Noncausal Page-Wootters circuits

TL;DR

This paper explores how combining the process matrix formalism with a multi-clock Page-Wootters approach reveals constraints on implementing noncausal quantum processes, advancing understanding of quantum causality.

Contribution

It introduces a framework integrating multiple quantum clocks into process matrices, analyzing implications for causal structures and constraints on noncausal process implementation.

Findings

Certain noncausal processes are not implementable with multiple quantum clocks.

The multi-clock approach constrains the causal reference frames of agents.

The framework links history states with process matrices, revealing causal constraints.

Abstract

One of the most fundamental open problems in physics is the unification of general relativity and quantum theory to a theory of quantum gravity. An aspect that might become relevant in such a theory is that the dynamical nature of causal structure present in general relativity displays quantum uncertainty. This may lead to a phenomenon known as indefinite or quantum causal structure, as captured by the process matrix formalism. Due to the generality of that framework, however, for many process matrices there is no clear physical interpretation. A popular approach towards a quantum theory of gravity is the Page-Wootters formalism, which associates to time a Hilbert space structure similar to spatial position. By explicitly introducing a quantum clock, it allows to describe time-evolution of systems via correlations between this clock and said systems encoded in history states. In this paper we combine the process matrix framework with a generalization of the Page-Wootters formalism in which one considers several agents, each with their own discrete quantum clock. We describe how to extract process matrices from scenarios involving such agents with quantum clocks, and analyze their properties. The description via a history state with multiple clocks imposes constraints on the implementation of process matrices and on the perspectives of the agents as described via causal reference frames. While it allows for scenarios where different definite causal orders are coherently controlled, we explain why certain noncausal processes might not be implementable within this setting.