量子费舍尔信息的体积不等式和不确定原理
A volume inequality for quantum Fisher information and the uncertainty principle
摘要：
设$A_1,...,A_N$为复自伴矩阵，$ρ$为密度矩阵。Robertson不确定原理$$ det(Cov_ρ(A_h,A_j)) \geq det(- \frac{i}{2} Tr(ρ[A_h,A_j])) $$给出了量子广义协方差的一个关于量子化子$[A_h,A_j]$的界。右侧矩阵是反对称的，因此在奇数情形$N=2m+1$中，界限是平凡的（等于零）。令$f$是任意归一化对称算子单调函数，并令$<\cdot, \cdot >_{ρ,f}$是相关的量子费舍尔信息。本文猜想一个不等式$$ det (Cov_ρ(A_h,A_j)) \geq det (\frac{f(0)}{2} < i[ρ, A_h],i[ρ,A_j] >_{ρ,f}) $$，它利用$i[ρ, A_h]$给出了任意自然数$N$的一个非平凡界。这个不等式已在$N=1,2$情形下得到证明.本文证明了实矩阵N=3的情况。
英文摘要：
Let $A_1,...,A_N$ be complex self-adjoint 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 conjecture 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 natural number $N$ using the commutators $i[ρ, A_h]$. 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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