Action-angle coordinates

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 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$\{v_{1}\}\subset$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 $\;Tw=\alpha v_{1}$. So

$Mw=\langle w,v_{1}\rangle v_{1}+Tw=(\langle w,v_{1}\rangle +\alpha )v_{1}.$

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).