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arXiv:1307.0470

Estimation of Phase and Diffusion: Combining Quantum Statistics and Classical Noise

Sergey Knysh, Gabriel A. Durkin

Published Jul 1, 2013• 13 citations
arXivHTMLPDF

Abstract

Coherent ensembles of $N$ qubits present an advantage in quantum phase estimation over separable mixtures, but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strength of diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spin systems using a novel mathematical formulation. For the first time, we show a closed form expression for the quantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probe state has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precision bound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noise limit is beaten only by a constant factor, independent of $N$. We provide upper and lower bounds to this factor, valid in low and high noise regimes. Unlike other noise types, it is shown for $N \gg 1$ that phase and diffusion can be measured simultaneously and optimally.

DOI
10.48550/arxiv.1307.0470

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Coherent ensembles of NNN qubits present an advantage in quantum phase estimation over separable mixtures, but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strength of diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spin systems using a novel mathematical formulation. For the first time, we show a closed form expression for the quantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probe state has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precision bound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noise limit is beaten only by a constant factor, independent of NNN. We provide upper and lower bounds to this factor, valid in low and high noise regimes. Unlike other noise types, it is shown for N≫1N \gg 1N≫1 that phase and diffusion can be measured simultaneously and optimally.
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.
Accurate phase estimation plays a pivotal role in quantum metrology, yet its precision is significantly affected by noise, particularly phase-diffusive noise caused by phase drift. To address this challenge, the joint estimation of phase and phase diffusion has emerged as an effective approach, transforming the problem into a multi-parameter estimation task. However, the incompatibility between optimal measurements for different parameters prevents single-copy measurements from reaching the fundamental precision limits defined by the quantum Cramer-Rao bound. Meanwhile, collective measurements performed on multiple identical copies can mitigate this incompatibility and thus enhance the precision of joint parameter estimation. This work experimentally demonstrates joint phase and phase-diffusion estimation using deterministic Bell measurements on a two-qubit system. A linear optical network is employed to implement both parameter encoding and deterministic Bell measurements, achieving improved estimation precision compared to any separable measurement strategy. This work proposes a new framework for phase estimation under phase-diffusive noise and underscores the substantial advantages of collective measurements in multi-parameter quantum metrology.
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.
The minimum achievable statistical uncertainty in the estimation of physical parameters is determined by the quantum Fisher information. Its computation for noisy systems is still a challenging problem. Using a variational approach, we present an equation for obtaining the quantum Fisher information, which has an explicit dependence on the mathematical description of the noise. This method is applied to obtain a useful analytical bound to the quantum precision in the estimation of phase-shifts under phase diffusion, which shows that the estimation uncertainty cannot be smaller than a noise-dependent constant.
We examine metrological scenarios where the parameter of interest is encoded onto a quantum state through the action of a noisy quantum gate and investigate the ultimate bound to precision by analyzing the behavior of the quantum Fisher information (QFI). We focus on qubit gates and consider the possibility of employing successive applications of the gate. We go beyond the trivial case of unitary gates and characterize the robustness of the metrological procedure introducing noise in the performed quantum operation, looking at how this affects the QFI at different steps (gate applications). We model the dephasing and tilting noise affecting qubit rotations as classical fluctuations governed by a von Mises--Fisher distribution. Compared to the noiseless case, in which the QFI grows quadratically with the number of steps, we observe a nonmonotonic behavior, and the appearance of a maximum in the QFI, which defines the ideal number of steps that should be performed in order to precisely characterize the action of the gate. Analyzing also the combination of different types of gate imperfections we defined a form of generic noise, finding also nontrivial regimes in which the combination of the two different noises leads to a more resilient QFI or, more remarkably, to an enhancement of the QFI with respect to the case in which only one type of noise is considered.