arXiv:1006.3978 · Journal of Inequalities and Applications

Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices

H. F. Chau, Yan Ting Lam

Published Jul 18, 2011• 4 citations

Abstract

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices--one by Childs et al. [J. Mod. Phys. 47 , 155 (2000)] and the other one by Chau [Quant. Inf. Comp. 11 , 721 (2011)]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.

DOI: 10.1186/1029-242x-2011-18

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arXiv:1006.3978 · Journal of Inequalities and Applications

Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices

H. F. Chau, Yan Ting Lam

Published Jul 18, 2011• 4 citations

arXiv HTML PDF

Abstract

DOI: 10.1186/1029-242x-2011-18

Related papers

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Jan 1, 20110 citations

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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].

It is proven that a necessary and sufficient condition for an

n

-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where

O

is a complex orthogonal matrix and

E

is a diagonal matrix with positive elements. This accomplishes a parametrization since

O

has ${n}^{2}\ensuremath{-}n$ real parameters and

E

has

n

of them. The proof is constructive, giving

O

and

E

. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.