Entropy and Holography through Adjunctions: A Bicategorical Perspective on Landauer's Principle

TL;DR

This paper introduces a bicategorical framework to model entropy and Landauer's principle, capturing the complexity of physical implementations and their holographic reconstruction through categorical structures.

Contribution

It develops a bicategory of open entropy systems that generalizes classical order-theoretic approaches, incorporating environmental dependence and many-to-many realizations.

Findings

Recover classical Landauer connection as a special case.

Identify bulk reconstruction from boundary data via a monad.

Characterize optimal information processes by minimal entropy production.

Abstract

We develop a bicategorical framework for entropy and Landauer's principle in which entropy-ordered state spaces are treated not merely through deterministic monotone maps, but through open many-to-many interfaces encoding feasible realizations between logical and thermodynamic descriptions. This leads to the bicategory of open entropy systems, whose objects are entropy posets, 1-morphisms are profunctorial feasibility relations, and 2-morphisms are refinements. In this setting, the classical order-theoretic Landauer connection is recovered as a representable special case, while the broader bicategorical language captures the openness, multiplicity, and environmental dependence of physical implementation, hence providing a more faithful language for many-to-many realizations between informational boundary states and thermodynamic bulk states. Additionally, the Landauer adjunction in this setting induces a boundary closure monad and a dual bulk interior operator, expressing categorically that bulk-mediated information processing is constrained by entropy and cannot increase recoverable information. We then show that this structure admits a holographic interpretation: the bulk accessible through a given interface is reconstructible from the boundary together with the induced idempotent monad. Via an Eilenberg-Moore construction, the visible bulk is identified, up to equivalence, with the closed sector of boundary data stable under the bulk-boundary round trip. The paper further formulates a quantitative enrichment in which interfaces carry dissipation costs and composition selects the least costly intermediate realization, thereby characterizing the optimal implementation of an information process as one that minimizes entropy production. In this way, the work tries to bring together entropy, Landauer's principle, and holographic reconstruction within a common categorical framework.