Generalized conditional expectations for quantum retrodiction and smoothing

Abstract

The inference of a hidden variable's historical value, based on observations before and after the fact, is a controversial subject in quantum mechanics. Here I address the controversy by proposing a formalism that unifies and generalizes some of the previous proposals for the task, including the quantum minimum-mean-square-error estimators proposed by Ohki, the generalized conditional expectation proposed by Accardi and Cecchini, the quantum smoothing theory proposed by Tsang, the optimal observables for parameter estimation proposed by Personick, Belavkin, and Grishanin, and the weak values proposed by Aharonov, Albert, and Vaidman. The formalism is based on Ohki's suggestion of a distance between two observables in the Heisenberg picture, which remains well defined for incompatible observables and serves as a more general foundation for quantum inference than Belavkin's nondemolition principle.

11 pages. v4: extended, published

References in corpus (12)

• Quantum-enhanced optical phase tracking
• Evading quantum mechanics
• Weak Values are Interference Phenomena
• Time-Symmetric Quantum Theory of Smoothing
• Can a quantum state over time resemble a quantum state at a single time?
• Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous time limit and state-variable approach to phase-locked loop design
• Unifying theory of quantum state estimation using past and future information
• Estimation of fluctuating magnetic fields by an atomic magnetometer
• Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach
• How to estimate past measurement interventions on a quantum system undergoing continuous monitoring
• Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing
• The Collapse Before a Quantum Jump Transition