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

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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.  .

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

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

1) span Ker(T). Clearly, in this case,   Ran(M).

2) Notice 1) is true if and only if Ker(T)  span , where   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  is nonempty, i.e. there exists some complex number α such that  . So

 

Therefore   lies in Ran(M).

Thus Ran(M) coincides with the linear span of  . 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

 

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

 

But this is exactly the same form as M from above, with the vectorial product state   replacing  . It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.

References

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