Eigenvalues, Separability and Absolute Separability of Two-Qubit States

Abstract

Substantial progress has recently been reported in the determination of the Hilbert-Schmidt (HS) separability probabilities for two-qubit and qubit-qutrit (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 question--rather than the diagonal entries, as originally used--comparable 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 large-scale numerical estimations of such eigenvalue-parameterized functions have been undertaken, it seems desirable also to study them in lower-dimensional specialized scenarios in which they can be exactly obtained. In this regard, we employ an Euler-angle parameterization of SO(4) derived by S. Cacciatori (reported in an Appendix)--in the manner of the SU(4)-density matrix parameterization of Tilma, Byrd and Sudarshan. We are, thus, able to find simple exact separability (inverse-sine-like) functions for two real two-qubit (rebit) systems, both having three free eigenvalues and one free Euler angle. We also employ the important Verstraete-Audenaert-de Moor bound to obtain exact HS probabilities that a generic two-qubit state is absolutely separable (that is, can not be entangled by unitary transformations). In this regard, we make copious use of trigonometric identities involving the tetrahedral dihedral angle arccos(1/3).

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

References in corpus (4)

• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• Two-Qubit Separability Probabilities and Beta Functions
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• A simple spectral condition implying separability for states of bipartite quantum systems

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• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
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• Formulas for Generalized Two-Qubit Separability Probabilities
• On the extreme points of sets of absolulely separable and PPT states
• Breaking absolute separability with quantum switch
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities