Indefinite Causal Order Reverses the Real-Complex Hierarchy

Abstract: Can causal relations be subject to quantum indefiniteness, similar to other physical properties? The process-matrix framework formalises this possibility: valid processes are defined by what local laboratories can implement, without assuming a global causal order. Standardly, the local labs are assumed to implement arbitrary quantum instruments. We ask what happens when symmetries restrict these local operations. Symmetry constraints, such as those arising from missing reference frames, superselection constraints, or the antiunitary symmetry defining real quantum theory, enlarge the admissible process cone. Do these extra processes generate genuinely new correlations? We prove a sharp dichotomy: no for any finite unitary symmetry, yes for real quantum theory. Recent work has shown that, under fixed and definite causal order, complex quantum theory is strictly richer than real quantum theory. Our work shows that this hierarchy is reversed under indefinite causal order: real quantum theory realizes strictly more process correlations than complex quantum theory.

• The paper demonstrates that under indefinite causal order, real quantum theory achieves enhanced process correlations that surpass those in complex quantum theory.
• It rigorously analyzes symmetry-restricted scenarios within the process-matrix framework using SDP algorithms and explicit reference-frame protocols.
• The study reveals that while finite unitary restrictions yield equivalent operational results, antiunitary (real) constraints allow for causal inequality violations and deeper quantum insights.

Indefinite Causal Order and the Real-Complex Hierarchy in Quantum Theory

Overview

This paper rigorously investigates the operational implications of symmetry-restricted quantum process theories with an emphasis on the longstanding question of the foundational role of real versus complex numbers in quantum mechanics. Within the process-matrix framework, which allows indefinite causal order, the authors analyze how restricting laboratory operations by symmetry—unitary or antiunitary—affects the achievable process correlations. The principal result is the discovery of an inversion in the operational hierarchy between real quantum theory (RQT) and complex quantum theory (QT) when causal order is indefinite: whereas standard results show that complex quantum theory is richer than real quantum theory under fixed or definite causal structure, the inclusion reverses for indefinite causal order, with RQT strictly containing more finite-dimensional process correlations than QT.

Process-Matrix Framework and Symmetry-Restricted Quantum Theories

The Oreshkov-Costa-Brukner process-matrix framework models scenarios where local operations are embedded in a global context without presupposing a fixed global causal order. A process matrix is an operator assigning probabilities to all possible local instruments implemented by the local laboratories, with physicality enforced by positivity and normalization constraints across all allowed instruments.

The operational content of the process matrix thus depends on the set of permissible laboratory actions. Allowing only symmetric operations (imposed by either lack of a reference frame or superselection rules) either leaves the process correlation set unchanged or, as demonstrated, may significantly alter it.

Two main classes of symmetry are considered:

• Finite unitary symmetries: Induced by compact groups represented unitarily on local Hilbert spaces. These enforce group covariance on all local labs, restricting operations to those compatible with the relevant symmetry.
• Antiunitary symmetries (e.g., complex conjugation): For RQT, all process matrices, states, effects, and operations are real in a fixed basis, reflecting time-reversal invariance.

The central technical theme is whether expanding the cone of admissible process matrices, resulting from weaker local constraints, genuinely increases the set of observable process correlations, or if the expansion is 'washed out' at the level of operational probabilities.

Main Results

1. Finite Unitary Symmetry Restrictions: Operational Neutrality

For symmetry constraints arising from finite unitary groups, the main theorem establishes:

$$\mathcal{C}^{\mathrm{proc,fin}}_{\mathsf{Tw}_G} = \mathcal{C}^{\mathrm{proc,fin}}_{\mathrm{QT}}$$

That is, in any finite-dimensional process-matrix scenario, restricting local operations to $G$-covariant (unitary symmetry-restricted) instruments does not create any new observable process correlations. All process correlations achievable in a $G$-twirled theory can be simulated within standard quantum theory by explicitly encoding reference frames and group registers. The proof provides a constructive method, leveraging reference-frame ancillas and explicit encoding, adapting and extending protocols from the literature on reference frames and resource theories of asymmetry.

2. Real Quantum Theory: Enlarged Correlation Set Under Indefinite Causality

For antiunitary symmetry, specifically real quantum theory, the strict inclusion result is:

$$\mathcal{C}^{\mathrm{proc,fin}}_{\mathrm{QT}} \subsetneq \mathcal{C}^{\mathrm{proc,fin}}_{\mathrm{RQT}}$$

Here, the process cone expansion has true operational consequences: there exist extremal RQT process strategies generating correlations that cannot be realized by any finite-dimensional QT process. This is witnessed using the "Lazy Guess Your Neighbour's Input" (LGYNI) causal inequality; explicit finite-dimensional RQT processes attain LGYNI values strictly greater than the tight quantum bound obtained by Liu and Chiribella for all finite-dimensional complex process strategies. The reported value $I^{\rm{RQT}}_{\mathrm{LGYNI}} \approx 0.860$ is compared to the quantum bound $I^{\mathrm{QT}}_{\mathrm{LGYNI}} \approx 0.819$, providing a direct, quantitative certificate of strict inclusion.

The mechanism is subtle: in the symmetry-restricted (real or twirled) theories, weaker local constraints admit process matrices with "OCB-forbidden" causal-loop support, which are locally tomographically inaccessible. These components can render physically-acceptable, and operationally novel, the locally accessible projections that would otherwise be inadmissible due to lacking global positivity.

3. Hierarchy Reversal: Comparison Across Causal Scenarios

Classically, and in quantum theory under definite multipartite causal structure, RQT is operationally weaker than (strictly included in) QT, as previously shown by Renou et al. and others. Under indefinite causal order, this inclusion reverses: process correlations accessible to real quantum processes strictly contain those of complex quantum processes. The authors provide a figure (not reproduced here) illustrating the sequence of inclusion reversals across causal scenarios.

Numerical Methods and Certificates

The paper details the use of see-saw SDP algorithms to search over RQT process matrices and real-instruments, with explicit verification of positivity, normalization, and all process constraints, including those arising from the specific structure of real quantum Choi matrices. The numerical gap found between RQT and QT in the LGYNI value is robust, i.e., well outside the maximal residuals or solver tolerances.

Implications

The work establishes that indefinite causal order fundamentally changes the operational hierarchy between real and complex quantum theory. Although twirled (unitary symmetry-restricted) theories are operationally equivalent to QT even under indefinite causality, this fails for antiunitary (real) symmetry. The results hold for arbitrary finite-dimensional local systems and arbitrary finite instrument dimensions, setting a new precedent for operational distinctions in quantum foundational research.

From a foundational standpoint, the strict operational gap found here implies that any reconstruction of quantum theory from physically-motivated principles must carefully account for the impact of causal structure assumptions. Failure of local tomography in symmetry-restricted theories becomes operationally significant only when causal order can be indefinite; thus, the concept of "locality" in process-theoretic reconstructions must be re-examined when dropping causal structure assumptions.

On the applied and future-theoretic side, these results raise the possibility of exploiting such operational gaps for information-theoretic or computational tasks in indefinite-causal-order contexts, should real quantum process correlations be physically accessible or simulatable.

Future Directions

Open questions and potential lines of research delineated by the authors include:

• Whether the QT–twirled theory equivalence persists for infinite or continuous groups (e.g., spatial rotations), where the reference-frame simulation argument may fail.
• Extension to quaternionic quantum theory and other GPTs, especially regarding causal inequalities and process matrices.
• The operational meaning and utilization of OCB-forbidden, locally inaccessible process matrix components for causality-based quantum protocols.
• Exploration of "antinomical" versus "noncausal" behaviors in higher-party scenarios, and their separation in RQT versus QT.
• Influence of modifications to locality or composition structure (as in recent works on real quantum theory variants) on the operational distinction under indefinite causal order.

Conclusion

This paper rigorously demonstrates that allowing indefinite causal order fundamentally alters the comparison between real and complex quantum theories. While finite unitary symmetry-restrictions are operationally inert at the level of process correlations, antiunitary (real) restrictions lead to strictly richer process behavior under indefinite causal order. These results revisit core questions regarding the role of complex numbers and the structure of quantum causality, and introduce new constraints for future axiomatic and information-theoretic analyses of quantum mechanics.