Decoherence to quantum theory from a causally-indefinite post-quantum theory

1. Problem Statement

The paper addresses the fundamental question of how standard quantum theory (CPTP) might emerge from a more fundamental "post-quantum" theory via a process analogous to decoherence, termed hyper-decoherence.

• Context: While decoherence explains the emergence of classicality from quantum mechanics, the reverse question—how quantum mechanics emerges from a post-quantum theory—remains open.
• The Obstacle: A previous no-go theorem by Lee and Selby (2018) established that no post-quantum theory can support hyper-decoherence to standard quantum theory if it satisfies two conditions:
  1. Causality: The theory must be causal (no signaling from future to past).
  2. Unique Purifications: Every mixed state must have a purification unique up to a reversible transformation on the environment.
• The Gap: Existing toy models attempting hyper-decoherence (e.g., Quartic Quantum Theory, Density Cubes) fail because they violate causality or lack well-defined processes. The authors seek a post-quantum theory that is physically reasonable (specifically, non-signalling) but evades the no-go theorem by relaxing the uniqueness of purifications.

2. Methodology

The authors utilize the framework of Process Theories (Symmetric Monoidal Categories) and higher-order quantum theory.

• Theoretical Framework: They define a process theory $\mathcal{C}$ and construct a higher-order theory $D(\mathcal{C})$ where systems are pairs $[H, H]$ representing input-output channels.
• The Post-Quantum Theory (QBox): They focus on QBox, the theory of second-order quantum operations (quantum supermaps) under the non-signalling tensor product.
  • Systems in QBox are of the form $\bigotimes_i [H_i, H_i]$.
  • Processes are supermaps taking non-signalling channels as input and outputting non-signalling channels.
  • Crucially, QBox allows for indefinite causal order (e.g., quantum combs, process matrices).
• Multi-Environment Structures: To handle causality in higher-order theories, the authors introduce multi-environment structures. Unlike standard theories with a unique discard effect, QBox admits a family of discard effects (process matrices). This allows them to define "deterministic" maps and "no-backwards-signalling" conditions without enforcing a single global causal order.
• Hyper-Decoherence Map Construction: They define a specific map, hypdec, acting on QBox systems.
  • Structure: It acts as the identity on the "top" (input) wire of the supermap and applies a completely depolarizing map to the "bottom" (output) wire.
  • Effect: This effectively discards the "past" half of the causal structure, leaving only the "future-evolving" half, thereby reducing the observer's power.

3. Key Contributions and Results

A. Existence of the Hyper-Decoherence Map

The authors prove that QBox supports a hyper-decoherence map to standard quantum theory (CPTP).

• The Map: The map hypdec is defined diagrammatically as:
  $$\text{hypdec} = \text{Id}_{\text{top}} \otimes \mathcal{D}_{\text{bottom}}$$
  where $\mathcal{D}$ is the completely depolarizing channel.
• Idempotency (Ax1): Applying the map twice yields the same result as applying it once.
• No-Backwards-Signalling (Ax2): The map prevents signaling from the future to the past, satisfying the causal constraints required for a physical transition.
• Equivalence (Theorem 1): The full sub-process theory of the idempotent splitting of QBox (generated by the hypdec systems) is monoidally equivalent to standard quantum theory (CPTP).

B. Satisfaction of Purity Axioms

The paper rigorously demonstrates that the map satisfies the critical axioms regarding purity, which are often the point of failure in such theories:

• Ax3 (Purity Copreservation): If a state in QBox hyper-decoheres to a pure state in CPTP, the original state in QBox must be pure.
  • Technical Insight: In QBox, "pure" states are defined as extremal points in the convex set of deterministic states (channels). The authors prove that if a channel $g$ (representing the state) maps the maximally mixed state to a pure state, $g$ must be an extremal channel (a constant preparation of a pure state).
• Ax4 (Maximal Mixtures): The maximally mixed state in QBox maps to the maximally mixed state in CPTP.

C. Evading the No-Go Theorem

The central theoretical breakthrough is explaining how this bypasses the Lee-Selby no-go theorem:

• Causality: QBox is non-signalling (satisfies the no-signalling condition), though it is not strictly "causal" in the sense of a fixed temporal order.
• Uniqueness of Purifications: This is the key evasion. In standard quantum theory, purifications are unique (Stinespring dilation). However, in QBox, purifications are not unique.
  • The authors show that two distinct non-signalling channels (purifications) can reduce to the same mixed state but cannot be connected by a reversible transformation on the environment.
  • This non-uniqueness arises because the "environment" in a higher-order theory involves temporal degrees of freedom, and the extremality of channels does not guarantee unique dilations in the same way unitaries do in standard quantum mechanics.

4. Significance and Implications

• Mechanism for Emergent Causality: The paper provides a concrete mechanism by which a theory with indefinite causal order (a feature expected in quantum gravity) can "decohere" into a theory with fixed causal order (standard quantum mechanics). This suggests that causality might be an emergent property resulting from a reduction in the observer's access to temporal degrees of freedom.
• Re-evaluation of Hyper-Decoherence Axioms: The authors argue that the success of this map challenges the strictness of the hyper-decoherence axioms, particularly regarding the role of purity. They suggest that the "purity copreservation" axiom might need refinement, as the definition of purity in higher-order theories (convex extremality) behaves differently than in standard theories.
• Dimensionality Trade-off: The hyper-decoherence process in QBox works by restricting the observer's access to one half of a bipartite system (effectively discarding a temporal dimension). The authors acknowledge this might seem "trivial" compared to spatial decoherence but argue it is a necessary feature for theories involving indefinite causal order.
• Future Directions: The work opens the door to constructing a "tower" of higher-order theories ($D(C)$, $D(D(C))$, etc.), where each level hyper-decoheres into the one below it. It also suggests that higher-order quantum theory might be the unique natural candidate for a post-quantum theory from which standard quantum theory emerges.

Conclusion

Hefford and Wilson demonstrate that QBox (a theory of non-signalling quantum supermaps) is a valid post-quantum theory that hyper-decoheres into standard quantum theory. They achieve this by relaxing the requirement for unique purifications, thereby evading the Lee-Selby no-go theorem. This result supports the hypothesis that standard quantum mechanics could emerge from a more fundamental theory of indefinite causal order through a process analogous to decoherence, driven by the limitations of observer access to temporal degrees of freedom.