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. [math]\displaystyle{ H = H_1 \otimes \cdots \otimes H_n }[/math].

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 [math]\displaystyle{ M = \sum_i v_i v_i^* }[/math], the range of M, Ran(M), is contained in the linear span of [math]\displaystyle{ \; \{ v_i \} }[/math]. On the other hand, we can also show [math]\displaystyle{ v_i }[/math] lies in Ran(M), for all i. Assume without loss of generality i = 1. We can write [math]\displaystyle{ M = v_1 v_1 ^* + T }[/math], where T is Hermitian and positive semidefinite. There are two possibilities:

1) span[math]\displaystyle{ \{ v_1 \} \subset }[/math]Ker(T). Clearly, in this case, [math]\displaystyle{ v_1 \in }[/math] Ran(M).

2) Notice 1) is true if and only if Ker(T)[math]\displaystyle{ \;^{\perp} \subset }[/math] span[math]\displaystyle{ \{ v_1 \}^{\perp} }[/math], where [math]\displaystyle{ \perp }[/math] denotes orthogonal complement. By Hermiticity of T, this is the same as Ran(T)[math]\displaystyle{ \subset }[/math] span[math]\displaystyle{ \{ v_1 \}^{\perp} }[/math]. So if 1) does not hold, the intersection Ran(T) [math]\displaystyle{ \cap }[/math] span[math]\displaystyle{ \{ v_1 \} }[/math] is nonempty, i.e. there exists some complex number α such that [math]\displaystyle{ \; T w = \alpha v_1 }[/math]. So

[math]\displaystyle{ M w = \langle w, v_1 \rangle v_1 + T w = ( \langle w, v_1 \rangle + \alpha ) v_1. }[/math]

Therefore [math]\displaystyle{ v_1 }[/math] lies in Ran(M).

Thus Ran(M) coincides with the linear span of [math]\displaystyle{ \; \{ v_i \} }[/math]. 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

[math]\displaystyle{ \rho = \sum_i \psi_{1,i} \psi_{1,i}^* \otimes \cdots \otimes \psi_{n,i} \psi_{n,i}^* }[/math]

where [math]\displaystyle{ \psi_{j,i} \psi_{j,i}^* }[/math] is a (un-normalized) pure state on the j-th subsystem. This is also

[math]\displaystyle{ \rho = \sum_i ( \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} ) ( \psi_{1,i} ^* \otimes \cdots \otimes \psi_{n,i} ^* ). }[/math]

But this is exactly the same form as M from above, with the vectorial product state [math]\displaystyle{ \psi_{1,i} \otimes \cdots \otimes \psi_{n,i} }[/math] replacing [math]\displaystyle{ v_i }[/math]. 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).