Thermodynamic Value of XOR-Game-Induced Side Information in a Szilard Engine

Abstract

In this work, we study the reversible work value of a binary side-information channel supplied to a Szilard engine. The working system is a classical thermal bit, while the feedback controller receives only a binary record correlated with its physical state. We show that a fixed average prediction accuracy does not determine a unique thermodynamic value, because different binary channels with the same accuracy can carry different amounts of mutual information. We derive the exact minimum and maximum reversible Szilard work compatible with a given prediction accuracy. The minimum is attained uniquely by the binary symmetric channel, whereas the maximum is attained by two asymmetric channels for which all errors are concentrated on one input value. We then show that the standard XOR-game embedding always produces the symmetric channel and therefore realizes the minimum possible Szilard work among all binary side-information channels with the same prediction accuracy. Finally, we contrast this information-to-work valuation with an event-routing battery valuation, which depends only on the task-success probability.

6 pages, 1 figure, v2 generalized to XOR games and reorganized, v3 minor changes, v4 major revision