The Aharonov-Bergmann-Lebowitz (ABL) rule of 1964 gives the probability for the outcome of an intermediate projective measurement on a quantum system conditioned on both an initial pre-selection and a final post-selection. Barandes (2026) argues from the textbook axioms that post-selection is an external culling operation on an ensemble rather than a boundary condition internal to any quantum system, that time evolution being first order cannot accommodate a final quantum state as an additional boundary condition, and that ABL-derived frequencies are therefore properties of post-selected ensembles, not of individual systems. The paper leaves three linked gaps open, none of them resolved in the literature to date: (1) whether retrocausal or Everettian frameworks, in which post-selection could act as an internal future boundary condition or as branch selection, can restore single-system readings of post-selected statistics - an escape route the paper explicitly declines to adjudicate; (2) a classification, in the form of necessary and sufficient conditions, of which post-selected ensemble observables admit rigorous derivations linking them to single-system observables under additional assumptions; (3) validated statistical protocols and hypothesis tests by which journals could certify a post-selection-based result as a genuine quantum effect rather than a selection artifact. The problem is to give a rigorous and operational account of when statistics from pre- and post-selected ensembles may be elevated to claims about individual quantum systems.

Background

The Aharonov-Bergmann-Lebowitz (ABL) rule (Aharonov, Bergmann and Lebowitz 1964) computes the probability of an outcome of a projective measurement performed at an intermediate time on a quantum system that has been pre-selected (prepared in a known initial state ∣Ψ⟩|\Psi\rangle at an earlier time) and will be post-selected (later found in a state ∣ϕf⟩|\phi_f\rangle). For a complete intermediate observable with projectors PbP_b, the rule assigns PABL(b∣Ψ,ϕf)=∣⟨ϕf∣Pb∣Ψ⟩∣2∑b′∣⟨ϕf∣Pb′∣Ψ⟩∣2.P_{\mathrm{ABL}}(b \mid \Psi, \phi_f) = \frac{|\langle \phi_f | P_b | \Psi \rangle|^2}{\sum_{b'} |\langle \phi_f | P_{b'} | \Psi \rangle|^2}. Pre- and post-selection together define a pre- and post-selected (PPS) ensemble: the subensemble of experimental runs, culled after the fact, whose final measurement yielded the post-selected outcome. The two-state vector formalism developed by Aharonov and collaborators treats a system between pre- and post-selection as described jointly by a forward-evolving and a backward-evolving quantum state; on this reading the final state functions as a genuine future boundary condition. Weak values - complex numbers of the form ⟨ϕf∣A∣Ψ⟩/⟨ϕf∣Ψ⟩\langle \phi_f | A | \Psi \rangle / \langle \phi_f | \Psi \rangle that arise from weakly coupled meters followed by post-selection - were introduced in this program. A long controversy surrounds the interpretation of PPS statistics: Albert, Aharonov and D'Amato (1985) argued that ABL-type statistics yield predictions in tension with the uncertainty principle for individual systems, while Bub and Brown (1986), Sharp and Shanks (1993), and Kastner (1999) argued that post-selected subensembles for different intervening measurements are simply different ensembles, so no counterfactual conclusions about individual systems follow; Vaidman and collaborators defend single-system readings and 'weak-measurement elements of reality' (Aharonov and Vaidman 1990; Vaidman 1996). The dispute remains unresolved (see Hance, Rarity and Ladyman 2023 for a recent statement of the ensemble-to-single-system inference problem in the weak-value setting). Barandes (2026) renews and sharpens the critique. Working from the textbook Dirac-von Neumann axioms, in which time evolution is first order in time and cannot accommodate a final quantum state as an additional boundary condition without violating the evolution rules, the paper argues that post-selection is an operation performed by an external agent that abruptly changes each system's evolution: it is not a boundary condition internal to any system or ensemble, so no feature of a system's state before the post-selection can physically depend on the later post-selected state - any such dependence would require 'clairvoyance' on the part of the system. The paper names two fallacies: the ensemble fallacy (identifying a single-system observable with an observable that emerges only for an ensemble, or drawing inferences between them 'without independent rigorous justification') and the post-selection fallacy (drawing erroneous conclusions from implicit or explicit post-selection, a form of the selection bias long studied in statistics and epidemiology; cf. Hernan, Hernandez-Diaz and Robins 2004). A footnote concedes that on certain retrocausal interpretations, or on the Everett (many-worlds) interpretation - where post-selection merely places the observer on a branch of the universal wave function with its own branch-history - such clairvoyance could be permissible, but these views are 'interpretation-dependent', and the paper does not adjudicate them. The question of whether a future projection can be an internal boundary condition also has a technical analogue in final-state-projection models of black-hole horizons and post-selected teleportation (Horowitz and Maldacena 2004; Lloyd et al. 2011), where a future boundary condition is implemented within the dynamics itself. The paper ends with a call for research journals to insist that authors show post-selection-based results are produced by quantum theory itself rather than by selection bias (citing Wharton and Price 2025 as existing self-corrective introspection), but supplies no statistical protocol for doing so.

Problem Statement

The problem is to establish rigorous, operational conditions under which statistics obtained from pre- and post-selected quantum ensembles may be elevated to claims about individual quantum systems - the open core left by Barandes (2026)'s critique of the ABL rule. The source paper's definition of the ensemble fallacy treats ensemble-to-single-system inference as fallacious only when drawn 'without independent rigorous justification', which leaves open what such justifications could be; the paper itself declines to adjudicate the interpretational escape route and attaches no completion rule to these questions. The open problem comprises the following three linked questions, all of which are required to resolve it (no single question is optional or substitutable for another). A resolution must give each question its own rigorous, operational content: for the first, precise definitions and a criterion validated on at least the canonical scenarios listed below; for the second, a stated verdict together with its supporting account; and for the third, at least one concrete, falsifiable protocol.

  • Classification of elevable observables. Give precise definitions of 'single-system observable' and 'ensemble observable' for PPS experiments, and establish rigorous conditions under which a post-selected ensemble observable admits a derivation linking it to a single-system observable under additional assumptions. The classification must be stated for arbitrary PPS ensemble observables and must be non-vacuous, with the canonical PPS scenarios of the literature - ABL conditional probabilities, the three-box paradox (a pre- and post-selected scenario in which the particle is found with certainty in box A if box A is opened and with certainty in box B if box B is opened), the certainty claims of Albert-Aharonov-D'Amato type, weak values of projectors and of general operators, and quantum Cheshire Cat property separations (pre- and post-selected protocols that claim a property of a particle, such as its polarization, is located away from the particle itself) - required as checked instances.
  • Interpretational status of post-selection as a boundary condition. Determine whether, within a retrocausal or Everettian framework in which post-selection could act as an internal future boundary condition or as branch selection, features of a system's past can legitimately depend on the future post-selected state - the escape route the source paper explicitly declines to adjudicate. The answer must cover the laboratory PPS ensembles in which the paper's first-order time-evolution argument applies. As noted in Background, final-state-projection models show that the idea of an internal future boundary condition is technically non-vacuous, but no account currently exists for laboratory PPS ensembles.
  • Statistical certification of post-selection-based claims. The paper's call for research journals to insist that authors show post-selection-based results are 'produced by quantum theory itself' rather than 'merely manifestations of selection bias' comes without any statistical protocol; whether and how selection-bias and collider-bias methods can distinguish genuine quantum effects from post-selection artifacts in PPS experiments remains open.

References

  1. Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, 'Time Symmetry in the Quantum Process of Measurement', Physical Review 134, B1410-B1416 (1964), doi:10.1103/PhysRev.134.B1410, https://doi.org/10.1103/PhysRev.134.B1410
  2. D. Z. Albert, Y. Aharonov, and S. D'Amato, 'Curious New Statistical Prediction of Quantum Mechanics', Physical Review Letters 54, 5-7 (1985), doi:10.1103/PhysRevLett.54.5, https://doi.org/10.1103/PhysRevLett.54.5
  3. J. Bub and H. Brown, 'Curious Properties of Quantum Ensembles Which Have Been Both Preselected and Post-Selected', Physical Review Letters 56, 2337-2340 (1986), doi:10.1103/PhysRevLett.56.2337, https://doi.org/10.1103/PhysRevLett.56.2337
  4. Y. Aharonov and L. Vaidman, 'Properties of a quantum system during the time interval between two measurements', Physical Review A 41, 11-20 (1990), doi:10.1103/PhysRevA.41.11, https://doi.org/10.1103/PhysRevA.41.11
  5. L. Vaidman, 'Weak-Measurement Elements of Reality', Foundations of Physics 26, 895-906 (1996), doi:10.1007/BF02148832, https://doi.org/10.1007/BF02148832
  6. W. D. Sharp and N. Shanks, 'The Rise and Fall of Time-Symmetrized Quantum Mechanics', Philosophy of Science 60, 488-499 (1993), doi:10.1086/289749, https://doi.org/10.1086/289749
  7. R. E. Kastner, 'The Three-Box "Paradox" and Other Reasons to Reject the Counterfactual Usage of the ABL Rule', Foundations of Physics 29, 851-863 (1999), doi:10.1023/A:1018865010735, https://doi.org/10.1023/A:1018865010735
  8. M. A. Hernán, S. Hernández-Díaz, and J. M. Robins, 'A Structural Approach to Selection Bias', Epidemiology 15, 615-625 (2004), doi:10.1097/01.ede.0000135174.63482.43, https://doi.org/10.1097/01.ede.0000135174.63482.43
  9. G. T. Horowitz and J. Maldacena, 'The black hole final state', Journal of High Energy Physics 02, 008 (2004), doi:10.1088/1126-6708/2004/02/008, https://arxiv.org/abs/hep-th/0310281
  10. S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti, and Y. Shikano, 'Quantum mechanics of time travel through post-selected teleportation', Physical Review D 84, 025007 (2011), doi:10.1103/PhysRevD.84.025007, https://arxiv.org/abs/1007.2615
  11. Jonte R. Hance, John Rarity, and James Ladyman, 'Weak values and the past of a quantum particle', Physical Review Research 5, 023048 (2023), doi:10.1103/PhysRevResearch.5.023048, https://arxiv.org/abs/2109.14060
  12. Ken Wharton and Huw Price, 'Bell Inequality Violations Without Entanglement? It's Just Postselection', arXiv:2508.13431 (2025), doi:10.48550/arXiv.2508.13431, https://arxiv.org/abs/2508.13431
  13. Jacob A. Barandes, 'The ABL Rule and the Perils of Post-Selection', arXiv:2602.07402 (2026), doi:10.48550/arXiv.2602.07402, https://arxiv.org/abs/2602.07402
  14. Jacob A. Barandes, 'The Trouble with Weak Values', arXiv:2602.09380 (2026), doi:10.48550/arXiv.2602.09380, https://arxiv.org/abs/2602.09380
  15. Huw Price, 'Retrieving the Baby: Reichenbach's Principle, Bell Locality, and Selection Bias', arXiv:2602.16985 (2026), doi:10.48550/arXiv.2602.16985, https://arxiv.org/abs/2602.16985
  16. Huw Price, 'Bell Correlations and Selection Bias', arXiv:2605.00406 (2026), doi:10.48550/arXiv.2605.00406, https://arxiv.org/abs/2605.00406
  17. Ouissal Moumou, 'Non-Commutation Chains in Pre- and Post-Selection Paradoxes', arXiv:2509.16258 (2025), doi:10.48550/arXiv.2509.16258, https://arxiv.org/abs/2509.16258
  18. Emily Adlam and Jacob A. Barandes, 'Against Many Worlds', arXiv:2607.17086 (2026), doi:10.48550/arXiv.2607.17086, https://arxiv.org/abs/2607.17086
  19. O. Cohen, 'Pre- and postselected quantum systems, counterfactual measurements, and consistent histories', Physical Review A 51, 4373-4380 (1995), doi:10.1103/PhysRevA.51.4373, https://doi.org/10.1103/PhysRevA.51.4373
  20. Jacob A. Barandes, 'Pilot-Wave Theories as Hidden Markov Models', arXiv:2602.10569 (2026), doi:10.48550/arXiv.2602.10569, https://arxiv.org/abs/2602.10569