Uniqueness of quantum state over time function

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.