Entropy Densities for Algebraic States

Fumio Hiai

Paolo Facchi A, Giovanni Gramegna, Arturo Konderak

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.

Andrzej Łuczak, Hanna Podsȩdkowska, Rafał Wieczorek

We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Rényi entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.

V. P. Belavkin, P. Staszewski

M. Fannes

Andrés Fernando Reyes-Lega

An algebraic approach to the study of entanglement entropy of quantum systems is reviewed. Starting with a state on a $C^*$-algebra, one can construct a density operator describing the state in the GNS representation state. Applications of this approach to the study of entanglement measures for systems of identical particles are outlined. The ambiguities in the definition of entropy within this approach are then related to the action of unitaries in the commutant of the representation and their relation to modular theory explained.