On Landauer's principle and bound for infinite systems

TL;DR

This paper establishes a quantum information-theoretic bound related to Landauer's principle for infinite systems using operator algebra and modular theory, linking free energy, Jones index, and quantized bounds.

Contribution

It introduces a new formula for incremental free energy of quantum channels in infinite systems, connecting it to the Jones index and extending Landauer's bound.

Findings

Lower bound for incremental free energy is half of Landauer's bound.

Bound is quantized and related to the Jones index.

In finite systems, the bound matches Landauer's original bound.

Abstract

Landauer's principle provides a link between Shannon's information entropy and Clausius' thermodynamical entropy. We set up here a basic formula for the incremental free energy of a quantum channel, possibly relative to infinite systems, naturally arising by an Operator Algebraic point of view. By the Tomita-Takesaki modular theory, we can indeed describe a canonical evolution associated with a quantum channel state transfer. Such evolution is implemented both by a modular Hamiltonian and a physical Hamiltonian, the latter being determined by its functoriality properties. This allows us to make an intrinsic analysis, extending our QFT index formula, but without any a priori given dynamics; the associated incremental free energy is related to the logarithm of the Jones index and is thus quantised. This leads to a general lower bound for the incremental free energy of an irreversible quantum channel which is half of the Landauer bound, and to further bounds corresponding to the discrete series of the Jones index. In the finite dimensional context, or in the case of DHR charges in QFT, where the dimension is a positive integer, our lower bound agrees with Landauer's bound.