Range criterion explained

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} .

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} , 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 without loss of generality 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 complement. 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} \rangle v _{1} + T w = ( \langle w, v _{1} \rangle + \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

\rho = \sum _{i} \psi _{1,i} \psi ^{*} _{1 ,i} ⊗ … ⊗ \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\rho = \sum _{i} ( \psi _{1,i} ⊗ … ⊗ \psi _{n,i} ) ( \psi _{1,i} ^{*} ⊗ … ⊗ \psi _{n,i} ^{*} ).

But this is exactly the same form as M from above, with the vectorial product state

\psi _{1,i} ⊗ … ⊗ \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).