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Russian Journal of Mathematical Physics

Commutative POV-Measures: from the Choquet Representation to the Markov Kernel and Back

Roberto Beneduci

Published Apr 1, 2018• 6 citations
Publisher

Abstract

We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.

DOI
10.1134/s1061920818020036

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We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.
Abstract We provide a proof based on transfinite induction that every weak Markov kernel is equivalent to a Markov kernel. We only assume the space where the weak Markov kernel is defined to be second countable and metrizable. That generalizes some previous results where the kernel is required to be defined on a standard Borel space (which is second countable and completely metrizable) and the framework is the theory of stochastic operators. This property of weak Markov kernels is at the root of the characterization of a commutative POVM as the fuzzification of a spectral measure through a Markov kernel. As a result, the characterization of commutative POVMs is also generalized. We then revisit the relationships between weak Markov kernels, Markov kernels, commutative POVMs and fuzzy observables.
This paper presents self-contained, constructive proofs of two new theorems about stochastic matrices, with direct relevance to quantum theory. The first theorem, herein called the Born Representation Theorem, shows that each entry of any stochastic matrix can be expressed as the trace of a pairwise product of matrices, where the first factor in the pairwise product belongs to a positive-operator-valued measure (POVM) and the second factor belongs to a projection-valued measure (PVM). As its name suggests, this theorem entails that the entries of any stochastic matrix can be expressed in terms of a generalized version of the quantum-theoretic Born rule. It follows as a corollary that if the POVM in this first theorem is a PVM, then the stochastic matrix is unistochastic, meaning that its entries are each the modulus square of the corresponding entry of a unitary matrix of the same size. The second theorem proved in this paper, called the Unistochastic Theorem, then shows that by dilating the underlying vector space by a bounded number of additional dimensions if necessary, each entry of any stochastic matrix can be expressed in terms of the trace of a pairwise product for which both factors belong to PVMs, and can thus be derived via marginalization from a larger unistochastic matrix. This second theorem therefore establishes a kind of primacy of unistochastic matrices over stochastic matrices, and hints at a close connection with unitary time evolution in quantum theory. The paper concludes with a brief discussion of potential applications to discrete-time deterministic processes and Markov chains.
Abstract After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H ⊗ K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.
We construct a family of bounded, finitely additive, covariant positive operator-valued measures (POVMs) on the real line by applying generalised Banach-Cesaro asymptotic averaging to locally finite covariant POVMs. In contrast to the strictly shift-invariant case, where countably additive operator measures are restricted to scalar multiples of a single fixed operator, the introduced covariant setting allows for non-commuting averaged operator values while successfully normalising the total mass. We develop an integration theory for complex-valued functions by taking the completion of simple integrable functions with respect to the continuous operator norm induced by the positive integral of their absolute values. The integration spaces for the original and the averaged measures are shown to possess distinct null classes. Finally, we introduce the operator-valued Fourier transform on these integration spaces and analyze the core algebraic constraints imposed by the covariance property on the exponential multipliers.
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.