New computable entanglement monotones from formal group theory

Abstract

We present a mathematical construction of new quantum information measures that generalize the notion of logarithmic negativity. Our approach is based on formal group theory. We shall prove that this family of generalized negativity functions, due their algebraic properties, is suitable for studying entanglement in many-body systems.Under mild hypotheses, the new measures are computable entanglement monotones. Also, they are composable: their evaluation over tensor products can be entirely computed in terms of the evaluations over each factor, by means of a specific group law.In principle, they might be useful to study separability and (in a future perspective) criticality of mixed states, complementing the role of Rényi's entanglement entropy in the discrimination of conformal sectors for pure states.

Permutation group entropy: A new route to complexity for real-valued processes

“…Among the many remarkable properties of entropy (17), we remind the following result, proved in [20]: Under mild technical hypotheses, Tsallis entropy is the most general traceform composable entropy. An entropy S(p) = S(p 1 , ..., p W ) is said to be trace form if S(p) = ∑ W i=1 g(p i ), i.e., it is the sum of univariate terms, such as the BGS entropy (1) but unlike the Rényi entropy (4).…”

“…The notion of logarithmic negativity introduced by Vidal and Werner in [52] can also be generalized by means of a mathematical formalism based on formal group theory. The main result is that there exists a "tower" of new, parametric information measures, each of them reducing to the logarithmic negativity in a suitable regime [17].…”

Multivariate Group Entropies, Super-exponentially Growing Complex Systems and Functional Equations

“…for a suitable G(t) ∈ R t . Besides, an n-dimensional generalization of both Definition 4 and equations ( 6)-( 7) has been proposed in [9].…”

“…The formal product Ψ(x, y) associated with the group law (36) has been computed in [9]. Thus, a similar analysis can be extended to the other functional equations related with the given ring structure.…”

“…A group entropy inherits in an obvious manner a functional equation, expressed by its own group law. We further investigate the set of functional equations related to group entropies by means of the algebraic structure of formal rings, recently introduced in [9]. A formal ring is essentially a formal group endowed with a second composition law, which can be naturally realized as a deformed product, compatible with the deformed sum expressed by the group law.…”