Non-Local Properties of Multi-Particle Density Matrices

quantum mechanics, quantum information, quantum non-locality, quantum entanglement As far as entanglement is concerned, two density matrices of n particles are equivalent if they are on the same orbit of the group of local unitary transformations, U (d1) ×??? × U (dn) (where the Hilbert space of particle r has dimension dr). We show that for n greater than or equal to two, the number of independent parameters needed to specify an n- particle density matrix up to equivalence is ∏rd 2 r- ∑rd 2 r + n – 1. For n spin – particles we also show how to characterise generic orbits, both by giving an explicit parametrisation of the orbits and by finding a finite set of polynomial invariants which separate the orbits