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arXiv:1801.00331

On Operator Valued Measures

Darian McLaren, Sarah Plosker, Christopher Ramsey

Published Dec 31, 2017• 9 citations
arXivHTMLPDF

Abstract

We consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikodym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesque decomposition theorem and Johnson's atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the "cleanness" relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.

DOI
10.48550/arxiv.1801.00331

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We consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikodym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesque decomposition theorem and Johnson's atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the "cleanness" relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.
It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
Given a positive operator-valued measure ννν acting on the Borel sets of a locally compact Hausdorff space XXX, with outcomes in the algebra B(H)\mathcal B(\mathcal H)B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H\mathcal HH, one can consider ννν-integrable functions X→B(H)X\rightarrow \mathcal B(\mathcal H)X→B(H) that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.
The problem of constructing a measure on a discrete set X taking values in a positive cone of bounded operators in a Hilbert space is considered. It is assumed that a projectionvalued function defined on a subset of X0 of the original set X is initially given. The aim of the study is to find such a scalar measure μ on the set X and the continuation of a projector-valued function from X0 to X, which results in an operator-valued measure having a projector-valued density relative to μ. In general, the problem is solved for |X| = 4 and |X0| = 2. As an example, we consider a function on X0 that takes values in a set of projections on coherent states. For this case, the question of the information completeness of the measurement determined by the constructed measure is investigated. In other words, is it possible to reconstruct a quantum state (a positive unit trace operator) from the values of the matrix trace from the product of a measure with a quantum state. It is shown that for the constructed measure it is possible to restore the quantum state only if it is a projection. A restriction on the probability distribution is also found, at which it can be obtained as a result of measuring a certain quantum state.
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.
Sufficient and necessary conditions are presented for the existence of ( N, M)-positive operator valued measures (( N, M)-POVMs) valid for arbitrary-dimensional quantum systems. Firstly, a sufficient condition for the existence of ( N, M)-POVMs is presented. It yields a simple relation determining an upper bound on the continuous parameter of an arbitrary ( N, M)-POVM, below which all its POVM elements are guaranteed to be positive semidefinite. Secondly, for arbitrary optimal ( N, M)-POVMs conditions on their existence are derived, which exhibit a close connection to the existence of sets of ( M − 1) N isospectral, traceless, orthonormal, and hermitian operators with particular common spectra. The specific form of the two possible types of spectra depends on whether M is smaller than the dimension of the quantum system under consideration or not. For the special case of M = 2 and dimensions, which are powers of two, these sets of operators always exist and can be expressed in terms of the Clifford algebra of tensor products of Pauli operators.