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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.
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 ...
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.
A positive operator-valued measure E then assigns each i a positive semidefinite m × m matrix . Naimark's theorem now states that there is a projection-valued measure on X whose restriction is E. Of particular interest is the special case when = where I is the identity operator.
Throughout, is a fixed Hilbert space. A projection-valued measure on a measurable space (,), where is a σ-algebra of subsets of , is a mapping: such that for all , is a self-adjoint projection on (that is, () is a bounded linear operator (): that satisfies () = and () = ()) such that = (where is the identity operator of ) and for every ,, the function defined by (), is a complex measure on ...
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 ...
[1] [2] Thus for instance if T is an operator, applying the squaring function s → s 2 to T yields the operator T 2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential e i t Δ . {\displaystyle e^{it\Delta }.}
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.