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Journal of Physics A Mathematical and General

An alternative to the Lüders generalization of the von Neumann projection, and its interpretation

Willem M. de Muynck

Published Jan 16, 1998• 21 citations
Publisher

Abstract

An alternative to the Lüders generalization of the von Neumann projection is formulated allowing a generalization to measurements represented by positive operator-valued measures. This generalization is based on the notions of non-orthogonal projections and bi-orthonormal systems in Hilbert-Schmidt space, and is closely related to phase-space representations of quantum mechanics. A number of applications will be discussed. It is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state, to be obtained by the measurement.

DOI
10.1088/0305-4470/31/2/005

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An alternative to the Lüders generalization of the von Neumann projection is formulated allowing a generalization to measurements represented by positive operator-valued measures. This generalization is based on the notions of non-orthogonal projections and bi-orthonormal systems in Hilbert-Schmidt space, and is closely related to phase-space representations of quantum mechanics. A number of applications will be discussed. It is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state, to be obtained by the measurement.
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.
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.
Positive operator valued measurements (POVMs) are the most general measurements that can be performed on quantum systems.A POVM is described by set of positive operators instead of orthogonal projections, and it can always be traced back to joint von Neumann measurements on the system and on an ancilla attached to it (Busch et. al, 1991;Peres, 1993).* Note on nomenclature: subscript letters denote the appropriate state spaces for vectors and density operators throughout the document.The same symbol with different subscripts intends to mean that the same quantum information is represented on different systems.
Notwithstanding it is well known that von Neumann's projection postulate is inapplicable to most realistic measurement procedures, it keeps haunting the foundations of quantum mechanics. In particular its applicability to EPR experiments is often assumed. In the present contribution this problem is considered from the point of view of a quantum mechanical theory of measurement, allowing a treatment of projection in EPR experiments as a special case of conditional preparation. The conditions are spelled out under which the postulate may be applicable.
The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.
We study possible realizations of generalized quantum measurements on measurement-assisted programmable quantum processors. We focus our attention on the realization of von Neumann measurements and informationally complete positive-operator-valued measures. Nielsen and Chuang [Phys. Rev. Lett. 79, 321 (1997)] have shown that two unitary transformations implementable by the same programmable processor require mutually orthogonal states. We show that two different von Neumann measurements can be encoded into nonorthogonal program states. Nevertheless, given the dimension of a Hilbert space of the program register the number of implementable von Neumann measurements is still limited. As an example of a programmable processor we use the so-called quantum-information distributor.