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 write $M\; =\; v\_1\; v\_1\; ^*\; +\; T$, 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

$M\; w\; =\; langle\; w,\; v\_1\; rangle\; v\_1\; +\; T\; w\; =\; (langle\; w,\; v\_1\; rangle\; +\; alpha\; )\; v\_1.$

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

$rho\; =\; sum\_i\; psi\_\{1,i\}\; psi\_\{1,i\}^*\; otimes\; cdots\; otimes\; psi\_\{n,i\}\; psi\_\{n,i\}^*$

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

$$

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 $psi\_\{1,i\}\; otimes\; cdots\; otimes\; psi\_\{n,i\}$ 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).