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.

The result

Consider a quantum mechanical system composed of n subsystems. The state space H.

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, the range of M, Ran(M), is contained in the linear span lies in Ran(M), for all i. Assume without loss of generality i = 1, where T is Hermitian and positive semidefinite. There are two possibilities:

1) spanKer(T) Ran(M).

2) Notice 1) is true if and only if Ker(T) span denotes orthogonal complement. By Hermiticity of T, this is the same as Ran(T) span. So if 1) does not hold, the intersection Ran(T) span. So

lies in Ran(M).

Thus Ran(M). 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

is a (un-normalized) pure state on the j-th subsystem. This is also

But this is exactly the same form as M. 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).