What grounds the autonomous non-unitary dynamics of quantum subsystems: the arrow-of-time input, the limits of dilation to unitary microphysics, and the discrimination of fundamental from environmental non-unitarity

Wallace's paper 'Quantum Systems Other Than the Universe' argues that quantum physics studies autonomous but not necessarily isolated subsystems, which generically require mixed states and non-unitary dynamics, and that the empirically successful practice of explaining this non-unitarity by dilation to larger unitary systems rests on four reductionist principles whose grounds and limits it leaves open. The problem is the four-part cluster the paper leaves unresolved: (1) which physical or cosmological principle supplies the time-asymmetric initial condition (the absence of delicate system-environment correlations) without which no dilation-based derivation of irreversible autonomous subsystem dynamics is possible; (2) to what extent empirically relevant non-unitary dynamics can be exhaustively explained by dilation to the known unitary microphysical theories, and in which regimes fundamental non-unitarity would be warranted; (3) whether an Open Systems View that treats subsystem dynamical equations as primitive can coherently account for interaction, manipulability, and compositional consistency; (4) what practicable experimental protocols could, in principle, decisively distinguish intrinsic fundamental non-unitarity from environment-induced effective non-unitarity. No resolution of these four strands is known in the literature to date.

Background

Formally, a closed quantum system evolves unitarily under the Schrödinger equation and, when unentangled, can be described by a pure state. A subsystem $S$ of a larger system $S+E$ (with $E$ an 'environment') must instead be described by a mixed density operator obtained by the partial trace over $E$ (averaging over the environment's degrees of freedom), and its reduced dynamics is generally non-unitary. A dilation is the reverse construction: embedding $S$ in a larger quantum system and deriving the dynamics of $S$ from that of the whole. Stinespring's dilation theorem shows that, at the formal level, any well-behaved linear (even non-unitary) dynamics of a system is the restriction of some unitary dynamics on a larger system; the physical content of dilation lies in requiring the larger system to be concretely specified and observable rather than fictional. The Nakajima-Zwanzig equation is an exact integro-differential equation for the reduced state of a subsystem; when the environment's degrees of freedom effectively equilibrate much faster than the system evolves (scale separation), it reduces to an autonomous equation for $S$ alone, generally with memory (non-Markovian), and under further timescale assumptions to a memoryless Markovian dynamics whose dissipative part must take the standard Lindblad form (the most general form of Markovian dissipative evolution of a density operator that preserves normalization and complete positivity, i.e. that maps density operators to valid density operators even when the system is coupled to an arbitrary external one).

Wallace (2024) draws two lessons. First, physics studies autonomous systems — systems admitting a self-contained effective dynamics, possibly stochastic — rather than isolated systems; autonomy is weaker than, and largely independent of, both isolation and unitarity (a decaying atom and a Brownian particle are paradigmatic autonomous but non-isolated systems). Second, this does not topple unitarity from its special place, because physics practice is committed to four principles of 'quantum reductionism': (i) non-unitarity implies interaction with an environment (denying this is a last resort); (ii) interaction is to be modelled by dilation to a larger quantum system; (iii) the relevant dilations are to systems that can be idealized as unitary; (iv) subsystem dynamics should ultimately be derivable from a short list of known unitary microphysical theories — nonrelativistic quantum mechanics, relativistic quantum mechanics, quantum optics, quantum electrodynamics (QED, the quantized-field theory of electrons and nuclei interacting with light), the Standard Model of particle physics, and low-energy quantum gravity. These principles replace the unjustified 'Fundamentality Assumption' (a theory should be understood as exactly true) and 'Cosmological Assumption' (a theory should be understood as describing the whole universe), and are justified methodologically and inductively by the successes of quantum statistical mechanics, condensed matter physics, and effective field theory (the framework treating a theory as valid only below characteristic energy scales).

The main empirically testable forms of fundamental non-unitarity are dynamical-collapse theories, which modify the Schrödinger equation by stochastic non-unitary terms (e.g. the Ghirardi-Rimini-Weber model, GRW, and its continuous-spontaneous-localization variant CSL) and the gravity-related Diósi-Penrose collapse model; searching for them requires eliminating environment-induced non-unitarity (decoherence, the loss of phase coherence between components of a superposition due to entanglement with unobserved degrees of freedom) as the explanation. Wallace contrasts his position with the Open Systems View of Cuffaro and Hartmann (2024), which takes open rather than closed quantum systems as fundamental and treats the dynamical equations of open systems as pertaining directly to the system itself, not as contractions of larger closed-system dynamics. Finally, Wallace observes that the exact Nakajima-Zwanzig derivation of autonomous dynamics from unitarity requires a time-asymmetric input: the initial state of system plus environment must lack the delicate correlations that would prevent effective equilibration — an assumption that can hold for initial but not, on pain of contradiction, for final states. Proposed groundings for this input include operationalist limits on state preparation, time-directed principles of causal reasoning, and cosmological boundary conditions of the 'Past Hypothesis' family (the posit that the universe began in a very low-entropy state); Wallace declines to adjudicate among them, or against the interventionist alternative of explaining irreversibility by coupling to an even larger, weakly non-unitary environment.

Problem Statement

Determine what grounds, how far, and how testably the empirically successful practice of deriving autonomous — generally non-unitary — subsystem dynamics by dilation to unitary microphysics extends. The problem has four linked parts, jointly answering the single question of whether the autonomy of quantum subsystems is anchored in unitary microphysics or must be taken as primitive:

(1) Arrow-of-time input. Identify (or show there cannot be) the precise physical or cosmological principle that supplies the time-asymmetric initial condition — the absence of delicate system-environment correlations — required by any derivation of irreversible autonomous subsystem dynamics from underlying unitary laws, and adjudicate whether the interventionist alternative (explaining irreversibility by coupling to an even larger, weakly non-unitary environment) avoids rather than merely postpones that input.

(2) Limits of dilation. Determine to what extent empirically relevant non-unitary subsystem dynamics can be exhaustively explained by dilation to the listed unitary microphysical theories (nonrelativistic quantum mechanics, relativistic quantum mechanics, quantum optics, QED, the Standard Model, low-energy quantum gravity), and characterize the empirical regimes, if any, in which positing fundamental non-unitarity would be warranted — including the assessment of black-hole evaporation (Hawking radiation appears to evolve pure states into mixed states, in apparent conflict with unitarity).

(3) Viability of the Open Systems View. Decide whether a view that treats open-system dynamical equations as primitive can give a coherent account of interaction (what it means for an environment to influence a system), manipulability (how environmental influence can be reduced or screened, as in quantum-computing error management and the elimination of decoherence in collapse-model tests), and compositional consistency (how the dynamics of subsystems relate to the dynamics of composites), or whether such a view must — and in what precise form — accept dilation-based explanations.

(4) Empirical discrimination. Specify practicable experimental protocols that would, in principle, decisively distinguish intrinsic fundamental non-unitarity from environment-induced effective non-unitarity, given that exhaustive elimination of every conceivable environmental coupling is in practice impossible; make explicit what counts as sufficient exclusion of the environmental couplings that known microphysics gives reason to expect, and how strong the evidence for new physics must be.

A complete answer resolves all four parts consistently: the parts constrain one another (a resolution of (3) fixes what (4) must establish; a resolution of (1) fixes what any dilation-based derivation can claim), so partial answers must state which strands they leave open and remain consistent with the others.

References

1. Wallace, D. (2024). Quantum Systems Other Than the Universe. arXiv:2406.13058, https://arxiv.org/abs/2406.13058. Published as a chapter in M. E. Cuffaro and S. Hartmann (eds.), Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press (2026), DOI 10.1093/9780198929277.003.0005, https://doi.org/10.1093/9780198929277.003.0005.
2. Cuffaro, M. E., and Hartmann, S. (2024). The Open Systems View. Philosophy of Physics 2(1), 6, DOI 10.31389/pop.90, https://doi.org/10.31389/pop.90.
3. Cuffaro, M. E., and Hartmann, S. (eds.) (2026). Open Systems: Physics, Metaphysics, and Methodology. Oxford University Press, ISBN 9780198929277, DOI 10.1093/9780198929277.001.0001, https://doi.org/10.1093/9780198929277.001.0001.
4. Cuffaro, M. E., and Hartmann, S. (2026). Quantum Theory is about Open Systems. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0002, https://doi.org/10.1093/9780198929277.003.0002.
5. Adlam, E. (2026). The Temporally Open Systems View. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0003, https://doi.org/10.1093/9780198929277.003.0003.
6. Ladyman, J., and Thébault, K. P. Y. (2026). Open Systems and Autonomy. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0011, https://doi.org/10.1093/9780198929277.003.0011.
7. Mjelva, J. K., Quirke, J., and Wilson, A. (2026). Open Systems as Metaphysically Fundamental: Some Questions. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0013, https://doi.org/10.1093/9780198929277.003.0013.
8. Gryb, S., and Sloan, D. (2026). How Closed is Cosmology? In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0014, https://doi.org/10.1093/9780198929277.003.0014.
9. Chen, E. K. (2026). Density Matrix Realism. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0015, https://doi.org/10.1093/9780198929277.003.0015.
10. Myrvold, W. C. (2026). Perspective Duality as a Physical Requirement. In Open Systems: Physics, Metaphysics, and Methodology, Oxford University Press, DOI 10.1093/9780198929277.003.0018, https://doi.org/10.1093/9780198929277.003.0018.
11. Mohammady, M. H., and Buscemi, F. (2025). Thermodynamic closure of quantum measurements and the limits of the indirect measurement model. arXiv:2502.14136, https://arxiv.org/abs/2502.14136.
12. Rivat, S. (2024). Open systems across scales. Synthese 205, 11, DOI 10.1007/s11229-024-04817-3, https://doi.org/10.1007/s11229-024-04817-3.
13. Bassi, A., Dorato, M., and Ulbricht, H. (2023). Collapse Models: A Theoretical, Experimental and Philosophical Review. Entropy 25(4), 645, DOI 10.3390/e25040645, https://doi.org/10.3390/e25040645.
14. Donadi, S., Piscicchia, K., Curceanu, C., Diósi, L., Laubenstein, M., and Bassi, A. (2021). Underground test of gravity-related wave function collapse. Nature Physics 17, 74-78, DOI 10.1038/s41567-020-1008-4, https://doi.org/10.1038/s41567-020-1008-4.