Uniqueness of quantum state over time function

TL;DR

This paper investigates the uniqueness of quantum state over time functions, showing that previous axioms do not guarantee uniqueness and proposing new axioms that establish the Fullwood-Parzygnat function as essentially unique.

Contribution

It introduces an alternative set of operationally motivated axioms that ensure the uniqueness of the quantum state over time function, specifically validating the Fullwood-Parzygnat proposal.

Findings

Previous axioms do not guarantee uniqueness of the state over time function.

New axioms ensure the Fullwood-Parzygnat function is essentially unique.

The framework extends to quantum states over multiple spacetime regions.

Abstract

A fundamental A fundamental asymmetry exists within the conventional framework of quantum theory between space and time, in terms of representing causal relations via quantum channels and acausal relations via multipartite quantum states. Such a distinction does not exist in classical probability theory. In effort to introduce this symmetry to quantum theory, a new framework has recently been proposed, such that dynamical description of a quantum system can be encapsulated by a static quantum state over time. In particular, Fullwood and Parzygnat recently proposed the state over time function based on the Jordan product as a promising candidate for such a quantum state over time function, by showing that it satisfies all the axioms required in the no-go result by Horsman et al. However, it was unclear if the axioms induce a unique state over time function. In this work, we demonstrate that the previously proposed axioms cannot yield a unique state over time function. In response, we therefore propose an alternative set of axioms that is operationally motivated, and better suited to describe quantum states over any spacetime regions beyond two points. By doing so, we establish the Fullwood-Parzygnat state over time function as the essentially unique function satisfying all these operational axioms.