SL-invariant entanglement measures in higher dimensions: the case of spin and
arXiv:1309.6235 · doi:10.1088/1751-8113/48/6/065303
Abstract
An SL-invariant extension of the concurrence to higher local Hilbert-space dimension is due to its relation with the determinant of the matrix of a two qudits state, which is the only SL-invariant of polynomial degree . This determinant is written in terms of antilinear expectation values of the local operators. We use the permutation invariance of the comb-condition for creating further local antilinear operators which are orthogonal to the original operator. It means that the symmetric group acts transitively on the space of combs of a given order. This extends the mechanism for writing -invariants for qubits to qudits. I outline the method, that in principle works for arbitrary dimension , explicitely for spin 1, and spin 3/2. There is an odd-even discrepancy: whereas for half odd integer spin a situation similar to that observed for qubits is found, for integer spin the outcome is an asymmetric invariant of polynomial degree .
10 pages, subject of the part on maximally entangled states split off, misprints and some of the formulas modified, revtex 4
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