Quantum Theory Can Decohere from a Causally-Indefinite Post-Quantum Theory
James Hefford, Matt Wilson
TL;DR
Can quantum theory emerge from a post-quantum theory with indefinite causal order? The authors propose $\mathsf{QBox}$, a second-order quantum theory, and construct a hyper-decoherence map $\mathtt{hypdec}$ that yields standard quantum theory $\mathsf{CPTP}$; they prove that the deterministic sub-theory of $\mathsf{Split}(\mathsf{QBox})$ is equivalent to $\mathsf{CPTP}$ via a fully faithful functor. Key results include that $\mathtt{hypdec}$ is idempotent and no-backwards-signalling, purifications in $\mathsf{QBox}$ are not unique (allowing a bypass of the Lee–Selby no-go), and the maximally mixed state is preserved under hyper-decoherence. Significance: this provides a concrete mechanism for causality to emerge from a non-causal higher-order theory, highlights the subtle role of purity, and guides future exploration of post-quantum theories and potential higher-order interference phenomena via a principled categorical framework.
Abstract
We find a process satisfying the axioms of hyper-decoherence which produces standard quantum theory from the theory of quantum boxes (higher-order quantum theory with the non-signalling tensor product). This hyper-decoherence map evades the no-go theorem of Lee and Selby by relaxing constraints on signalling to the past and the uniqueness of purifications. We discuss some natural opposing conclusions: that the existence of this map might be evidence of a genuine hyper-decoherence process producing causal quantum theory from its causally-indefinite higher-order theory; or that it serves as an indication that the axioms of hyper-decoherence might need careful re-consideration, especially regarding the subtle albeit central role that purity plays.