Unifying theory of quantum state estimation using past and future information

Abstract

Quantum state estimation for continuously monitored dynamical systems involves assigning a quantum state to an individual system at some time, conditioned on the results of continuous observations. The quality of the estimation depends on how much observed information is used and on how optimality is defined for the estimate. In this work, we consider problems of quantum state estimation where some of the measurement records are not available, but where the available records come from both before (past) and after (future) the estimation time, enabling better estimates than is possible using the past information alone. Past-future information for quantum systems has been used in various ways in the literature, in particular, the quantum state smoothing, the most-likely path, and the two-state vector and related formalisms. To unify these seemingly unrelated approaches, we propose a framework for partially-observed quantum system with continuous monitoring, wherein the first two existing formalisms can be accommodated, with some generalization. The unifying framework is based on state estimation with expected cost minimization, where the cost can be defined either in the space of the unknown record or in the space of the unknown true state. Moreover, we connect all three existing approaches conceptually by defining five new cost functions, and thus new types of estimators, which bridge the gaps between them. We illustrate the applicability of our method by calculating all seven estimators we consider for the example of a driven two-level system dissipatively coupled to bosonic baths. Our theory also allows connections to classical state estimation, which create further conceptual links between our quantum state estimators.

48 pages, 9 figures

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Cited by in corpus (15)

• Pontryagin-Optimal Control of a non-Hermitian Qubit
• Quantum state smoothing as an optimal estimation problem with three different cost functions
• Optimal Zeno Dragging for Quantum Control: A Shortcut to Zeno with Action-based Scheduling Optimization
• Operational meanings of a generalized conditional expectation in quantum metrology
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• Quantum state smoothing cannot be assumed classical even when the filtering and retrofiltering are classical
• Inequivalent ways to apply semi-classical smoothing to a quantum system
• Noise-Canceling Quantum Feedback: non-Hermitian Dynamics with Applications to State Preparation and Magic State Distillation
• A Retrodictive Approach to Quantum State Smoothing
• Completely positive trace-preserving maps for higher-order unraveling of Lindblad master equations
• Quantum state-preparation control in noisy environment via most-likely paths
• Quantum backaction effects in sequential measurements
• Counterfactual quantum measurements
• Quantum state smoothing when Alice assumes the wrong type of monitoring by Bob