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arXiv:0712.1208

Quantum covariance, quantum Fisher information and the uncertainty principle

Paolo Gibilisco, Fumio Hiai, Dėnes Petz

Published Dec 7, 2007• 6 citations
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Abstract

In this paper the relation between quantum covariances and quantum Fisher informations are studied. This study is applied to generalize a recently proved uncertainty relation based on quantum Fisher information. The proof given hereconsiderably simplifies the previously proposed proofs and leads to more general inequalities.
DOI
10.48550/arxiv.0712.1208

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Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B), in a state ρ\rhoρ, if the observables A,BA,BA,B are not compatible, namely if the commutator [A,B][A,B][A,B] is not zero. In this paper we prove an uncertainty principle in Schr{\"o}dinger form where the bound for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B) depends on the area spanned by the commutators [ρ,A][\rho,A][ρ,A] and [ρ,B][\rho,B][ρ,B] with respect to an arbitrary quantum version of the Fisher information.