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The measure class [clarification needed] of μ and the measure equivalence class of the multiplicity function x → dim H x completely characterize the projection-valued measure up to unitary equivalence. A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem.
A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used.
Depending on the source, the resolution of the identity is defined, either as a projection-valued measure , [4] or as a one-parameter family of projection-valued measures {} with < <. [ 5 ]
Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a projection-valued measure. One formulation of the spectral theorem expresses the operator A as an integral of the coordinate function over the operator's spectrum σ ( A ) {\displaystyle \sigma (A)} with ...
The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate. A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case.
In functional analysis and quantum measurement theory, a positive-operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum ...
This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of projection-valued measures. The residual spectrum of a normal operator is empty. [3]
In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum ...