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Quantum Information and Measurement (QIM)

Trade-offs in the simultaneous quantum-limited estimation of phase and phase diffusion

Magdalena Szczykulska, Tillmann Baumgratz, Animesh Datta

Published Jan 1, 2017• 0 citations
Publisher

Abstract

We examine the precision attainable in the joint estimation of phase and diffusion by fixed particle number states within the quantum Cramer-Rao bound framework. We provide analytical understanding of the large and small diffusion regimes.
DOI
10.1364/qim.2017.qt6a.15

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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.
Phase diffusion invariably accompanies all phase estimation strategies – quantum or classical. A precise esti- mation of the former can often provide valuable understanding of the physics of the phase generating phenom- ena 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 si- multaneous local estimation of phase and phase diffusion within the quantum Crame ́r-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 measure- ments 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 Crame ́r-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.