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 unentangled, where τ is the normalized trace. The basic properties of l, introduced by Vidal & Tarrach [1] 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 6= 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 ’s are computed. General upper and lower bounds for CF are given, and then improved for bipartite systems.