A Robertson-type Uncertainty Principle and Quantum Fisher Information

2007/07/09 by Gibilisco, Paolo, Imparato, Daniele, Isola, Tommaso
#46L30 #46L60 #62B10 #94A17 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.0707.1231

Abstract

Let A₁,...,A_N be complex selfadjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle det (Cov_ρ(A_h,A_j)) ≥ det (- (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 _ρ,f be the associated quantum Fisher information. In this paper we prove the inequality det (Cov_ρ(A_h,A_j)) ≥ det ((f(0))/(2) lt; i[ρ, A_h],i[ρ,A_j] gt;_ρ,f) that gives a non-trivial bound for any N ∈ \mathbb N using the commutators [ρ,A_h].

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