Maximum channel entropy principle and microcanonical channels

Abstract

The thermal state plays a number of significant roles throughout physics, information theory, quantum computing, and machine learning. It arises from Jaynes' maximum-entropy principle as the maximally entropic state subject to linear constraints, and is also the reduced state of the microcanonical state on the system and a large environment. We formulate a maximum-channel-entropy principle, defining a thermal channel as one that maximizes a channel entropy measure subject to linear constraints on the channel. We prove that thermal channels exhibit an exponential form reminiscent of thermal states. We study examples including thermalizing channels that conserve a state's average energy, as well as Pauli-covariant and classical channels. We propose a quantum channel learning algorithm based on maximum channel entropy methods that mirrors a similar learning algorithm for quantum states. We then demonstrate the thermodynamic relevance of the maximum-channel-entropy channel by proving that it resembles the action on a single system of a microcanonical channel acting on many copies of the system. Here, the microcanonical channel is defined by requiring that the linear constraints obey sharp statistics for any i.i.d. input state, including for noncommuting constraint operators. Our techniques involve convex optimization methods to optimize recently introduced channel entropy measures, typicality techniques involving noncommuting operators, a custom channel postselection technique, as well as Schur-Weyl duality. As a result of potential independent interest, we prove a constrained postselection theorem for quantum channels. The widespread relevance of the thermal state throughout physics, information theory, machine learning, and quantum computing, inspires promising applications for the analogous concept for quantum channels.