A Robertson-type Uncertainty Principle and Quantum Fisher Information

Gibilisco, Paolo; Imparato, Daniele; Isola, Tommaso

A Robertson-type Uncertainty Principle and Quantum Fisher Information

Authors

Gibilisco, Paolo

Imparato, Daniele

Isola, Tommaso

Description

Let $A_1,...,A_N$ be complex selfadjoint matrices and let $ρ$ be a density matrix. The Robertson uncertainty principle $$ det (Cov_ρ(A_h,A_j)) \geq det (- \frac{i}{2} Tr (ρ[A_h,A_j])) $$ gives a bound for the quantum generalized covariance in terms of the commutators $ [A_h,A_j]$. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case $N=2m+1$. Let $f$ be an arbitrary normalized symmetric operator monotone function and let $<\cdot, \cdot >_{ρ,f}$ be the associated quantum Fisher information. In this paper we prove the inequality $$ det (Cov_ρ(A_h,A_j)) \geq det (\frac{f(0)}{2} < i[ρ, A_h],i[ρ,A_j] >_{ρ,f}) $$ that gives a non-trivial bound for any $N \in {\mathbb N}$ using the commutators $[ρ,A_h]$.
17 pages (approx.)