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

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

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

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

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

:$ho\; =\; sum\_i\; (\; psi\_\{1,i\}\; otimes\; cdots\; otimes\; psi\_\{n,i\}\; )\; (\; psi\_\{1,i\}\; ^*\; otimes\; cdots\; otimes\; psi\_\{n,i\}\; ^*\; ).$

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).