Operational time-symmetric formulation of quantum theory: axiomatic derivation, infinite-dimensional extension, physical realizability of generalized time-reversal symmetries, and predefined-time-free completion

Oreshkov and Cerf's operational formulation of time reversal (Nature Physics 11, 853 (2015)) rebuilt quantum theory's operational framework so that time reversal becomes an empirically consistent symmetry: valid operations may involve both pre- and post-selection, states and effects are pairs of positive semidefinite operators, and a Wigner-analogue theorem classifies all probability-preserving symmetries, revealing broader classes than the unitary/antiunitary ones, parameterized by a generally non-unitary operator $S$. The work left open several foundational questions -- posed as scattered forward-looking remarks rather than as one bundled agenda -- that remain unresolved: (1) deriving the generalized formulation from a concise set of operational axioms; (2) extending the pair representation, normalization, composition and update rules to infinite-dimensional Hilbert spaces; (3) determining whether the broader, non-unitary-$S$ symmetries are realized in nature -- e.g. in post-selection-based or gravity-coupled regimes -- or are forbidden by dynamical constraints such as universal unitary evolution and energy invariance; and (4) completing the extension, remarked in the paper's Discussion, to an operational quantum theory without predefined time. Later work developed part of (4) -- the authors' no-predefined-time formulation, time-symmetric process-matrix frameworks, and categorical treatments -- but no strand is closed: no axiomatic derivation, no infinite-dimensional construction, and no realization or no-go result for the non-unitary-$S$ classes exist. The problem is open.

Background

An operational formulation of quantum theory describes experiments without presupposing the theory's mathematical skeleton: devices with input and output systems are composed into circuits, and every device is characterized only by the statistics of its outcomes. In the standard formulation, a preparation of a system with a $d$-dimensional Hilbert space is represented by a density operator $\rho$ (a trace-one positive semidefinite operator), a measurement by a POVM -- a family of positive semidefinite operators $E_j$ summing to the identity -- and the joint probability of a preparation and a measurement outcome is given by Born's rule $p=\mathrm{Tr}(\rho E_j)$. The framework respects the causality axiom: probabilities of earlier outcomes cannot depend on later settings. Wigner's theorem then classifies the symmetries of this structure: any transformation of the pure states preserving transition probabilities is unitary or antiunitary, so time reversal is represented antiunitarily.

Oreshkov and Cerf (2015) observed that this conclusion rests on an operational asymmetry: Born's rule applies to a measurement following a preparation, not to the reverse order, and the standard notion of a valid operation excludes procedures that use post-selection (conditioning data on later outcomes), even though such procedures are routine in laboratories. Taking as valid any closed-box procedure — any procedure performable in a laboratory 'box', characterized only through the statistics of its input and output systems, possibly involving both pre- and post-selection — they derived a generalized formulation in which a preparation corresponds to a pair of positive semidefinite operators $(\rho;\bar\rho)$ with $\rho\le\bar\rho$ and $\mathrm{Tr}(\bar\rho)=1$, and a measurement to a pair $(E;\bar E)$ with $E\le\bar E$ and $\mathrm{Tr}(\bar E)=d$; the joint probability of events is

$p=\frac{\mathrm{Tr}(\rho E)}{\mathrm{Tr}(\bar\rho\bar E)}$

when $\mathrm{Tr}(\bar\rho\bar E)\neq0$ (with the convention that the probability is defined to be zero when the denominator vanishes — a null event, the pair simply never being found connected). In this framework the causality axiom is not part of the definition of an operation but a non-fundamental empirical constraint, tied to the form of the future boundary measurement of the universe.

Two structural facts follow. First, the sets of states and effects, viewed as real functions on each other via the probability rule, are not closed under convex combinations: a convex mixture of two states does not correspond to any procedure implementable separately in the preparation and measurement boxes -- realizing it requires joint post-selection across both ends of the circuit, so the convex-mixture functionals are not locally implementable. Second, defining a symmetry as a transformation preserving the probabilities of all circuits, the authors proved an analogue of Wigner's theorem: every such symmetry is generated by an invertible operator $S$ acting as $\rho\mapsto S\rho S^\dagger$ or $\rho\mapsto S\rho^{T}S^\dagger$ (with $T$ the transposition), in two types -- Type I mapping states to states and effects to effects, and Type II exchanging states and effects, as time reversal does. If isolated systems must follow unitary Schrödinger dynamics with energy spectra invariant under time reversal for arbitrary Hamiltonians, $S$ is forced to be unitary and the standard antiunitary picture is recovered; but since the generalized formulation also permits more general-than-unitary evolutions, symmetries with non-unitary $S$ are conceivable in principle, and the paper asks whether they are realized in nature -- through effective post-selection in laboratory scenarios, or in regimes where quantum theory couples to gravity, where post-selection-like dynamics have already been proposed for black holes and closed timelike curves.

The paper itself is finite-dimensional, defers the infinite-dimensional extension to future 'suitable modifications of the representation conventions', leaves open the criterion that distinguishes a valid operation from an arbitrary subset of events (the axiomatic status of the causality axiom being unclear), and notes that the time-symmetric approach suits theories without background time, building on the process-matrix framework -- the formalism in which operations on space-time regions are described without assuming a predefined global time or causal order. The operational completion of this program is the subject of the present problem.

Problem Statement

The problem is to settle the four foundational questions that the operational formulation of time reversal leaves open, treated as one program whose common objective is to determine whether the operationally time-symmetric formulation of quantum theory is a coherent, complete and physically instantiated framework -- that is, whether Wigner's antiunitary conclusion is an artifact of the standard representation choices, or the broader operational symmetry classes it reveals are physically real. The four linked deliverables are: (1) Axiomatic status: can the generalized formulation -- the pair representation of states and effects, its normalization conventions, composition and update rules, and its non-convex state and effect spaces -- be fully derived from a concise set of operational axioms of the kind used to reconstruct standard quantum theory? (2) Infinite dimensions: can the generalized representation, normalization conditions, composition and update rules be extended to infinite-dimensional Hilbert spaces, controlling domains, traces and normalizations? (3) Physical status of the broader symmetries: are the symmetry transformations with non-unitary $S$ realized in nature -- for example through effective post-selection in laboratory scenarios or in regimes where quantum theory couples to gravity -- or do dynamical constraints such as universal unitary evolution and energy-spectrum invariance forbid them? (4) Background-free completion: can the generalized, time-symmetric framework be fully extended to an operational quantum theory without any predefined time? A complete answer resolves each strand by a derivation, a construction, a realization, a no-go theorem, or an impossibility result, and states how the four resolutions jointly support the common objective. The four strands are jointly mandatory and independent: resolving any single strand contributes to the common objective but does not by itself constitute a complete answer.

References

1. Ognyan Oreshkov and Nicolas J. Cerf, 'Operational formulation of time reversal in quantum theory', Nature Physics 11, 853-858 (2015). DOI: 10.1038/nphys3414, https://doi.org/10.1038/nphys3414 (arXiv:1507.07745, https://arxiv.org/abs/1507.07745)
2. Ognyan Oreshkov and Nicolas J. Cerf, 'Operational quantum theory without predefined time', New Journal of Physics 18, 073037 (2016). DOI: 10.1088/1367-2630-18/7/073037, https://doi.org/10.1088/1367-2630-18/7/073037 (arXiv:1406.3829, https://arxiv.org/abs/1406.3829)
3. Ognyan Oreshkov, 'Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics', Quantum 3, 206 (2019). DOI: 10.22331/q-2019-12-02-206, https://doi.org/10.22331/q-2019-12-02-206 (arXiv:1801.07594, https://arxiv.org/abs/1801.07594)
4. Julian Wechs, Cyril Branciard, and Ognyan Oreshkov, 'Existence of processes violating causal inequalities on time-delocalised subsystems', Nature Communications 14, 1471 (2023). DOI: 10.1038/s41467-023-36893-3, https://doi.org/10.1038/s41467-023-36893-3 (arXiv:2201.11832, https://arxiv.org/abs/2201.11832)
5. Luke Mrini and Lucien Hardy, 'Indefinite Causal Structure and Causal Inequalities with Time-Symmetry', arXiv:2406.18489, https://arxiv.org/abs/2406.18489
6. John H. Selby, Maria E. Stasinou, Stefano Gogioso, and Bob Coecke, 'Time symmetry in quantum theories and beyond', arXiv:2209.07867, https://arxiv.org/abs/2209.07867
7. John H. Selby, Maria E. Stasinou, Matt Wilson, and Bob Coecke, 'Generalised Process Theories', arXiv:2502.10368, https://arxiv.org/abs/2502.10368
8. Giulio Chiribella, Erik Aurell, and Karol Życzkowski, 'Symmetries of quantum evolutions', Physical Review Research 3, 033028 (2021). DOI: 10.1103/PhysRevResearch.3.033028, https://doi.org/10.1103/PhysRevResearch.3.033028 (arXiv:2101.04962, https://arxiv.org/abs/2101.04962)
9. Giulio Chiribella and Zixuan Liu, 'Quantum operations with indefinite time direction', Communications Physics 5, 190 (2022). DOI: 10.1038/s42005-022-00967-3, https://doi.org/10.1038/s42005-022-00967-3 (arXiv:2012.03859, https://arxiv.org/abs/2012.03859)
10. Yu Guo, Zixuan Liu, Hao Tang, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, and Giulio Chiribella, 'Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device', Physical Review Letters 132, 160201 (2024). DOI: 10.1103/PhysRevLett.132.160201, https://doi.org/10.1103/PhysRevLett.132.160201 (arXiv:2210.17046, https://arxiv.org/abs/2210.17046)
11. Yi-Te Huang, Hsiang-Wei Huang, Jhen-Dong Lin, Adam Miranowicz, Neill Lambert, Guang-Yin Chen, Franco Nori, and Yueh-Nan Chen, 'Experimental simulation of postselected closed timelike curves for decoding scrambled quantum information', Physical Review Research 8, 023084 (2026). DOI: 10.1103/tm83-sxpm, https://doi.org/10.1103/tm83-sxpm (arXiv:2501.16335, https://arxiv.org/abs/2501.16335)
12. Eliot Jean, Ralph Silva, and V. Vilasini, 'An equivalence between time-symmetry and cyclic causality in quantum theory', arXiv:2508.02463, https://arxiv.org/abs/2508.02463
13. Ding Jia, 'Quantum theories from principles without assuming a definite causal structure', Physical Review A 98, 032112 (2018). DOI: 10.1103/PhysRevA.98.032112, https://doi.org/10.1103/PhysRevA.98.032112 (arXiv:1808.00898, https://arxiv.org/abs/1808.00898)
14. Arthur J. Parzygnat and Francesco Buscemi, 'Axioms for retrodiction: achieving time-reversal symmetry with a prior', Quantum 7, 1013 (2023). DOI: 10.22331/q-2023-05-23-1013, https://doi.org/10.22331/q-2023-05-23-1013 (arXiv:2210.13531, https://arxiv.org/abs/2210.13531)
15. Seok Hyung Lie and James Fullwood, 'Multipartite Quantum States over Time from Two Fundamental Assumptions', Physical Review Letters 135, 230204 (2025). DOI: 10.1103/lbf3-snp8, https://doi.org/10.1103/lbf3-snp8 (arXiv:2410.22630, https://arxiv.org/abs/2410.22630)
16. Alexander B. Boyd and Paul M. Riechers, 'Thermodynamic Advantage of Quantum Time-Reversal', arXiv:2501.04865, https://arxiv.org/abs/2501.04865
17. Sreenath K. Manikandan and Andrew N. Jordan, 'Time reversal symmetry of generalized quantum measurements with past and future boundary conditions', Quantum Studies: Mathematics and Foundations 6, 241-268 (2019). DOI: 10.1007/s40509-019-00182-w, https://doi.org/10.1007/s40509-019-00182-w (arXiv:1801.04364, https://arxiv.org/abs/1801.04364)
18. Evgeniy O. Kiktenko, 'Exploring postselection-induced quantum phenomena with time-bidirectional state formalism', Physical Review A 107, 032419 (2023). DOI: 10.1103/PhysRevA.107.032419, https://doi.org/10.1103/PhysRevA.107.032419 (arXiv:2210.01583, https://arxiv.org/abs/2210.01583)
19. Andrea Di Biagio, Pietro Donà, and Carlo Rovelli, 'The arrow of time in operational formulations of quantum theory', Quantum 5, 520 (2021). DOI: 10.22331/q-2021-08-09-520, https://doi.org/10.22331/q-2021-08-09-520 (arXiv:2010.05734, https://arxiv.org/abs/2010.05734)