SL-invariant entanglement measures in higher dimensions: the case of spin 1 and 3/2

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 d × d two qudits state, which is the only SL-invariant of polynomial degree d. This determinant is written in terms of antilinear expectation values of the local SL(d) 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, which in principle works for arbitrary dimension d, explicitly 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 2d.