Algebraic Measures of Entanglement

We study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗ Hk. The rank of u is the minimal number p of pure states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C) and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for (C); in particular, we show that the maximal rank of a tensor in (C) is equal to 4.