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arXiv · 1611.09586

Volume of the set of locally diagonalizable bipartite states

Lin Zhang·Seunghun Hong
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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.

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Lin Zhang, Seunghun Hong. 2016-11-29. Volume of the set of locally diagonalizable bipartite states. https://doi.org/10.1088/1751-8121/aad3de

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