Range criterion

In quantum mechanics, in particular quantum information, the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a separability criterion.

In quantum mechanics, in particular quantum information, the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a separability criterion. 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. 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.

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