Optimal estimation of joint parameters in phase space

Abstract

We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cram\'er-Rao bound, providing the theoretical benchmark on the estimation and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.Comment: 8 pages, 3 figures; v2: references added and sections added to the supplemental material; v3: minor changes (published version

Bounds on quantum multiple-parameter estimation with Gaussian state

“…Using the same homodyne measurement scheme proposed in Ref. [11], the sum the two resulting variances is Fig. 2 displays the three bounds B R,S,M versus the squeezing parameter r. We see that which bound is tighter depends on the actual values of r, and the bound B M from the homodyne measurement is always higher than the theoretical RLD and SLD bounds.…”

“…Next, we jointly estimate the two conjugate parameters λ R and λ I of the displacement operator D(λ) = e λa † 1 −λ * a1 with a measurement on the displaced state ρ = D(λ)ρ 0 D † (λ) [11]. If we take the two-mode squeezed thermal state ρ 0 = S 2 (r)(ρ νT ⊗ ρ νT )S † 2 (r) as the input, where…”

“…[10,11]). For single-parameter estimation problems, it is known that the SLD QFI is always smaller than the RLD QFI, and thus gives a tighter bound for parameter sensitivity [4].…”

Role of entanglement in calibrating optical quantum gyroscopes

“…While the QCRB for a single parameter can generally be attained, this is not always true of the QCRB for multiple parameters [22,23]. Where multiple parameters are being estimated, it matters whether the generators of translation of the parameters commute or not, with implications for the optimal strategies of the parameter estimation procedures [24][25][26][27][28][29]. Even though multimode entanglement can be used to improve the estimation of multiple phase parameters beyond the classical SNL [13], this is not always the case.…”

Monitoring dispersive samples with single photons: the role of frequency correlations

“…While typically all the information needed can be e ciently collected through a single parameter [4,5], there are instances in which two parameters or more are necessary to capture the physical process under study [6][7][8]. Such parameters might not be associated to compatible observables, hence trade-o may appear in attempts at simultaneously measuring them at the ultimate quantum precision, especially when restrictions are imposed on the resources or, in other words, to the available Hilbert space [7,[9][10][11][12][13][14][15] These trade-o can be interpreted, and often circumvented, by understanding the estimation process under the geometrical standpoint by identifying the physical carrier of information with their state vectors [13]; however, quantum probes only partly approximate such geometric entities, since these typically describe one degree of freedom at the time. The interaction with the sample might actually depend on other degrees of freedom, on which we might have limited control.…”