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arXiv:1206.0043 · Physical Review A

Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry

Philip J. D. Crowley, Animesh Datta, Marco Barbieri, Ian A. Walmsley

Published Feb 27, 2014• 141 citations
arXivHTMLPDF

Abstract

Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cram\'er-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.

DOI
10.1103/physreva.89.023845

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Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cram'er-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.
Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not attainable, leading to trade-offs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a trade-off in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cramer-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters, and has implications for sophisticated sensing scenarios such as quantum imaging.
Optical interferometry is a cornerstone technique for precise phase measurements across various fields. In many applications, for example, biological imaging, it often necessitates stringent limits on light intensity to prevent adverse effects on light-sensitive samples, a condition known as dose-limited regimes. Maximizing the precision per dose is therefore crucial. In quantum metrology, quantum correlations enable high precision in phase estimation while adhering to dose constraints. Nevertheless, photon loss, including absorption by a sample, substantially diminishes the benefits of quantum enhancement in interferometry. In this work, we experimentally investigate a dose-efficient approach to quantum phase estimation using sequential strategies in the presence of loss. Performance of sequential strategies with and without control is evaluated through quantum Fisher information (QFI) per dose. Experimental results show that both sequential strategies exceed the classical limit and outperform the parallel strategy using unbalanced N00N states. Notably, the control-enhanced sequential strategy attains superior QFI per dose, approaching the quantum limit. These results highlight the promise of sequential strategy for imaging and sensing in resource-constrained scenarios, marking a significant step toward practical and efficient quantum metrology in lossy environments.
With the rapid development of quantum technologies in recent years, the need for high-sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high-precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work, we derive the quantum Fisher information matrix associated with the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of photons. We find the optimal measurement attaining the ultimate precision for both parameters in a single setup and perform an experiment based on a pair of photons with an unknown degree of indistinguishability entering a two-port interferometer.
Astronomical interferometry is a cornerstone technique for high-resolution stellar imaging and observational astrophysics, extracting spatial information from the coherence of light collected by separated telescopes. Since the degree of coherence is complex, a genuine imaging task requires the joint recovery of the modulus and the relative phase, instead of independent singleparameter estimations. We investigate the simultaneous estimation of both parameters based on direct interferometry scheme and continuou-svariable quantum teleportation scheme. We find that in simultaneous estimation the direct interferometry scheme consistently yields a lower quantum Cramér-Rao bound, demonstrating its superiority over the continuous-variable quantum teleportation scheme. Furthermore, we establish the conditions under which the classical Cramér-Rao bound for Gaussian measurements saturates the quantum Cramér-Rao bound, identifying heterodyne detection as a near-optimal measurement scheme in the large mean photon number regime. An analysis of transmission loss reveals that the direct interferometry scheme yields superior precision in the short-baseline regime, whereas the continuous-variable quantum teleportation scheme outperforms it at longer baselines.
Abstract Phase diffusion invariably accompanies all phase estimation strategies, quantum or classical. A precise estimation of the former can often provide valuable understanding of the physics of the phase generating phenomena itself. We theoretically examine the performance of fixed-particle number probe states in the simultaneous estimation of phase and collective phase diffusion. We derive analytical quantum limits associated with the simultaneous local estimation of phase and phase diffusion within the quantum Cramér-Rao bound framework in the regimes of large and small phase diffusive noise. The former is for a general fixed-particle number state and the latter for Holland-Burnett states, for which we show quantum-enhanced estimation of phase as well as phase diffusion. We next investigate the simultaneous attainability of these quantum limits using projective measurements acting on a single copy of the state in terms of a trade-off relation. In particular, we are interested how this trade-off varies as a function of the dimension of the state. We derive an analytical bound for this trade-off in the large phase diffusion regime for a particular form of the measurement, and show that the maximum of 2 , set by the quantum Cramér-Rao bound, is attainable. Further, we show numerical evidence that as diffusion approaches zero, the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime and suggest that the trade-off for estimating one parameter with quantum-limited precision leads to a complete lack of precision for the other parameter as the diffusion strength approaches zero. Finally, we provide numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett probe states.
By using a systematic optimization approach, we determine quantum states of light with definite photon number leading to the best possible precision in optical two-mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit, thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.