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arXiv:1502.07924

Quantum parameter estimation using multi-mode Gaussian states

Dominik Šafránek, Antony R. Lee, Ivette Fuentes

Published Feb 27, 2015• 1 citation
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Abstract

Gaussian states are of increasing interest in the estimation of physical parameters because they are easy to prepare and manipulate in experiments. In this article, we derive formulae for the optimal estimation of parameters using two- and multi-mode Gaussian states. As an application of our result, we derive the optimal Gaussian probe states for the estimation of the parameter characterizing a one-mode squeezing channel.

DOI
10.48550/arxiv.1502.07924

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Gaussian states are of increasing interest in the estimation of physical parameters because they are easy to prepare and manipulate in experiments. In this article, we derive formulae for the optimal estimation of parameters using two- and multi-mode Gaussian states. As an application of our result, we derive the optimal Gaussian probe states for the estimation of the parameter characterizing a one-mode squeezing channel.
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Abstract We discuss the ultimate precision bounds on the multiparameter estimation of single- and two-mode pure Gaussian states. By leveraging on previous approaches that focused on the estimation of a complex displacement only, we derive the Holevo Cramér–Rao bound (HCRB) for both displacement and squeezing parameter characterizing single and two-mode squeezed states. In the single-mode scenario, we obtain an analytical bound and find that it degrades monotonically as the squeezing increases. Furthermore, we prove that heterodyne detection is nearly optimal in the large squeezing limit, but in general the optimal measurement must include non-Gaussian resources. On the other hand, in the two-mode setting, the HCRB improves as the squeezing parameter grows and we show that it can be attained using double-homodyne detection.
Gaussian quantum states and channels are pivotal across many branches of quantum science and their applications, including the processing and storage of quantum information, the investigation of thermodynamics in the quantum regime, and quantum computation. The great advantage is that Gaussian states are experimentally accessible via their first and second statistical moments. In this work, we investigate parameter estimation for Gaussian states, in which the probe-state preparation stage involves two noncommutative Gaussian measurements on the position and momentum observables, introducing tunable parameters. The influence of these noncommutative Gaussian measurements is investigated through the quantum Fisher information (QFI). We showed that the QFI for characterizing Gaussian channels can be increased by adjusting the uncertainty parameters in the preparation of the probe state. Furthermore, if the probe is initially in a thermal state, probe-state preparation may generate quantum coherence in its energy basis. We showed that not only does the amount of coherence affect the improvement of the QFI, but also the rate of change of the coherence with respect to the parameter to be estimated. The proposed probe-state protocol is applied to two paradigmatic single-mode Gaussian channels, the attenuator and amplification channels, which are building blocks of Gaussian quantum information. Our results contribute to the use of coherence in quantum metrology and are experimentally feasible in quantum-optical devices.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
We calculate the quantum Cram\'er-Rao bound for the sensitivity with which one or several parameters, encoded in a general single-mode Gaussian state, can be estimated. This includes in particular the interesting case of mixed Gaussian states. We apply the formula to the problems of estimating phase, purity, loss, amplitude, and squeezing. In the case of the simultaneous measurement of several parameters, we provide the full quantum Fisher information matrix. Our results unify previously known partial results and constitute a complete solution to the problem of knowing the best possible sensitivity of measurements based on a single-mode Gaussian state.