Superchannel without Tears: A Generalized Occam's Razor for Quantum Processes

TL;DR

This paper develops a unified, consistent framework for superchannels in quantum theory, connecting various representations and enabling analysis of complex quantum dynamics.

Contribution

It introduces a generalized Occam's razor and tensor-network methods to unify superchannel theory, resolving inconsistencies and incomplete structures in existing frameworks.

Findings

Unified foundation for superchannels established

Connections between Choi formulations clarified

Tools for analyzing non-Markovian quantum dynamics developed

Abstract

Quantum channels function as the operational primitives of quantum theory, while superchannels describe the most general transformations acting upon them. Yet the prevailing framework for superchannels is both internally inconsistent, owing to the coexistence of distinct Choi operator constructions, and structurally incomplete, lacking the analogue of representations that ground channel theory. We resolve these issues by combining tensor-network methods with a generalized Occam's razor introduced here, establishing a unified foundation for superchannels. Our framework establishes the connections between competing Choi formulations, develops the Kraus, Stinespring, and Liouville representations for superchannels, and provides a simplified derivation of the realization theorem that identifies the minimal memory required to implement a given transformation. These structural tools also enable characterizations of superchannels that destroy quantum correlations or causal structure, opening a systematic route to non-Markovian quantum dynamics.