Low Rank Separable States Are A Set Of Measure Zero Within The Set of Low Rank States

It is well known that the set of separable pure states is measure 0 in the set of pure states. We extend this fact and show that the set of rank r separable states is measure 0 in the set of rank r states provided r is less than ∏p i=1 ni + p − ∑ ni. Recently quite a few authors have looked at low rank separable and entangled states. (See [1] and the references therein and [2] and [3] Therefore it makes sense to determine the size of the set of rank r separable states within the set of rank r states. For rank 1, it is well known that the separable states are a set of measure zero. This contrasts with the maximal rank case, where the separable states not only are not measure zero, but contain an open set. The purpose of this note is to show that the rank 1 result is also true for many other low ranks. In particular, suppose we have p particles modelled on the Hilbert space C1 ⊗ · · · ⊗ Cp . Then the following is true. Theorem 1 Let Sr be the set of rank r separable matrices on C n1 ⊗ · · · ⊗ Cp and Dr the set of all rank r density matrices. Sr is measure 0 in Dr, for all r < N + p− ∑ ni, where N = n1 · · ·np This is an extension of recent results of Chen. In [3], he showed in the bipartite case whereH = C⊗C that Sr is measure 0 inDr if r < 2m−3. As a result of Theorem 1, we see that Sr is in fact measure 0 inDr if r < m −2m+2 = (m− 1). Since the number of possible ranks is m, we see that in this case the proportion of those r for which Sr is measure 0 in Dr is (1 − (1/m )). In the case of p-qubits, Sr is measure 0 in Dr if r < 2 p − 2p + p = 2 − p. Since the number of possible ranks in this case is 2, we see that the proportion of those r for which Sr is measure 0 in Dr is 1− (p/2 ). Thus we see that the ranks of ”low” rank matrices can be quite high. The proof will use Sard’s Theorem [4]to show the set of ranges of separable rank r density matrices is measure 0 in the set of ranges of rank r density matrices, i.e. within the set of r-dimensional subspaces of C ; i.e. within