Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian Geometry

Abstract

We establish a direct connection between the power of a unitary map in $d$-dimensions ($d<\infty$) to generate quantum coherence and the geometry of the set ${\cal M}_d$ of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on ${\cal M}_d$ which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra one can define, on physical grounds, an associated measure of quantum coherence. We show that the average quantum coherence generated by a unitary map acting on a uniform ensemble of quantum states in the algebra (the so-called coherence generating power of the map) is proportional to the distance between a pair of maximally abelian subalgebras in ${\cal M}_d$ connected by the unitary transformation itself. By embedding the Grassmannian into a projective space one can pull-back the standard Fubini-Study metric on ${\cal M}_d$ and define in this way novel geometrical measures of quantum coherence generating power. We also briefly discuss the associated differential metric structures.