A generalized skew information and uncertainty relation

A generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by J.I.Fujii. In addition, we prove the trace inequality conjectured by S.Luo and Z.Zhang. Finally we point out that Theorem 1 in {\it S.Luo and Q.Zhang, IEEE Trans.IT, Vol.50, pp.1778-1782 (2004)} is incorrect in general, by giving a simple counter-example.Comment: to appear in IEEE TI

Schr\"odinger uncertainty relation with Wigner-Yanase skew information

We shall give a new Schr\"odinger type uncertainty relation for a quantity representing a quantum uncertainty, introduced by S.Luo in \cite{Luo1}. Our result improves the Heisenberg uncertainty relation shown in \cite{Luo1} for a mixed state.Comment: to appear in Phys.Rev.

Schr\"odinger uncertainty relation, Wigner-Yanase-Dyson skew information and metric adjusted correlation measure

In this paper, we give a Schr\"odinger-type uncertainty relation using the Wigner-Yanase-Dyson skew information. In addition, we give Schr\"odinger-type uncertainty relation by use of a two-parameter extended correlation measure. Moreover, we give the further generalization for Schr\"odinger-type uncertainty relation by metric adjusted correlation measure. These results generalize our previous result in [Phys. Rev. A, Vol.82(2010), 034101].Comment: Section 3 was revise

A Robertson-type Uncertainty Principle and Quantum Fisher Information

Let $A_1,...,A_N$ be complex selfadjoint 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 $_{\rho,f}$ be the associated quantum Fisher information. In this paper we prove the inequality $$det (Cov_\rho (A_h,A_j)) \geq det (\frac{f(0)}{2} _{\rho,f})$$ that gives a non-trivial bound for any $N \in {\mathbb N}$ using the commutators $[\rho,A_h]$.Comment: 17 pages (approx.

A volume inequality for quantum Fisher information and the uncertainty principle

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 $_{\rho,f}$ be the associated quantum Fisher information. In this paper we conjecture the inequality $$det (Cov_\rho(A_h,A_j)) \geq det (\frac{f(0)}{2} _{\rho,f})$$ 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