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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク POVM ： ウィキペディア英語版 POVM In functional analysis and quantum measurement theory, a positive-operator valued measure (POVM) is a measure whose values are non-negative self-adjoint operators on a Hilbert space, and whose integral is the identity operator. It is the most general formulation of a measurement in the theory of quantum physics. The need for the POVM formalism arises from the fact that projective measurements on a larger system, by which we mean measurements that are performed mathematically by a ''projection-valued measure'' (PVM), will act on a subsystem in ways that cannot be described by a PVM on the subsystem alone. POVMs are used in the field of quantum information. In rough analogy, a POVM is to a PVM what a density matrix is to a pure state. Density matrices are needed to specify the state of a subsystem of a larger system, even when the larger system is in a pure state (see purification of quantum state); analogously, POVMs on a physical system are used to describe the effect of a projective measurement performed on a larger system. Historically, the term probability-operator measure (POM) has been used as a synonym for POVM, although this usage is now rare. == Definition == In the simplest case, a POVM is a set of Hermitian positive semidefinite operators $\backslash$ on a Hilbert space $\backslash mathcal$ that sum to the identity operator, :$\backslash sum\_^n\; F\_i\; =\; \backslash operatorname\_H.$ This formula is a generalization of the decomposition of a (finite-dimensional) Hilbert space by a set of orthogonal projectors, $\backslash$, defined for an orthogonal basis $\backslash$ by: :$\backslash sum\_^N\; E\_i\; =\; \backslash operatorname\_H,\; \backslash quad\; E\_i\; E\_j\; =\; \backslash delta\_\; E\_i,E\_i=\backslash left|\backslash phi\_\backslash right\backslash rangle\; \backslash left\backslash langle\; \backslash phi\_\backslash right|.$ An important difference is that the elements of a POVM are not necessarily orthogonal, with the consequence that the number of elements in the POVM, n, can be larger than the dimension, N, of the Hilbert space they act in. In general, POVMs can be defined in situations where the outcomes of measurements take values in a non-discrete space. The relevant fact is that measurement determines a probability measure on the outcome space: Definition. Let (''X'', ''M'') be measurable space; that is ''M'' is a σ-algebra of subsets of ''X''. A POVM is a function ''F'' defined on ''M'' whose values are bounded non-negative self-adjoint operators on a Hilbert space ''H'' such that F(''X'') = I''H'' and for every ξ $\backslash in$ ''H'', :$E\; \backslash mapsto\; \backslash langle\; F(E)\; \backslash xi\; \backslash mid\; \backslash xi\; \backslash rangle$ is a non-negative countably additive measure on the σ-algebra ''M''. This definition should be contrasted with that of the projection-valued measure, which is similar, except that for projection-valued measures, the values of ''F'' are required to be projection operators. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「POVM」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.