Higher-order quantum processes respecting closed labs in a spacetime have quantum controlled causal order

• URL: http://arxiv.org/abs/2605.08351v1
• Date: Fri, 08 May 2026 18:01:12 GMT
• Title: Higher-order quantum processes respecting closed labs in a spacetime have quantum controlled causal order
• Abstract summary: A longstanding open question is what is the largest class of abstract processes that admit physical realisations without post-selection.<n>We provide a rigorous answer using a top-down approach grounded in relativistic causality principles.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In quantum causality and quantum information, there is a vast landscape of abstract quantum protocols permitting cyclic or non-acyclic causal structures between operations, including frameworks for indefinite causal order and higher-order quantum processes such as process matrices. A longstanding open question is what is the largest class of abstract processes that admit physical realisations without post-selection. In this work, we provide a rigorous answer using a top-down approach grounded in relativistic causality principles. Building on the framework of causal boxes, which characterise the most general quantum information-processing protocols compatible with fixed background spacetimes, we formalise additional constraints (Acting Once + Local Order) capturing the closed-laboratory assumptions of the process matrix framework at a fine-grained spacetime level. We prove that any protocol in a classical acyclic spacetime satisfying these conditions is behaviourally equivalent to a quantum circuit with quantum control of causal order (QC-QC), providing a top-down derivation of QC-QCs from physical principles. Our results show that QC-QCs constitute precisely the class of higher-order quantum processes, including those with indefinite order, that can be physically realised within classical spacetime, ruling out more general non-causal processes under the closed-labs assumption. This clarifies the relationship between abstract higher-order process matrix frameworks and experimentally accessible quantum protocols, as well as the interplay between coarse-grained cyclic and fine-grained acyclic operational causal structures. We also develop characterisation techniques for process box protocols that lead to new causality-based open questions concerning spacetime quantum protocols and relativistic quantum experiments.

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