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Eigenvalues, Separability and Absolute Separability of Two-Qubit States | | Paul B. Slater
; | | Date: |
2 May 2008 | | 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). | | Source: | arXiv, 0805.0267 | | Services: | Forum | Review | PDF | Favorites | |
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