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

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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.  H = H 1 ⊗ ⋯ ⊗ H n {\displaystyle 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

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In general, if a matrix M is of the form  M = ∑ i v i v i ∗ {\displaystyle M=\sum _{i}v_{i}v_{i}^{*}} , the range of M, Ran(M), is contained in the linear span of  { v i } {\displaystyle \;\\{v_{i}\\}} . On the other hand, we can also show  v i {\displaystyle 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 {\displaystyle M=v_{1}v_{1}^{*}+T} , where T is Hermitian and positive semidefinite. There are two possibilities:

1) span { v 1 } ⊂ {\displaystyle \\{v_{1}\\}\subset } Ker(T). Clearly, in this case,  v 1 ∈ {\displaystyle v_{1}\in } Ran(M).

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

M w = ⟨ w , v 1 ⟩ v 1 + T w = ( ⟨ w , v 1 ⟩ + α ) v 1 . {\displaystyle Mw=\langle w,v_{1}\rangle v_{1}+Tw=(\langle w,v_{1}\rangle +\alpha )v_{1}.}

Therefore  v 1 {\displaystyle v_{1}} lies in Ran(M).

Thus Ran(M) coincides with the linear span of  { v i } {\displaystyle \;\\{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

ρ = ∑ i ψ 1 , i ψ 1 , i ∗ ⊗ ⋯ ⊗ ψ n , i ψ n , i ∗ {\displaystyle \rho =\sum _{i}\psi _{1,i}\psi _{1,i}^{*}\otimes \cdots \otimes \psi _{n,i}\psi _{n,i}^{*}}

where  ψ j , i ψ j , i ∗ {\displaystyle \psi _{j,i}\psi _{j,i}^{*}} is a (un-normalized) pure state on the j-th subsystem. This is also

ρ = ∑ i ( ψ 1 , i ⊗ ⋯ ⊗ ψ n , i ) ( ψ 1 , i ∗ ⊗ ⋯ ⊗ ψ n , i ∗ ) . {\displaystyle \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  ψ 1 , i ⊗ ⋯ ⊗ ψ n , i {\displaystyle \psi _{1,i}\otimes \cdots \otimes \psi _{n,i}} replacing  v i {\displaystyle v_{i}} . 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).