Indefinite Causal Structure and Causal Inequalities with Time-Symmetry

TL;DR

This paper develops a time-symmetric process matrix formalism for indefinite causal structures in quantum theory, expanding the set of processes and causal inequalities, and enabling new device-independent certifications.

Contribution

It introduces a time-symmetric framework for indefinite causal structure, broadening the scope of processes and inequalities beyond previous asymmetric models.

Findings

Larger set of causal inequalities identified

New opportunities for device-independent certification

Equivalence with processes having Indefinite Causal Order and Time Direction

Abstract

Time-reversal symmetry is a prevalent feature of microscopic physics, including operational quantum theory and classical general relativity. Previous works have studied indefinite causal structure using the language of operational quantum theory, however, these rely on time-asymmetric conditions to constrain both operations and the process matrix. Here, we use time-symmetric, operational probabilistic theory to develop a time-symmetric process matrix formalism for indefinite causal structure. This framework allows for more processes than previously considered and a larger set of causal inequalities. We demonstrate that this larger set of causal inequalities offers new opportunities for device-independent certification of causal non-separability by violating new inequalities. Additionally, we determined that the larger class of time-symmetric processes found here is equivalent to those with Indefinite Causal Order and Time Direction (ICOTD) considered by Chiribella and Liu, thereby providing a description of these processes in terms of process matrices.