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arXiv:1205.3294 · Journal of Modern Optics

Phase properties of operator valued measures in phase space

T. S. Subeesh, Vivishek Sudhir

Published Mar 1, 2013• 3 citations

Abstract

The Wigner phase operator (WPO) is identified as an operator valued measure (OVM) and its eigenstates are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner phase distribution. It is then argued that the WPO cannot represent a projective measurement of the phase; but is in fact to be interpreted as an operator valued measure for the phase. The non-positivity of the latter can be overcome by defining a positive operator valued measure (POVM) via a proper filter function, based on the view that phase measurements are coarse-grained in phase space, leading to the well known Q-distribution. The identification of the Q phase operator as a POVM is in good agreement with the earlier observation regarding the relation between operational phase measurement schemes and the Q-distribution. The Q phase POVM can be dilated in the sense of Gelfand–Naimark, to an operational setting of interference at a beam-splitter with another coherent state – this results in a von Neumann projector with well-defined phase in the expanded Hilbert space of the two modes.

DOI: 10.1080/09500340.2013.793418

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The Wigner phase operator (WPO) is identified as an operator valued measure (OVM) and its eigenstates are obtained. An operator satisfying the canonical commutation relation with the Wigner phase operator is also constructed and this establishes a Wigner distribution based operator formalism for the Wigner phase distribution. It is then argued that the WPO cannot represent a projective measurement of the phase; but is in fact to be interpreted as an operator valued measure for the phase. The non-positivity of the latter can be overcome by defining a positive operator valued measure (POVM) via a proper filter function, based on the view that phase measurements are coarse-grained in phase space, leading to the well known Q-distribution. The identification of the Q phase operator as a POVM is in good agreement with the earlier observation regarding the relation between operational phase measurement schemes and the Q-distribution. The Q phase POVM can be dilated in the sense of Gelfand–Naimark, to an operational setting of interference at a beam-splitter with another coherent state – this results in a von Neumann projector with well-defined phase in the expanded Hilbert space of the two modes.

In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.

Recently, quantum models of carrier transport in semiconductors have been proposed on the basis of the Wigner formulation of quantum mechanics. Such models are demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. However, the Wigner function cannot be interpreted as a quantum analog of a distribution function, since it is not a probability density. In this work we show how this difficulty could be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures [3–5,7].Such a theory is demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. The Wigner function cannot be strictly interpreted as a quantum analog of a distribution function, because it is not a probability density. In this communication we show that this difficulty can be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures rather than on Projector-Valued (PV) measures. The introduction of this more general class of mathematical objects accounts for the stochastic effect of measure errors in the effective evaluation of observables described by PV measures [1, 2]. Following Prugovecki [3], we show that, on an appropriately defined stochastic phase space, it is possible to introduce a “quantum distribution function” whose marginals give the desired macroscopic quantities, and which does not present the disadvantages of the Wigner function. We present the resulting quantum Boltzmann equations for an ensemble of free carriers, and discuss extensions to the case of carriers in a semiconductor.

Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.

We present a correspondence between positive operator valued measures (POVMs) and sets of generalized coherent states. POVMs describe quantum observables and, similar to quantum states, quantum observables can also be mixed. We show how the formalism of generalized coherent states leads to a useful characterization of extremal POVMs. We prove that covariant phase-space observables related to squeezed states are extremal, while those related to number states are not extremal. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.