G.A. Raggio
Spectral conditions on the state of a composite quantum system implying its separability
(82K, revtex4)

ABSTRACT.  Simple conditions on the spectrum of the state (density operator) of a composite finite quantum system which guarantee its separability (unentangledness) are obtained. 
This is applied to thermal equilibrium (Gibbs) states $\rho_{\beta} = 
\exp ( - \beta H)/tr(\exp(-\beta H))$, and it is shown that there is an interval $I=[\beta_c^-, \beta_c^+]$ such that $\beta \in I$ implies that 
$\rho_{\beta}$ is separable, and if an interval $J$ contains $I$ properly then there is $\beta ' \in J$ with $\rho_{\beta '}$ entangled. 
It is shown that for every unitarily invariant, convex, continuous functional $F$ on states which isolates the normalized trace, there is a critical $C_F$ such that $F( \rho )\leq C_F$ implies $\rho$ is separable, and for each possible $C > C_F$ there is an entangled state $\phi$ with 
$F( \phi )=C$. Upper and lower bounds on $C_F$ are given. 
Some $C_F$'s are computed for bipartite systems of two qubits or a qubit and a qutrit.