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 $M = \sum_i v_i v_i^*$, the range of ''M'', ''Ran(M)'', is contained in the linear span of $\; \$. 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 write
$M = v_1 v_1 ^* + T$, where ''T'' is Hermitian and positive semidefinite. There are two possibilities:

1) ''span''$\ \subset$''Ker(T)''. Clearly, in this case, $v_1 \in$ ''Ran(M)''.

2) Notice 1) is true if and only if ''Ker(T)''$\;^ \subset$ ''span''$\^$, where $\perp$ denotes orthogonal complement. By Hermiticity of ''T'', this is the same as ''Ran(T)''$\subset$ ''span''$\^$. So if 1) does not hold, the intersection ''Ran(T)'' $\cap$ ''span''$\$ 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 $\; \$. 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

:$\rho = \sum_i \psi_ \psi_^* \otimes \cdots \otimes \psi_ \psi_^*$

where $\psi_ \psi_^*$ is a (un-normalized) pure state on the ''j''-th subsystem. This is also

:$\rho = \sum_i ( \psi_ \otimes \cdots \otimes \psi_ ) ( \psi_ ^* \otimes \cdots \otimes \psi_ ^* ).$

But this is exactly the same form as ''M'' from above, with the vectorial product state $\psi_ \otimes \cdots \otimes \psi_{n,i}$ 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).

Quantum information science