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Communications in Mathematical Physics

Bogoliubov inequalities for infinite systems

J. C. Garrison, Jack Wong

Published Mar 1, 1972• 13 citations
Publisher
DOI
10.1007/bf01877542

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The quantum-mechanical two-sided Bogoliubov inequality provides upper and lower bounds for the free energy required to separate a system of interacting particles into independent subsystems. The bounds can be calculated straightforwardly from the ensemble average of the interface energy, bypassing the direct evaluation of the free energy. In this work, we generalize the two-sided Bogoliubov inequality to arbitrary von Neumann algebras by employing the Araki-Uhlmann relative entropy and the framework of unbounded perturbation theory of KMS states. Furthermore, we obtain variational expressions for the relative free energy that extend existing bounded-perturbation principles to the unbounded setting. Crucially, these mathematical developments yield a physically well-founded thermodynamic criterion for the quantification of entanglement in infinite-dimensional systems.
Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived.They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N -particle system and the corresponding Bogoliubov-Duhamel inner product.The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to N for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian.The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter.