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Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy

On extension of the family of projections to positive operator-valued measure

Anton O. Alekseev, Grigori Gennadievich Amosov

Published Jan 1, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.21638/spbu01.2023.101

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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.
Abstract In this paper, the problem of constructing a measure which accepts values within a positive cone of bounded operators in a Hilbert space is considered. The assumption is made that a projection-valued function defined on a subset X0 of the original set X is initially given. The goal of this paper is to find such a scalar measure μ on set X as well as the extension of a projection-valued function from X0 to X, and, as a result, to obtain an operator-valued measure which has a projection-valued density relative to μ. In general, the solution to this problem is found for |X| = 4 and |X0| = 2. As an example, a function on X0 is considered which accepts values within a set of projections onto coherent states. For this case, the problem which concerns the informational completeness of the measurement determined by the constructed measure is investigated. In other words, the possibility is considered of restoring a quantum state (a positive unit-trace operator) based on trace values of the matrix formed from the product of a measure and a quantum state. In the case of the constructed measure, it is demonstrated that it is possible to restore a quantum state only if it represents a projection. A restriction imposed on the probability distribution is also found; with this restriction satisfied, the probability distribution can be obtained as a result of measuring a certain quantum state.
We study a projective unitary representation of the product G=G˜×G, where G is a locally compact Abelian group and G^ is its dual consisting of characters on G. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations of G. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
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.