Spectral conditions on the state of a composite quantum system implying its separability

The separability modulus l(ρ) of a state ρ of an arbitrary finite composite quantum system is the largest t in [0, 1] such that t . ρ + (1 - t) . τ is separable, where τ is the normalized trace. The basic properties of l, introduced by Vidal and Tarrach in another guise, are briefly established. With these properties, we obtain conditions on the spectrum of a state which imply that it is separable. As a consequence, we show that for any Hamiltonian H the thermal equilibrium states e-H/T/Tr(e-H/T) are separable if T is large enough. Also, for F a unitarily invariant, convex continuous real-valued function on states, for which F(ρ) > F(τ) whenever ρ ≠ τ, there is a critical CF such that F(ρ) ≤ CF implies that ρ is separable, and for each possible c > CF there are entangled states φ with F(φ) = c. This class includes all strictly convex unitarily invariant continuous functions, and also every non-trivial partial eigenvalue-sum. Some CF are computed. General upper and lower bounds for CF are given, and then improved for bipartite systems