Geometric picture of entanglement and Bell inequalities

Abstract

We work in the real Hilbert space H s of hermitian Hilbert-Schmidt operators and show that the entanglement witness which shows the maximal violation of a generalized Bell inequality (GBI) is a tangent functional to the convex set S ⊂ H s of separable states. This violation equals the euclidean distance in H s of the entangled state to S and thus entanglement, GBI and tangent functional are only different aspects of the same geometric picture. This is explicitly illustrated in the example of two spins, where also a comparison with familiar Bell inequalities is presented.

On information-theoretic characterizations of physical theories

“…Let K denote the set of convex combinations of pure product states; in other words, K is the space of separable states of B(C 2 ⊗ C 2 ), and corresponds to a compact convex subset of R 15 . (For some results on the geometry of K, see (Bertlmann, Narnhofer, & Thirring, 2002). )…”

Entanglement, Bell Inequalities and Decoherence in Particle Physics

“…A more general separability-entanglement criterion, valid in any dimensions, does exist; it is formulated by a so-called generalized Bell inequality, see Ref. [93]. Now let us return to the question of entanglement and separability of our kaon quantum state described by density matrix ρ N (t) (106), (93) as it evolves in time.…”

“…[93]. Now let us return to the question of entanglement and separability of our kaon quantum state described by density matrix ρ N (t) (106), (93) as it evolves in time.…”

“…Proposition: Bertlmann-Durstberger-Hiesmayr [48] • The state represented by the density matrix ρ N (t) (106), (93) becomes mixed for 0 < t < ∞ but remains entangled. Separability is achieved asymptotically t → ∞ with the weight e −λt .…”

“…It means that for t > 0 and λ = 0 the density matrix ρ(t) does not correspond to a pure state anymore. Finally, in order to arrive at a proper density matrix for the kaon system, conditioned on having not decayed up to time t , we have to divide ρ(t) (93) by the trace Trρ(t) , see Sect. 9.1.…”