Complete Characterization of Arbitrary Quantum Measurement Processes

Abstract

We examine two simple and feasible practical schemes allowing the complete determination of any quantum measuring arrangement. This is illustrated with the example of parity measurement.PACS numbers: 03.65. Bz, 42.50.Ar, 42.50.Dv In the most standard picture of quantum mechanics the statistics of every measurement are governed by the projection of the system state on the orthogonal eigenstates of a set of commuting self-adjoint operators representing the measured observables. This implicitly assumes that the measurement is performed on a closed system.A more complete and realistic picture must encompass the possibility of controllable as well as unpredictable couplings of the observed system with external agents. This is to say that the measurement is performed, in general, on an open system. Among other consequences, this extends the idea of observables beyond self-adjoint operators, introducing generalized measurements described by positive operator measures (POMs).In particular, this occurs when a standard measurement is preceded by an interaction of the observed system with other degrees of freedom that are in a fixed and known initial state [1]. On the other hand, the process can also involve uncontrollable influences (usually undesired) of outer degrees of freedom. This is frequently the case with couplings with reservoirs and other mechanisms leading to losses and decoherence effects, for instance. In many practical situations it is not possible to predict which external variables are involved or the way they affect the performance of the measurement. In other words, to some extent, the real measurement differs from the intended one in an unpredictable way.In this paper we present two simple and feasible practical procedures that allow us to determine completely any quantum measurement process. The objective of such a characterization is to obtain in practice the actual POM governing the statistics. This would allow one to ascertain to what extent the planned performance is reached by revealing undesired deviations.The system, which is the object of the observation, and the external variables involved are described by the Hilbert spaces H s and H a , respectively. Since the total Hilbert space H H s ≠ H a represents by definition a closed system, the system-environment interaction can always be implemented by a unitary operator U acting on H . The system is initially in an arbitrary state with density matrix r s , while the external variables will be in some state r a that does not depend on r s . After the interaction, some set of compatible observables are measured on the output state Ur s r a U y . The statistics of the measurement are given by the projection on a set of orthonormal vectors jk͘ whose span need not coincide with H s :where P ͑k͒ is the probability of the outcome k. This is a standard ideal measurement because H s ≠ H a is a closed system. Since r a is fixed and does not depend on r s , the information provided by the measurement can be regarded as information about r s . This interpr...

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