A volume inequality for quantum Fisher information and the uncertainty principle

TL;DR

This paper proposes a conjectured volume inequality relating quantum Fisher information and the uncertainty principle, extending bounds to cases where the traditional Robertson inequality is trivial, and proves it for three real matrices.

Contribution

The paper introduces a new conjectured inequality involving quantum Fisher information that generalizes the uncertainty principle for all matrix sizes and proves it for the case N=3.

Findings

Conjectured a new volume inequality for quantum Fisher information.

Proved the inequality for the case N=3 for real matrices.

Extends the uncertainty principle to non-trivial bounds for all N.

Abstract

Let $A_1,...,A_N$ be complex self-adjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$ det(Cov_\rho(A_h,A_j)) \geq det(- \frac{i}{2} Tr(\rho [A_h,A_j])) $$ 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 $<\cdot, \cdot >_{\rho,f}$ be the associated quantum Fisher information. In this paper we conjecture the inequality $$ det (Cov_\rho(A_h,A_j)) \geq det (\frac{f(0)}{2} < i[\rho, A_h],i[\rho,A_j] >_{\rho,f}) $$ that gives a non-trivial bound for any natural number $N$ using the commutators $i[\rho, A_h]$. The inequality has been proved in the cases $N=1,2$ by the joint efforts of many authors. In this paper we prove the case N=3 for real matrices.