Reverse Data-Processing Theorems and Computational Second Laws

TL;DR

This paper explores the analogy between thermodynamic second laws and data-processing inequalities, establishing a framework that characterizes computationally isolated systems in both classical and quantum contexts.

Contribution

It generalizes Cover's analogy, providing a formal framework and proving that data-processing inequalities are both necessary and sufficient for system isolation.

Findings

Data-processing inequalities characterize computationally isolated systems.

The framework applies to both classical and quantum systems.

An information-theoretic entropy principle analogous to thermodynamics is established.

Abstract

Drawing on an analogy with the second law of thermodynamics for adiabatically isolated systems, Cover argued that data-processing inequalities may be seen as second laws for "computationally isolated systems," namely, systems evolving without an external memory. Here we develop Cover's idea in two ways: on the one hand, we clarify its meaning and formulate it in a general framework able to describe both classical and quantum systems. On the other hand, we prove that also the reverse holds: the validity of data-processing inequalities is not only necessary, but also sufficient to conclude that a system is computationally isolated. This constitutes an information-theoretic analogue of Lieb's and Yngvason's entropy principle. We finally speculate about the possibility of employing Maxwell's demon to show that adiabaticity and memorylessness are in fact connected in a deeper way than what the formal analogy proposed here prima facie seems to suggest.