An equivalence between time-symmetry and cyclic causality in quantum theory

TL;DR

This paper establishes an operational equivalence between time-symmetric multi-time state formalism and cyclic causal influences via post-selected closed timelike curves in quantum theory, linking two foundational approaches.

Contribution

It extends the P-CTC framework to include time-labelled objects and proves their equivalence with multi-time states, revealing a deep connection between time symmetry and cyclic causality in quantum foundations.

Findings

Operationally equivalent mappings between MTS and P-CTC-assisted objects.

Extension of P-CTC framework to include explicit temporal structure.

Discussion of resource-theoretic partial order for MTS.

Abstract

Understanding the relationship between the time-symmetric nature of physical laws and the apparent directionality of causality is a central question in quantum foundations. The standard operational formulation, widely used in quantum information, imposes a definite, acyclic causal order on agents' operations, contrasting with time-symmetric dynamics. Two prominent extensions of this framework are the multi-time state (MTS) formalism, which incorporates time symmetry via arbitrary pre- and post-selection, and the post-selected closed timelike curve (P-CTC) framework, which enables cyclic causal influences through post-selection on maximally entangled states. While prior work has noted structural connections between MTS and P-CTCs, it remained unclear whether an operational equivalence exists, or whether constructive mappings can be established between their most general objects. In this work, we address this gap by extending the P-CTC framework to define time-labelled P-CTC assisted combs, a more general class of P-CTC-assisted objects that support open processing slots and explicit temporal structure. We prove that for every (possibly mixed) MTS, there exists an operationally equivalent time-labelled P-CTC-assisted comb, and vice versa. The equivalence is shown via explicit mappings, while discussing the number and dimensionality of the P-CTCs involved. We also explore a resource-theoretic view of MTS, defining a partial order under free transformations that do not use P-CTCs. We conclude by discussing future directions informed by the operational equivalence between time symmetry and cyclic causality established here.