and . We hence recover the well known result that in this case the state is optimal and the QFI from Eq. (LABEL:eq:quantum_fisher_info_m_plus_n_particles) yields .
Whereas the scaling with respect to the average energy cannot be better than quadratic, even for arbitrary Gaussian transformations beyond the examples presented so far, one can still improve the constant prefactor in this scaling. For example, assuming and , the particle numbers and in the state guarantee additional optimally scaling terms in the QFI while avoiding losses originating from . The latter is nonzero whenever contains superpositions of Fock states that can be converted into each other by changes of at most two excitations, such as and , or and .
Notice that in all of the examples considered so far the optimal states are not entangled, but can be regarded as highly squeezed and hence nonclassical. The final state may become entangled due to the transformation, but, as seen from (18) and (19), optimal scaling may arise from terms (here ) that do not produce entanglement. Indeed, entanglement of the initial state is not necessary for optimality (a feature that was also noted in a related context in (34)), although one may find entangled states that do admit Heisenberg scaling in precision such as the state , which also has minimal tracing loss. For the entangled state above, homodyne measurements would be required for optimal scaling, whereas the measurements for the previous examples need to be number resolving. The choice of input state and measurement is thus also a matter of what is practical in a given experimental setup.
We have shown that the ultimate limits for estimating small parameters encoded in the linear transformations of the mode operators describing a set of noninteracting harmonic oscillators scale inversely proportional to the average input energy of the modes. Using a perturbative approach for these Bogoliubov transformations, applicable when the parameter of interest is small, e.g., as in the case of nonuniformly moving cavities, we have derived analytical formulas for the QFI, which apply to initial overall pure states of all modes. We have provided a lower bound on the losses when some of these modes cannot be controlled. Finally, we have identified the class of states that yield optimally scaling precision while exhibiting minimal tracing losses. This provides a significant advancement beyond previous analysis of this problem (24), which was limited to pure Gaussian states. An investigation of mixed initial states, such as the mixed Gaussian states that were investigated recently (35), goes beyond the scope of the current paper, but is certainly of interest.
Our results open up the possibility to explore optimality in a range of specific applications, where information about the Bogoliubov coefficients is available. Examples include field modes in non-uniformly moving cavities (25); (26), analog gravity phenomena (14), and effects in superconducting materials (15); (17); (36).
Acknowledgements.We are grateful to Hans J. Briegel, Vedran Dunjko, Antony R. Lee, and Pavel Sekatski for valuable discussions and comments. This work was supported by the Austrian Science Fund (FWF) through Grants No. SFB FoQuS F4012 and No. P24273-N16.
Appendix A Explicit calculations
In this appendix we present explicit proofs of some key results presented in the main text. In particular, we give a detailed derivation of the QFI for the estimation of small parameters , encoded in Bogoliubov transformations, and we discuss the optimality of Heisenberg scaling.
a.1 Reduced state QFI
We start with the QFI for a mixed state , which we will show to be given by
where , , and have small-parameter expansions of the form
with . The QFI of the state may be expressed via the Bures distance as
where the Uhlmann fidelity is given by
With Eq. (22) at hand, the expansion in terms of can be seen to be of the form
where we have made use of the fact that is pure. Now we can take a closer look at the term linear in , i.e.,
where we have used . On the other hand, the unitarity of the transformation requires
Since , it follows that all other terms on the right-hand side of Eq. (31) must vanish. We hence find
Reinserting this into Eq. (29), we obtain the Uhlmann fidelity
which in turn yields the QFI of Eq. (21).
a.2 Relation to pure state QFI
Next we show that the QFI of the mixed state , arising from tracing out some of the modes from the pure state , that is, , may be related to the QFI of the latter pure state via the expression