Quantitative theory of redundant records of consistent histories: the objective quantum past for mixed states and imperfect environments

Riedel, Zurek and Zwolak (Phys. Rev. A 93, 032126 (2016)) proposed that the classical past becomes objective in the same way pointer states do in quantum Darwinism: many environment fragments independently hold records of a history, and the resulting 'redundant consistency' selects a small quasiclassical subset of the consistent histories, narrowing the set-selection problem of the decoherent-histories program. They proved that redundant records imply redundant consistency, with the converse holding only for pure global states, and explicitly deferred the quantitative theory to a companion paper ('The objective past of a quantum universe—part 2') that was never published. The open problem is to build this quantitative theory of the objective quantum past beyond pure states and ideal non-interacting environments: a redundancy measure for records of multi-time histories, equivalence conditions under mixed states and interacting (scrambling) environments with scaling laws for imperfect records, and a classification of the residual projector ('gauge') freedoms in selecting redundantly consistent histories.

Background

Quantum mechanics is unitary: a closed system's state evolves with no built-in notion of measurement outcome. Two research programs explain why the world nevertheless appears classical. Decoherence theory observes that when a system $S$ interacts with an environment $E$, certain robust 'pointer states' survive the monitoring while superpositions of them are rapidly destroyed, and information about the pointer states is copied into many environment fragments. Quantum Darwinism (Zurek, Nat. Phys. 5, 181 (2009); Ollivier, Poulin and Zurek, PRL 93, 220401 (2004)) makes the copying quantitative: the mutual information $I(S:F)$ between the system and a small fragment $F$ of the environment reaches (approximately) the full classical information content $H(S)$ already for fragments far smaller than the whole environment — a plateau of $I(S:F)$ as a function of fragment size. Because many independent observers can each learn the pointer state from their own fragment without disturbing the system, the pointer state is objective. Redundancy is quantified, for example, by counting how many distinct fragments each carry (nearly) all the information about the pointer observable, or by the distinguishability of the fragment states conditioned on the pointer value, measured by the quantum Chernoff information (the asymptotically optimal error exponent for discriminating two quantum states; Zwolak, Riedel and Zurek, PRL 112, 140406 (2014)).

The consistent- (or decoherent-) histories formalism (Griffiths, J. Stat. Phys. 36, 219 (1984); Gell-Mann and Hartle, PRD 47, 3345 (1993)) addresses the complementary, multi-time question: which sequences of events ('histories') in a closed quantum system admit classical probabilistic reasoning. A homogeneous (chain-form) history is a time-ordered string of projection operators; a set of histories is consistent when the decoherence functional $D(\alpha,\beta)$ — the overlap of the branch states $C_\alpha|\psi\rangle$ and $C_\beta|\psi\rangle$ produced by the two histories' projector strings acting on the global state — vanishes for $\alpha\neq\beta$, so that the histories' probabilities are additive and Boolean logic applies within the set. Consistency alone is nearly worthless as a classicality criterion: almost all consistent sets are flagrantly non-classical, and combining reasoning from different consistent sets yields contradictions. Identifying the physically meaningful set is the set-selection problem (Dowker and Kent, PRL 75, 3038 (1995)), the multi-time analogue of the preferred-basis problem.

Riedel, Zurek and Zwolak (2016) combined the two programs. Using Finkelstein's partial-trace condition (PRD 47, 5430 (1993)) — a history is partial-trace consistent with respect to an environment fragment $F$ when the branches are orthogonal within $F$ after tracing out the rest of the universe, i.e. the fragment has been the agent of the history's decoherence — they defined redundant consistency: partial-trace consistency with respect to many disjoint fragments. They proved that redundant records (many fragments each holding a distinguishable imprint of the history) imply redundant consistency, and that the converse holds when the global state is pure; in their spin-environment model, uniqueness of the branch decomposition follows from triorthogonal-decomposition uniqueness (the theorem that a state of three or more subsystems with a product decomposition into mutually orthogonal local vectors admits no non-trivial alternative decomposition; Riedel, arXiv:1310.4473 (2013)). Redundant consistency thus selects, from the multitude of consistent sets, a small quasiclassical subset: an 'objective past' that many observers can independently reconstruct from their own fragments and agree upon.

The paper itself left the program explicitly incomplete. Its acknowledged open issues define this problem: (i) the quantitative connection between redundant consistency and the mature redundancy measures of quantum Darwinism was deferred to a companion paper ('The objective past of a quantum universe—part 2', announced as in preparation) that never appeared; (ii) records were treated mainly in the ideal limit — pure initial environment states and non-interacting fragments — while realistic environments start mixed (Zwolak, Quan and Zurek studied mixed environments only for instantaneous states, PRL 103, 110402 (2009)) and interact, scrambling the information they carry; for a mixed global state, consistency no longer implies the existence of records at all, and only heuristic fidelity scalings were given; (iii) redundant consistency fixes the branch vectors but not the projectors: many projector choices reproduce the same branches, and these residual gauge-like freedoms were left unclassified; (iv) the treatment was restricted to homogeneous chain-form histories and pure-state derivations, with inhomogeneous, branch-dependent, and relativistic or continuum settings explicitly out of scope; (v) the environment modeling remains the illustrative CNOT/collisional-photon treatments, with realistic physical modeling deferred.

Problem Statement

Develop the quantitative quantum-Darwinian theory of records of multi-time histories — the 'objective past' program announced by Riedel, Zurek and Zwolak but never delivered — without their idealizing assumptions. The problem has three linked deliverables that jointly serve one objective: establishing when many observers, each accessing their own fragment of a decohering environment, can independently reconstruct the same classical past, and to what extent that requirement fixes the quasiclassical history set. (1) Define a redundancy measure for records of histories: a quantity, specified for a set of consistent histories and an environment fragment, that quantifies how much information about the multi-time history the fragment carries, has an explicit, checkable coarse-graining behaviour (monotone in a stated direction under coarse graining of the history set, or invariant up to a stated tolerance under refinement), and reduces, for single-time histories, to the established quantum-Darwinism redundancy measures (mutual-information plateau, or quantum-Chernoff distinguishability of conditional fragment states). (2) Prove which of the pure-state equivalences survive realistic conditions: characterize when redundant consistency implies locally accessible redundant records for initially mixed subenvironments, for interacting (information-scrambling) fragments, and for mixed global states — in particular whether any strengthened consistency condition restores the implication that fails for mixed states — and derive quantitative scaling laws for imperfect, partially redundant records of multi-time histories, at least in a definite model class (e.g. repeated-interaction spin or collisional-photon environments with mixed subenvironment states). (3) Characterize the residual gauge freedoms: classify the projector choices that leave the branch vectors of a redundantly consistent set unchanged (e.g. trivial refinements, projectors onto records versus onto pointer states, auxiliary-system decompositions of fragments into temporal subsystems), covering all such freedoms and not merely the listed examples, and determine in what sense redundant record accessibility makes the quasiclassical history set unique. All three deliverables are required for completion; within (2), each of the three named environment regimes, the strengthened-consistency question, and the scaling-law requirement must all be addressed, the scaling laws being satisfied within any definite model class. Extensions to inhomogeneous or branch-dependent histories and to relativistic or continuum settings are recorded open directions of the source but lie outside this problem's completion boundary.

References

1. C. Jess Riedel, Wojciech H. Zurek, and Michael Zwolak, "Objective past of a quantum universe: Redundant records of consistent histories", Phys. Rev. A 93, 032126 (2016). DOI: 10.1103/PhysRevA.93.032126, https://doi.org/10.1103/PhysRevA.93.032126 (arXiv:1312.0331, https://arxiv.org/abs/1312.0331)
2. R. B. Griffiths, "Consistent histories and the interpretation of quantum mechanics", J. Stat. Phys. 36, 219 (1984). DOI: 10.1007/BF01015734, https://doi.org/10.1007/BF01015734
3. Murray Gell-Mann and James B. Hartle, "Classical equations for quantum systems", Phys. Rev. D 47, 3345 (1993). DOI: 10.1103/PhysRevD.47.3345, https://doi.org/10.1103/PhysRevD.47.3345
4. F. Dowker and A. Kent, "Properties of Consistent Histories", Phys. Rev. Lett. 75, 3038 (1995). DOI: 10.1103/PhysRevLett.75.3038, https://doi.org/10.1103/PhysRevLett.75.3038
5. J. Finkelstein, "Definition of decoherence", Phys. Rev. D 47, 5430 (1993). DOI: 10.1103/PhysRevD.47.5430, https://doi.org/10.1103/PhysRevD.47.5430
6. Wojciech H. Zurek, "Quantum Darwinism", Nat. Phys. 5, 181 (2009). DOI: 10.1038/nphys1202, https://doi.org/10.1038/nphys1202
7. Harold Ollivier, David Poulin, and Wojciech H. Zurek, "Objective Properties from Subjective Quantum States: Environment as a Witness", Phys. Rev. Lett. 93, 220401 (2004). DOI: 10.1103/PhysRevLett.93.220401, https://doi.org/10.1103/PhysRevLett.93.220401
8. M. Zwolak, H. T. Quan, and W. H. Zurek, "Quantum Darwinism in a Mixed Environment", Phys. Rev. Lett. 103, 110402 (2009). DOI: 10.1103/PhysRevLett.103.110402, https://doi.org/10.1103/PhysRevLett.103.110402
9. M. Zwolak, C. J. Riedel, and W. H. Zurek, "Amplification, Redundancy, and Quantum Chernoff Information", Phys. Rev. Lett. 112, 140406 (2014). DOI: 10.1103/PhysRevLett.112.140406, https://doi.org/10.1103/PhysRevLett.112.140406 (arXiv:1312.5373, https://arxiv.org/abs/1312.5373)
10. C. Jess Riedel, "Local Records and Global Entanglement: A Unique Multi-Partite Generalization of the Schmidt Decomposition", arXiv:1310.4473 (2013), https://arxiv.org/abs/1310.4473
11. James B. Hartle, "Decoherent Histories Quantum Mechanics Starting with Records of What Happens", arXiv:1608.04145 (2016), https://arxiv.org/abs/1608.04145
12. Philipp Strasberg, "Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach", SciPost Phys. 15, 024 (2023). DOI: 10.21468/SciPostPhys.15.1.024, https://doi.org/10.21468/SciPostPhys.15.1.024 (arXiv:2301.02563, https://arxiv.org/abs/2301.02563)
13. Philipp Strasberg, Teresa E. Reinhard, and Joseph Schindler, "First Principles Numerical Demonstration of Emergent Decoherent Histories", Phys. Rev. X 14, 041027 (2024). DOI: 10.1103/PhysRevX.14.041027, https://doi.org/10.1103/PhysRevX.14.041027 (arXiv:2304.10258, https://arxiv.org/abs/2304.10258)
14. Jiaozi Wang and Philipp Strasberg, "Decoherence of Histories: Chaotic Versus Integrable Systems", Phys. Rev. Lett. 134, 220401 (2025). DOI: 10.1103/m8vq-l449, https://doi.org/10.1103/m8vq-l449 (arXiv:2406.15577, https://arxiv.org/abs/2406.15577)
15. Irene Valladares Duque and Philipp Strasberg, "Toy Model Challenging Prevailing Definitions of Classicality", EPL 152, 28001 (2025). DOI: 10.1209/0295-5075/ae1639, https://doi.org/10.1209/0295-5075/ae1639 (arXiv:2412.08444, https://arxiv.org/abs/2412.08444)
16. Akram Touil, Bin Yan, and Wojciech H. Zurek, "Consensus About Classical Reality in a Quantum Universe", arXiv:2503.14791 (2025), https://arxiv.org/abs/2503.14791
17. Adamantia Zampeli, Georgios E. Pavlou, and Petros Wallden, "Contrary Inferences for Classical Histories within the Consistent Histories Formulation of Quantum Theory", Found. Phys. 56, 3 (2026). DOI: 10.1007/s10701-025-00902-w, https://doi.org/10.1007/s10701-025-00902-w (arXiv:2205.15893, https://arxiv.org/abs/2205.15893)
18. Nataliya Arefyeva and Evgeny Polyakov, "Emergence of non-Markovian decoherent histories in an integrable environment: A 'tape recorder' model for local quantum observables", Phys. Rev. A 113, 062225 (2026). DOI: 10.1103/3ct7-y9tn, https://doi.org/10.1103/3ct7-y9tn (arXiv:2509.00845, https://arxiv.org/abs/2509.00845)
19. Benoît Ferté, Davide Farci, and Xiangyu Cao, "Decoherent histories with(out) objectivity in a (broken) apparatus", Phys. Rev. Lett. 136, 090404 (2026). DOI: 10.1103/8qzx-xpfz, https://doi.org/10.1103/8qzx-xpfz (arXiv:2508.16482, https://arxiv.org/abs/2508.16482)
20. Philipp Strasberg, Joseph Schindler, Jiaozi Wang, and Andreas Winter, "Approximate Decoherence, Recoherence and Records in Isolated Quantum Systems", arXiv:2601.19703 (2026), https://arxiv.org/abs/2601.19703
21. Nick Ormrod, Tein van der Lugt, Yìlè Yīng, and Jarosław K. Korbicz, "Decoherence without the state: A causal quantum Darwinist approach", arXiv:2605.07090 (2026), https://arxiv.org/abs/2605.07090
22. C. Jess Riedel, "Wavefunction branches demand a definition!", Quantum Views 9, 85 (2025). DOI: 10.22331/qv-2025-06-16-85, https://doi.org/10.22331/qv-2025-06-16-85 (arXiv:2506.15663, https://arxiv.org/abs/2506.15663)
23. Jordan K. Taylor and Ian P. McCulloch, "Wavefunction branching: when you can't tell pure states from mixed states", Quantum 9, 1670 (2025). DOI: 10.22331/q-2025-03-25-1670, https://doi.org/10.22331/q-2025-03-25-1670 (arXiv:2308.04494, https://arxiv.org/abs/2308.04494)