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arXiv · 0707.1231

A Robertson-type Uncertainty Principle and Quantum Fisher Information

Paolo Gibilisco·Daniele Imparato·Tommaso Isola
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Abstract

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 $[ρ,A_h]$.

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Paolo Gibilisco, Daniele Imparato, Tommaso Isola. 2007-07-09. A Robertson-type Uncertainty Principle and Quantum Fisher Information. https://arxiv.org/abs/0707.1231

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