In the framework of Layton, Oppenheim and Weller-Davies ("A one-world interpretation of quantum mechanics", 2025), an ontically classical system and a quantum system interact through a single shared diffusion noise, and the coupled dynamics - including unitary evolution, wavefunction collapse and the Born rule - is fully characterized by a positive function g on Hilbert space realized as a change of measure on classical paths. The authors leave open, with an explicit statement that there is "no proof in either direction", whether every such classical-quantum description can be obtained as an effective reduction of a fully quantum theory (a unitary quantum dilation) for all times, or whether genuinely post-quantum hybrid dynamics exist that no underlying quantum theory can reproduce. The surrounding literature (consistency classifications of hybrid dynamics, partial embeddability results from decoherence limits, and recent derivations of non-Markovian classical-quantum dynamics from decohered quantum systems) leaves the question open: partial positive results exist, but neither a general dilation theorem covering the full class of admissible g nor a non-embeddable counterexample is known.

Background

The measurement problem - explaining how definite classical outcomes emerge from unitary quantum evolution - is usually attacked by deriving classicality from quantum theory. Layton, Oppenheim and Weller-Davies (2025) invert this strategy: they postulate an objective, "ontic" (possessing definite values independent of observation) classical system ZtZ_t existing alongside a quantum system ∣ψ⟩t|\psi\rangle_t, require that ordinary probability theory governs their interaction, and show that unitary evolution, collapse and the Born rule then follow from four axioms. In their framework the classical system obeys a stochastic differential equation driven by a noise ξt\xi_t, and the quantum state evolves linearly under the same shared noise. Consistency forces the classical probability distribution to be obtainable from the pure-noise evolution by a change of measure - a standard tool of stochastic calculus in which path probabilities are reweighted by a positive function - implemented by a function gg on Hilbert space that must be a martingale (a stochastic process whose expectation value is constant in time) and normalized. Requiring gg to be invariant under unitary changes of basis singles out standard quantum mechanics; other admissible choices of gg define consistent "quantum-like" theories, for example real-amplitude quantum theory, which are post-quantum in the sense of not obviously arising from quantum theory.

Because the two sectors share one noise process, the quantum state conditioned on the classical trajectory remains pure at all times - there is no decoherence (no irreversible suppression of quantum superpositions into a statistical mixture), hence "one world" rather than the branching worlds of the Everett interpretation. The framework is a non-Markovian (memory-dependent, so the future evolution depends on the interaction history) member of the broader family of hybrid classical-quantum (CQ) dynamics, in which a classical phase-space degree of freedom and a quantum system couple through evolution that is linear, completely positive (CP: positivity-preserving even when an arbitrary auxiliary system is coupled on), and trace-preserving. The memoryless case of this family was classified by Oppenheim, Sparaciari, Šoda and Weller-Davies, who showed there are exactly two classes of such dynamics. This family underlies Oppenheim's "post-quantum" theory of classical gravity, in which spacetime is fundamentally classical but evolves stochastically while coupled to quantum matter, and a proven general feature of all such dynamics is a trade-off between decoherence of the quantum system and diffusion of the classical one.

Since noise in physical theories usually arises from tracing out an unobserved environment, it is natural to ask whether the one-world description is an effective theory: could a fully quantum system, evolving unitarily on a larger Hilbert space and partially averaged over, reproduce the hybrid dynamics? This is a quantum dilation question - a dilation being an embedding of an effective open-system dynamics into unitary dynamics on an enlarged system, as in the Stinespring dilation of quantum channels. The answer is known to be positive in certain cases: Layton and Oppenheim showed that when a quantum subsystem is strongly decohered by its environment, a double scaling limit (Planck's constant and the decoherence time both tending to zero at fixed ratio) yields exactly the CP classical-quantum generators; Tomizuka and Takeda (2026) derived generically non-Markovian CQ dynamics from a hidden quantum model in which a mediator is coupled to unobserved environmental degrees of freedom, subject to a positivity criterion (positive semidefinite semi-Wigner operator, equivalently positivity of nonlocal kernels), recovering the Markovian Oppenheim-class dynamics in the short-memory limit; and Tilloy parameterized general hybrid dynamics as continuous measurement with feedback, which is suggestive of quantum realizations. None of these covers the full one-world class for all times. The source paper conjectures the general answer may be negative - roughly, a classical system that records non-commuting quantum observables while keeping the quantum state pure and conditioned may be impossible to sustain for arbitrarily long times, and complete positivity is a weaker constraint for classical systems than for quantum ones - while explicitly stating that there is "no proof in either direction".

Problem Statement

For the class of hybrid dynamics defined by the axioms of Layton, Oppenheim and Weller-Davies (2025) - an ontically classical system and a quantum system driven by a single shared diffusion noise, with the coupled evolution characterized by a change of measure implemented by a positive function gg on Hilbert space satisfying the martingale and normalization conditions - determine whether every such classical-quantum description can be obtained as an effective reduction of a fully quantum theory for all times. Equivalently: does every admissible choice of gg (including non-unitarily-invariant, post-quantum choices) admit an underlying quantum theory, that is, a unitary dynamics on a larger Hilbert space with environmental degrees of freedom whose reduction reproduces both the classical trajectory statistics (via the change of measure) and the purity of the quantum state conditioned on the classical trajectory? A complete answer takes one of two forms: (i) a general dilation construction, together with an exact characterization of the hybrid dynamics it covers, showing universality over the full class; or (ii) a proof that admissible hybrid dynamics exist which no underlying quantum theory can reproduce for all times, exhibiting the witness dynamics and identifying the obstruction (for example, a time scale beyond which a classical system cannot keep recording non-commuting quantum observables while the quantum state remains pure). A classification giving necessary and sufficient conditions on gg, or on the induced dynamics, for quantum embeddability - with explicit examples on both sides - likewise resolves the question.

References

  1. Isaac Layton, Jonathan Oppenheim, Zachary Weller-Davies, "A one-world interpretation of quantum mechanics", arXiv:2510.09149 (2025), DOI: 10.48550/arXiv.2510.09149, https://arxiv.org/abs/2510.09149
  2. Jonathan Oppenheim, "A Postquantum Theory of Classical Gravity?", Physical Review X 13, 041040 (2023), DOI: 10.1103/PhysRevX.13.041040, arXiv:1811.03116, https://arxiv.org/abs/1811.03116
  3. Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, Zachary Weller-Davies, "The two classes of hybrid classical-quantum dynamics", arXiv:2203.01332 (2022), https://arxiv.org/abs/2203.01332
  4. Isaac Layton, Jonathan Oppenheim, "The Classical-Quantum Limit", PRX Quantum 5, 020331 (2024), DOI: 10.1103/PRXQuantum.5.020331, arXiv:2310.18271, https://arxiv.org/abs/2310.18271
  5. Antoine Tilloy, "General quantum-classical dynamics as measurement based feedback", SciPost Physics 17, 083 (2024), DOI: 10.21468/SciPostPhys.17.3.083, arXiv:2403.19748, https://arxiv.org/abs/2403.19748
  6. Shogo Tomizuka, Hiroki Takeda, "Emergence of Non-Markovian Classical-Quantum Dynamics from Decoherence", arXiv:2604.06891 (2026), https://arxiv.org/abs/2604.06891
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  10. Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, Zachary Weller-Davies, "Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity", Nature Communications 14, 7910 (2023), DOI: 10.1038/s41467-023-43348-2, arXiv:2203.01982, https://arxiv.org/abs/2203.01982
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