Physical boundary of indefinite causal order

The process-matrix framework (Oreshkov–Costa–Brukner) generalizes quantum theory to allow operations without a fixed causal order between them. Some 'processes' are causally nonseparable — they cannot be explained by any definite (even probabilistic) ordering of events — raising the question of which are physically realizable.

What is known: Causal witnesses detect nonseparability, and implementations of the quantum switch realize coherent control of order. Nonseparability and violation of a causal inequality are different properties. Vilasini–Renner's published spacetime constraints distinguish sharply localized laboratories from delocalized implementations; the physical boundary depends on the laboratory and spacetime model.

Preprint theorem (May 2026): Salzger–Vilasini show that, in a classical acyclic background spacetime and the causal-box framework, processes satisfying fine-grained Acting Once and Local Order closed-laboratory conditions are behaviorally equivalent to quantum-controlled-order circuits (QC-QCs). This is a characterization under explicit assumptions, not a no-go theorem for all indefinite causal order or a characterization of quantum gravity.

Separate accepted-paper progress: Mothe–Abbott–Branciard classify non-influenceable causal order and corresponding controlled-order circuits. APS lists their paper as accepted in PRX Quantum on 20 August 2026 (preprint July 2025), not published in Physical Review Research.

Remaining questions require a specified change of model: relax a closed-lab condition, allow different localization or continuous relativistic implementations, or specify a nonclassical background. Characterize the new achievable processes and observable signatures; do not keep the characterized closed-lab regime open by omitting its assumptions. Related: F10.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: QC-QC characterization in classical acyclic spacetime with causal boxes and fine-grained Acting Once/Local Order closed-lab conditions; not an unrestricted quantum-gravity or ICO no-go result. Evidence