Metrics of unitary matrices and their applications in quantum information theory

Olivia R. MacLean

Guillaume Aubrun, Stanisław Szarek

H. F. Chau, Chi-Kwong Li, Yiu‐Tung Poon +1 more

Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$ unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on ${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].

Debashree Chakraborty

H. F. Chau

By studying the minimum resources required to perform a unitary transformation, families of metrics and pseudo-metrics on unitary matrices that are closely related to a recently reported quantum speed limit by the author are found. Interestingly, this family of metrics can be naturally converted into useful indicators of the degree of non-commutativity between two unitary matrices.

Dorje C. Brody

Given a pure state vector | x ⟩ and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.