Uncertainty Principle and Quantum Fisher Information - II

Abstract

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

v2, Minor revision. v3, changes made to conform to version accepted on J. Math. Phys

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