Range criterion

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".

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^*, it is obvious that 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 w.l.o.g. "i = 1". We can writeM = 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 compliment. 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 angle v_1 + T w = ( langle w, v_1 angle + 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

: ho = 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

: ho = 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).


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