The Equivalence of Schr\"{o}dinger and Heisenberg Pictures in Quantum Cellular Automata

TL;DR

This paper proves that for quantum cellular automata, the Heisenberg and Schr"{o}dinger descriptions are mathematically equivalent, resolving a long-standing open problem in quantum theory.

Contribution

It establishes the equivalence of Heisenberg and Schr"{o}dinger templates for quantum cellular automata using representation theory and category theory.

Findings

Proves the existence of a Schr"{o}dinger template for every Heisenberg template.

Establishes the equivalence of the two templates in QCA.

Uses advanced mathematical frameworks to solve a foundational problem.

Abstract

Quantum cellular automata (QCA) are discrete models of space and time homogeneous quantum field theories (QFTs) and regarded as natural candidates for quantum simulation. Description of a QCA over the separable Hilbert space of finite, unbounded configurations (UFC Hilbert space) with unitary state evolution is the {\it Schr\"{o}dinger template}, and over the incomplete infinite tensor product algebra (ITPA) with evolution by algebra automorphism is the {\it Heisenberg template}. Whether every Heisenberg template admits an equivalent Schr\"{o}dinger template is a foundational question, and one that has persisted as an open problem. In the present paper we prove that for every Heisenberg template an equivalent Schr\"{o}dinger template exists. We frame the question from a representation theory standpoint, using constructs and results from the representation theory of finite and countably infinite dimensional vector spaces and from category theory to answer it. With the previously known existence of a Heisenberg template for every Schr\"{o}dinger template, our result establishes the equivalence of the templates.