Range criterion

The result

Consider a quantum mechanical system composed of n subsystems. The state space H of such a system is the tensor product of those of the subsystems, i.e. H = H_1 \otimes \cdots \otimes H_n.

For simplicity we will assume throughout that all relevant state spaces are finite-dimensional.

The criterion reads as follows: If ρ is a separable mixed state acting on H, then the range of ρ is spanned by a set of product vectors.

Proof

In general, if a matrix M is of the form M = \sum_i v_i v_i^, the range of M, Ran(M), is contained in the linear span of ; { v_i }. On the other hand, we can also show v_i lies in Ran(M), for all i. Assume without loss of generality i = 1. We can write M = v_1 v_1 ^ + T, where T is Hermitian and positive semidefinite. There are two possibilities:

1. span{ v_1 } \subsetKer(T). Clearly, in this case, v_1 \in Ran(M).

2. Notice 1) is true if and only if Ker(T);^{\perp} \subset span{ v_1 }^{\perp}, where \perp denotes orthogonal complement. By Hermiticity of T, this is the same as Ran(T)\subset span{ v_1 }^{\perp}. So if 1) does not hold, the intersection Ran(T) \cap span{ v_1 } is nonempty, i.e. there exists some complex number α such that ; T w = \alpha v_1. So

:M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1.

Therefore v_1 lies in Ran(M).

Thus Ran(M) coincides with the linear span of ; { v_i }. The range criterion is a special case of this fact.

A density matrix ρ acting on H is separable if and only if it can be written as

:\rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^*

where \psi_{j,i} \psi_{j,i}^* is a (un-normalized) pure state on the j-th subsystem. This is also

: \rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* ).

But this is exactly the same form as M from above, with the vectorial product state \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} replacing v_i. It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.

References

• P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", Physics Letters A 232, (1997).