Multigraded Poincare series for mixed states of two qubits and the boundary of the set of separable states

Details Multigraded Poincare series for mixed states of two qubits and the boundary of the set of separable states Multigraded Poincare series for mixed states of two qubits and the boundary of the set of separable states

2006-04-26

arxiv.org/abs/quant-ph/0604190v2

Physics

Quantum Physics

6 pages, no figures. Exposition improved. Abstract modified. The last section corrected and expanded giving more accurate desc

Scientific paper

Let M be the set of mixed states and S the set of separable states of the two-qubit system, and G = SU(2) x SU(2) the group of local unitary transformations (ignoring the overall phase factor). We compute the multigraded Poincare series for the algebra of G-invariant polynomial functions on the affine space of all Hermitian operators of trace 1. We check that this series is consistent with the list of invariants computed by Makhlin. By using the recent result of Augusiak et al., we show that the boundary of S decomposes naturally into two pieces. We prove that the part of this boundary which is contained in the relative interior of M is a smooth manifold.

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