Non-local properties of multi-particle density matrices

Abstract

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, $U(d_1)\times ...\times U(d_n)$ (where the Hilbert space of particle $r$ has dimension $d_r$). 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 ${\Pi }_r{d}_r^2-{\sum }_r{d}_r^2+n-1$. For $n$ spin-$\frac{1}{2}$ 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.

13 pages RevTeX

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• Quantifying Entanglement

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