A volume inequality for quantum Fisher information and the uncertainty principle

Abstract

Let $A_1,...,A_N$ be complex self-adjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$det(Co{v}_{\rho }(A_h,A_j))\ge det(-\frac{i}{2}Tr(\rho [A_h,A_j]))$$ 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 $$det(Co{v}_{\rho }(A_h,A_j))\ge det(\frac{f(0)}{2}<i[\rho ,A_h],i[\rho ,A_j]{>}_{\rho ,f})$$ 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.

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Cited by in corpus (6)

• Introduction to quantum Fisher information
• Uncertainty principle with quantum Fisher information
• Generalized Wigner-Yanase Skew Information and the Affiliated Inequality
• Quantumness of pure-state ensembles via coherence of Gram matrix based on generalized $\alpha$-$z$-relative Rényi entropy
• From quasi-entropy to skew information
• Uncertainty principle for Wigner-Yanase-Dyson information in semifinite von Neumann algebras