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 \} \subset
Ker(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 ==