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数理解析研究所講究録

On a continuity of quantum statistical models in the infinite-dimensional Hilbert space (Statistical-theoretic approach to high-dimensional quantum tomography)

冬彦 田中

Published Aug 1, 2014• 0 citations
OpenAlex

Abstract

Let us consider functions from a locally compact metric space to trace- class operators on a separable Hilbert space.It is a basic framework in constructing quantum statistical models.In theoretical development, Holevo gave one definition of continuity of those functions in a more general setting.A regularity condition of quantum statistical models is derived from this definition but it seems complicated.We show that it is rewritten in a simple form.

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Let us consider functions from a locally compact metric space to trace- class operators on a separable Hilbert space.It is a basic framework in constructing quantum statistical models.In theoretical development, Holevo gave one definition of continuity of those functions in a more general setting.A regularity condition of quantum statistical models is derived from this definition but it seems complicated.We show that it is rewritten in a simple form.
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
. The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups. 1. introduction In Quantum Mechanics a physical system is associated with a Hilbert space H: the states are described by positive trace-class trace-one operators T on H, the physical quantities by self-adjoint operators A on H and the physical content of the theory is given by the expectation values Tr (AT ). The state T is completely determined by Tr (Q n T ) for Q n running on a suitable set fQ n g of observables and, for arbitrary operator A, Tr (AT ) can be computed in terms of Tr (Q n T ). In order to implement this scheme one has to estimate Tr (Q n T ) experimentally, facing the problems arising from statistical errors and instrumental noise. Moreover, the number of experimental data is clearly nite, while A and T are operators on an innite dimensional Hilbert space and the set fQ n g is innite. The problem of determ...
The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations. It is shown that as a result of some specific properties of the von Neumann entropy (as a function on the set of density operators) several results on the output entropy of positive maps can be obtained, which cannot be derived from the general properties of entropy type functions. In particular, it is proved that global continuity of the output entropy of a positive map follows from its finiteness. A characterization of positive linear maps preserving continuity of the entropy (in the following sense: continuity of the entropy on an arbitrary subset of input operators implies continuity of the output entropy on this subset) is obtained. A connection between the local continuity properties of two completely positive complementary maps is considered. Bibliography: 21 titles.
A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase-space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.