We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data processing of the outcomes of a quantum measurement described by a positive operator-valued measure. We determine the optimal processing that minimizes the statistical error of the estimation.
The expectation value of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) covariance group.
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of N independent and identically prepared two-level systems (qubits).
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.
A review of the fundamental ideas and methods of the optimal reception and processing of quantum signals is given. Estimation via an operator based on the usual and generalized measurements (e.g., quasi-measurements) are discussed. The theory of operator estimation enables one to obtain Bayes limits for the usual estimates. The change from usual measurements to quasi-measurements generally improves performance. We show that some quasi-measurements can be realized as indirect measurements, and introduce an operator measure II(db) which describes the quasi-measurements on the space of measurements. Estimation based on quasi-measurements is described by operator measures Q(du) on the space of estimates. Minimization with respect to Q(du) is minimization simultaneously for all quasi-measurements and all estimates. For the M-ary hypothesis testing problem finding the optimum reduces to finding non-negative definite Hermitian operators Q1,…, Q m satisfying Q1+…+Q m = 1 where 1 is the identity operator, which extremize. Optimal Bayes quantum estimation is discussed. In the case of Gaussian quantum signal and minimum variance estimation, finding the quasi-measurements leads to optimal linear estimation. Further suboptimal methods for finding the operator in more general cases are discussed.