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Register a new userSL-invariant entanglement measures in higher dimensions: the case of spin $1$ and $3/2$
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Authors
Andreas Osterloh
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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 $dtimes d$ 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 $SL(2)$-invariants for qubits to qudits. I outline the method, that in principle works for arbitrary dimension $d$, 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 $2d$.
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An algorithm is proposed that serves to handle full rank density matrices, when coming from a lower rank method to compute the convex-roof. This is in order to calculate an upper bound for any polynomial SL invariant multipartite entanglement measure E. Here, it is exemplifyed how this algorithm works, based on a method for calculating convex-roofs of rank two density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-two case. The algorithm is therefore quasi exact as far as the two rank case is concerned, and it also gives hints where it should include more states in the decomposition of the density matrix. Focusing on the threetangle, I show the results the algorithm gives for two states, one of which being the $GHZ$-Werner state, for which the exact convex roof is known. It overestimates the threetangle in the state, thereby giving insight into the optimal decomposition the $GHZ$-Werner state has. As a proof of principle, I have run the algorithm for the threetangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.
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