Hilbert space representations of decoherence functionals and quantum measures
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AbstractWe show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.
2011 ◽
Vol 2011
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pp. 1-6
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2017 ◽
Vol 11
(01)
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pp. 1850004
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2009 ◽
Vol 61
(1)
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pp. 124-140
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1973 ◽
Vol 49
(3)
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pp. 707-713
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