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arXiv:quant-ph/0204001 · Foundations of Physics

On Relations Between Probabilities Under Quantum and Classical Measurements

Andrei Yu. Khrennikov, Elena R. Loubenets

Published Mar 15, 2004• 15 citations
arXivHTMLPDF
DOI
10.1023/b:foop.0000019631.84010.a6

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Plausibility measures are structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations. So far, the theory of plausibility measures has only been developed for classical sample spaces. In this paper, we generalize the theory to test spaces, so that they can be applied to general operational theories, and to quantum theory in particular. Our main results are two theorems on when a plausibility measure agrees with a probability measure, i.e. when its comparative relations coincide with those of a probability measure. For strictly finite test spaces we obtain a precise analogue of the classical result that the Archimedean condition is necessary and sufficient for agreement between a plausibility and a probability measure. In the locally finite case, we prove a slightly weaker result that the Archimedean condition implies almost agreement.
In this paper it was proved that the quantum relative entropy D(k) can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of measurements. The sequence of measurements depends on , but is independent of the choice of . In classical statistical theory the relative entropy D(pkq) is an information quantity which means the statistical eÆciency in order to distinguish a probability measure p of a measurable space from another probability measure q of the same measurable space. The states correspond to measures on measurable space. When p; q are discrete probabilities, the relative entropy (called also information divergence) introduced by Kullback and Leibler is dened by [1]: D(pkq) := X i p i log p i q i : In general, when p; q are measures on measurable space the relative entropy is dened by: D(pkq) := Z log dq dp (!)p( d!); where dq dp (!) is Radon-Nikodym derivative of q with respect to p. In this paper, we co...