Geometry of density matrix states

Luis J. Boya and Kuldeep Dixit
Phys. Rev. A 78, 042108 – Published 17 October 2008

Abstract

We reconsider the geometry of pure and mixed states in a finite quantum system. The ranges of eigenvalues of the density matrices delimit a regular symplex (hypertetrahedron TN) in any dimension N; the polytope isometry group is the symmetric group SN+1, and splits TN in chambers, the orbits of the states under the projective group PU(N+1). The type of states correlates with the vertices, edges, faces, etc., of the polytope, with the vertices making up a base of orthogonal pure states. The entropy function as a measure of the purity of these states is also easily calculable; we draw and consider some isentropic surfaces. The Casimir invariants acquire then also a more transparent interpretation.

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  • Received 7 August 2008

DOI:https://doi.org/10.1103/PhysRevA.78.042108

©2008 American Physical Society

Authors & Affiliations

Luis J. Boya* and Kuldeep Dixit

  • Department of Physics, University of Texas, Austin, Texas 78712, USA

  • *luisjo@unizar.es; Present address: Departamento de Física Teórica, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
  • kuldeep@physics.utexas.edu

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Issue

Vol. 78, Iss. 4 — October 2008

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