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 ⊗ ⋯ ⊗ 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

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.

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