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

Rank-preserving geometric means of positive semi-definite matrices

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.Comment: To appear in Linear Algebra and its Application

Riemannian geometry and matrix geometric means

AbstractThe geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authors do not readily extend to three matrices and it has been a long-standing problem to define a natural geometric mean of three positive definite matrices. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of A and B as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining A and B. This suggests some natural definitions for a geometric mean of three positive definite matrices. We explain the necessary geometric background and explore the properties of some of these candidates

Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute.Comment: the present version is very close to the published one. It contains some corrections with respect to the previous arxiv submssio

A general framework for extending means to higher orders

Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to define and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. We consider uniqueness properties and preservation properties of these extensions, properties which provide validation to our approach. As our targeted application, we consider the positive operators on a Hilbert space under the Thompson metric and apply our methods to derive higher order extensions of a variety of standard operator means such as the geometric mean, the Gauss mean, and the logarithmic mean. That the operator logarithmic mean admits extensions of all higher orders provides a positive solution to a problem of Petz and Temesi [SIAM J. Matrix Anal. Appl. 27 (2005)]