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arXiv:2412.16023 · SciPost Physics

Knowledge-dependent optimal Gaussian strategies for phase estimation

Ricard Ravell Rodríguez, Simon Morelli

Published May 27, 2025• 0 citations
arXivHTMLPDF

Abstract

When estimating an unknown phase rotation of a continuous-variable system with homodyne detection, the optimal probe state strongly depends on the value of the estimated parameter. In this article, we identify the optimal pure single-mode Gaussian probe states depending on the knowledge of the estimated phase parameter before the measurement. We find that for a large prior uncertainty, the optimal probe states are close to coherent states, a result in line with findings from noisy parameter estimation. However, with increasingly precise estimates of the parameter, it becomes beneficial to put more of the available energy into the squeezing of the probe state. Surprisingly, there is a clear jump, where the optimal probe state changes abruptly to a squeezed vacuum state, which maximizes the Fisher information for this estimation task. We use our results to study repeated measurements and compare different methods to adapt the probe state based on the changing knowledge of the parameter according to the previous findings.

DOI
10.21468/scipostphys.18.5.167

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When estimating an unknown phase rotation of a continuous-variable system with homodyne detection, the optimal probe state strongly depends on the value of the estimated parameter. In this article, we identify the optimal pure single-mode Gaussian probe states depending on the knowledge of the estimated phase parameter before the measurement. We find that for a large prior uncertainty, the optimal probe states are close to coherent states, a result in line with findings from noisy parameter estimation. However, with increasingly precise estimates of the parameter, it becomes beneficial to put more of the available energy into the squeezing of the probe state. Surprisingly, there is a clear jump, where the optimal probe state changes abruptly to a squeezed vacuum state, which maximizes the Fisher information for this estimation task. We use our results to study repeated measurements and compare different methods to adapt the probe state based on the changing knowledge of the parameter according to the previous findings.
The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators XP+PX, i.e., a non-Gaussian measurement, is required to be fully optimal.
Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators X^P^+P^X^\hat X\hat P + \hat P\hat XX^P^+P^X^ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.
We consider an instance of ``black-box'' quantum metrology in the Gaussian framework, where we aim to estimate the amount of squeezing applied on an input probe, without previous knowledge on the phase of the applied squeezing. By taking the quantum Fisher information (QFI) as the figure of merit, we evaluate its average and variance with respect to this phase in order to identify probe states that yield good precision for many different squeezing directions. We first consider the case of single-mode Gaussian probes with the same energy, and find that pure squeezed states maximize the average quantum Fisher information (AvQFI) at the cost of a performance that oscillates strongly as the squeezing direction is changed. Although the variance can be brought to zero by correlating the probing system with a reference mode, the maximum AvQFI cannot be increased in the same way. A different scenario opens if one takes into account the effects of photon losses: coherent states represent the optimal single-mode choice when losses exceed a certain threshold and, moreover, correlated probes can now yield larger AvQFI values than all single-mode states, on top of having zero variance.
We address the problem of continuous-variable quantum phase estimation in the presence of linear disturbance at the Hamiltonian level by means of Gaussian probe states. In particular we discuss both unitary and random disturbance by considering the parameter which characterizes the unwanted linear term present in the Hamiltonian as fixed (unitary disturbance) or random with a given probability distribution (random disturbance). We derive the optimal input Gaussian states at fixed energy, maximizing the quantum Fisher information over the squeezing angle and the squeezing energy fraction, and we discuss the scaling of the quantum Fisher information in terms of the output number of photons, nout{n}_{\mathsf{out}}nout​. We observe that, in the case of unitary disturbance, the optimal state is a squeezed vacuum state and the quadratic scaling is conserved. As regards the random disturbance, we observe that the optimal squeezing fraction may not be equal to one and, for any nonzero value of the noise parameter, the quantum Fisher information scales linearly with the average number of photons. Finally, we discuss the performance of homodyne measurement by comparing the achievable precision with the ultimate limit imposed by the quantum Cram\'er--Rao bound.
Summary form only given. The use of quantum states of light in optical interferometry improves the precision in the estimation of a phase shift, paving the way for applications in quantum metrology, computation and cryptography. Sub-shot noise phase sensing can for example be achieved by injecting a squeezed vacuum into an interferometer . However, this approach leads to enhanced sensitivity only for small phase shifts. In this work we aim for ab initio sub-shot noise estimation of an unknown phase shift using a pre-determined squeezed probe and an adaptive measurement approach. We experimentally investigate the performances of such protocol under the realistic assumption of thermalization of the probe state. Indeed, adaptive phase estimation schemes with squeezed states and Bayesian processing of homodyne data have been shown to be asymptotically optimal in the pure case, thus approaching the quantum Cramér-Rao bound. In our protocol we take advantage of the enhanced sensitivity of homodyne detection in proximity of the optimal phase which maximizes the homodyne Fisher information. A squeezed thermal probe state (signal) undergoes an unknown phase shift. The first estimation step involves interference on a beam splitter of the signal and a local oscillator followed by homodyne detection. Homodyne data is then processed to compute a rough estimation of the phase through Bayesian inference. The rough estimation is fed back to the local oscillator in order to match the optimal relative phase with the signal. A second estimation step leads to the final estimation of the phase shift. Thermalization of the probe state prevents the attainability of the quantum Cramér-Rao bound. Nevertheless, we show that the studied adaptive scheme still saturates the classical Cramér-Rao bound, showing sub-shot noise behaviour and therefore extracting the maximum available information from homodyne data. In contrast to previous approaches, our scheme is optimized for Gaussian states.
We propose two-stage optimized strategies for full-period optical phase estimation with single-mode Gaussian states and Gaussian measurements under a fixed energy constraint. In the first stage (Stage I), displaced squeezed probes and heterodyne measurements provide coarse localization of the phase to a window on the circle. In the second stage (Stage II), squeezed-vacuum probes with adaptive homodyne measurements perform efficient phase estimation inside the selected window. We derive a generalized Cramér-Rao bound for this family of two-stage Gaussian strategies, which contains the contribution from local parameter estimation in Stage II plus an overshoot penalty from coarse localization errors in Stage I. For E <= 25 photons and squeezing limited to 12 dB, protocols using displaced squeezed states in Stage I reduce the optimized two-stage bound relative to protocols using coherent states in Stage I, and remain within a factor of 3 to 30 of the idealized local squeezed-vacuum quantum Cramér-Rao bound.