Volume of the set of locally diagonalizable bipartite states

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.

22 pages, LaTeX, no figure, accepted version

References in corpus (5)

• Volume of the quantum mechanical state space
• Entangled bases with fixed Schmidt number
• Volumes of conditioned bipartite state spaces
• Invariance of separability probability over reduced states in 4x4 bipartite systems
• Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient

Cited by in corpus (3)

• Geometry of generalized Pauli channels
• A variant of Horn's problem and derivative principle
• One application of Duistermaat-Heckman measure in quantum information theory