State change, quantum probability, and information in operational phase-space measurement

Abstract

State change, quantum probability, and information gain in the operational phasespace measurement are formulated by means of positive operator-valued measure (POVM) and operation. The properties of the operational POVM and its marginal POVM which yield the quantum probability distributions of the measurement outcomes obtained by the operational phase-space measurement are investigated. The Naimark extension of the operational POVM can be expressed in terms of the relative-position states and the relative-momentum states in the extended Hilbert space. An observable quantity measured in the operational phase-space measurement becomes a fuzzy or unsharp observable. The state change of a physical system caused by the operational phase-space measurement is described by the operation which is obtained explicitly for the position and momentum measurements and for the simultaneous measurement of position and momentum. Using the results, the entropy change of the measured physical system and the information gain in the operational phase-space measurement are investigated. It is found that the average value of the entropy change is equal to the Shannon mutual information extracted from the outcomes exhibited by the measurement apparatus.

Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries

“…is called the operational phase-space probability distribution [104][105][106]. Here, ρ is the density matrix of the quantum state of the system and ρ u is the density matrix (given by the projector |u u| for a pure state) of the so-called "quantum ruler" state which characterizes the measurement device.…”

Inverted spectroscopy and interferometry for quantum-state reconstruction of systems withSU(2) symmetry

“…In brief, one applies a phase-space displacement [specifically, a rotation in the SU(2) case] to the initial quantum state and then measures the probability to find the displaced system in a specific state (the so-called "quantum ruler" state). Repeating this procedure with identically prepared systems for many phase-space points [many rotation angles in the SU(2) case], one determines a function on the phase space (the so-called operational phase-space probability distribution [8][9][10]). In particular, by measuring the population of the ground state, one obtains the so-called Q function.…”

Retrodictively optimal localizations in phase space

“…In several recent publications [3,4,5] we have discussed how to characterise the accuracy of, and disturbance caused by such measurements. One approach to the problem is that based on the concept of an "unsharp observable" [6,7,8,9]. This approach has recently been criticised by Uffink [10].…”

“…These functions are operational distributions of the type defined by Wódkiewicz [6,9,14,15]. They are the distributions which result when the filter reference state (or "quantum ruler") is an arbitrary squeezed vacuum state [16].…”