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Physical Review A

Positive-operator-valued measures in the Hamiltonian formulation of quantum mechanics

D. Arsenović, Nikola Burić, Duska B Popovic, Milan Radonjić, Slobodan Prvanović

Published Jun 12, 2015• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.91.062114

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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.
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 propose a procedure for defining all single-photon observables in terms of Positive-Operator Valued Measures (POVMs), in particular spin and position. We identify the suppression of 000-helicity photon states as a projection from an extended Hilbert space onto the photon Hilbert space. We show that all single-photon observables are in general described by POVMs, obtained by applying this projection to opportune Projection-Valued Measures (PVMs), defined on the extended Hilbert space. The POVMs associated to momentum and helicity reduce to PVMs, unlike those associated to position and spin, this fact reflecting the intrinsic unsharpness of these observables. We finally extensively study the preparation uncertainty relations for position and momentum and the probability distribution of spin, exploring single photon Gaussian states for several choices of spin and polarization.
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
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.