Volume of the set of locally diagonalizable bipartite states

TL;DR

This paper explores the geometry of locally diagonalizable bipartite quantum states, providing new formulas for their volume and conditions for diagonalizability, especially in the qubit-qubit case, and partitioning these states into local unitary orbits.

Contribution

It introduces the Hilbert-Schmidt volume of locally diagonalizable states and a necessary and sufficient condition for local diagonalizability in qubit-qubit systems, along with a modified volume formula for local unitary orbits.

Findings

Derived the Hilbert-Schmidt volume of locally diagonalizable states.

Established a necessary and sufficient condition for local diagonalizability in qubit-qubit systems.

Modified Harish-Chandra's volume formula for local unitary orbits.

Abstract

The purpose of this article is to investigate the geometry of the set of locally diagonalizable bipartite quantum states. We have the following new results: the Hilbert-Schmidt volume of all locally diagonalizable states, and a necessary and sufficient condition for local diagonalizability in the qubit-qubit case. Besides, we partition the set of all locally diagonalizable states as local unitary orbits (or coadjoint orbits) of diagonal forms. It is well-known that the Riemannian volume of a coadjoint orbit for a regular point in a specified Weyl chamber can be calculated by Harish-Chandra's volume formula. By modifying Harish-Chandra's volume formula, we give, for the first time, a specific formula for the Riemannian volume of a local unitary orbit of a regular point in a specified Weyl chamber. Several open questions are presented as well.