Unitary Matrices

Ch. Zacharatou Jarlskog

We present further properties of a previously proposed recursive scheme for parametrization of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study the invariant phases of unitary matrices. The case of four-by-four unitary matrices is investigated in detail. We also address the question of how to construct symmetric unitary matrices (i.e., unitary matrices U that satisfy the condition Uij=Uji) using the recursive approach.

Ted Hurley

A unitary matrix is shown to be a product of certain {\em basic} unitary matrices and the product is unique up to order. A basic unitary matrix itself is a product of {\em minimal basic} matrices. A unitary $n\times n$ matrix can be expressed as a product of at most $n$ basic matrices. A basic matrix is defined in terms of an idempotent matrix; the idempotent matrix used in the definition of a minimal basic unitary matrix is a {\em density} matrix. This gives builders for unitary matrices. Quantum logic gates are represented by unitary matrices thus giving unique building material for quantum logic gates.

Brian P. Anderson

Julian Schwinger

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.