Eigenvalues, separability and absolute separability of twoqubit states
Abstract
Substantial progress has recently been reported in the determination of the HilbertSchmidt (HS) separability probabilities for twoqubit and qubitqutrit (real, complex and quaternionic) systems. An important theoretical concept employed has been that of a separability function. It appears that if one could analogously obtain separability functions parameterized by the eigenvalues of the density matrices in questionrather than the diagonal entries, as originally usedcomparable progress could be achieved in obtaining separability probabilities based on the broad, interesting class of monotone metrics (the Bures, being its most prominent [minimal] member). Though largescale numerical estimations of such eigenvalueparameterized functions have been undertaken, it seems desirable also to study them in lowerdimensional specialized scenarios in which they can be exactly obtained. In this regard, we employ an Eulerangle parameterization of SO(4) derived by S. Cacciatori (reported in an Appendix)—in the manner of the SU(4)densitymatrix parameterization of Tilma, Byrd and Sudarshan. We are, thus, able to find simple exact separability (inversesinelike) functions for two real twoqubit (rebit) systems, both having three free eigenvalues and one free Euler angle. We also employ the important VerstraeteAudenaertde Moor bound to obtain exact HS probabilities that a generic twoqubit state is absolutely separable (that is, cannot be entangled by unitary transformations). In this regard, we make copious use of trigonometric identities involving the tetrahedral dihedral angle ϕ=cos^{1}({1}/{3}).
 Publication:

Journal of Geometry and Physics
 Pub Date:
 January 2009
 DOI:
 10.1016/j.geomphys.2008.08.008
 arXiv:
 arXiv:0805.0267
 Bibcode:
 2009JGP....59...17S
 Keywords:

 Quantum Physics
 EPrint:
 30 pages, 7 figures, formula (48) for the Haar measure corrected by changing sin x_3 to sin^ x_3