HEXAGONbeta

Classical Extensions and Strongly Positive Realizations of Decoherent History Probabilities

Primarily human-written textHuman understanding: all partsquant-ph — Quantum Physicsmath-ph — Mathematical Physics

Contributed by Zhenyu Sun ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Probabilities on several decoherent families of histories may agree on shared events yet admit no classical extension to the Boolean algebra that the families generate. We characterize the strongly positive decoherence functionals that realize given local probabilities on a finite algebra. A covariance decomposition allows any signed extension to be prescribed at any attainable rank and gives the resulting increase in minimum trace. For two contexts, agreement on their intersection suffices for a strongly positive realization. On the bipartite graph of context atoms, the minimum negative mass of a signed extension is (W1−1)/2(W_1-1)/2, where W1W_1 is the transport cost for graph distance. A graph Laplacian formula gives the minimum trace. Its minimizers form an affine copy of a rectangular contraction ball whose extreme points have the smallest realization rank. For a pure-state implementation with one intermediate projective decomposition and a final postselection, the optimal success probability of a realization is the reciprocal of the least trace of a diagonal matrix dominating it. In a family of classical examples, this probability tends to zero for the unique trace minimizer even though another realization succeeds with probability one.

References

  1. Dowker, H. F.; Halliwell, J. J. Quantum Mechanics of History: The Decoherence Functional in Quantum Mechanics. Physical Review D, vol. 46, no. 4, pp. 1580–1609. 1992DOI
  2. Griffiths, Robert B. Consistent Quantum Theory. Cambridge University Press. 2002DOI
  3. Isham, C. J. Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets. International Journal of Theoretical Physics, vol. 36, pp. 785–814. 1997arXivDOI
  4. Halliwell, J. J. Incompatible Multiple Consistent Sets of Histories and Measures of Quantumness. Physical Review A, vol. 96, no. 1, pp. 012123. 2017arXivDOI
  5. Abramsky, Samson; Brandenburger, Adam. The Sheaf-Theoretic Structure of Non-Locality and Contextuality. New Journal of Physics, vol. 13, no. 11, pp. 113036. 2011arXivDOI
  6. Heindorf, Lutz; Shapiro, Leonid B. Nearly Projective Boolean Algebras. Springer, vol. 1596. 1994. With an appendix by Sakaé FuchinoDOI
  7. Milovich, David. Amalgamating Many Overlapping Boolean Algebras. 2016. arXiv:1607.07944v1arXiv
  8. Dowker, Fay; Henson, Joe; Wallden, Petros. A Histories Perspective on Characterizing Quantum Non-Locality. New Journal of Physics, vol. 16, no. 3, pp. 033033. 2014arXivDOI
  9. Craig, David; Dowker, Fay; Henson, Joe; Major, Seth; Rideout, David; Sorkin, Rafael D. A Bell Inequality Analog in Quantum Measure Theory. Journal of Physics A: Mathematical and Theoretical, vol. 40, no. 3, pp. 501–523. 2007arXivDOI
  10. Gudder, Stan. Hilbert Space Representations of Decoherence Functionals and Quantum Measures. Mathematica Slovaca, vol. 62, no. 6, pp. 1209–1230. 2012arXivDOI
Show all 32 referencesShow fewer references
  1. Dowker, Fay; Wilkes, Henry. An Argument for Strong Positivity of the Decoherence Functional in the Path Integral Approach to the Foundations of Quantum Theory. AVS Quantum Science, vol. 4, no. 1, pp. 012601. 2022arXivDOI
  2. Zhou, Li; Ying, Shenggang; Yu, Nengkun; Ying, Mingsheng. Strassen's Theorem for Quantum Couplings. Theoretical Computer Science, vol. 802, pp. 67–76. 2020DOI
  3. de Barros, J. Acacio; Dzhafarov, Ehtibar N.; Kujala, Janne V.; Oas, Gary. Measuring Observable Quantum Contextuality. Quantum Interaction, vol. 9535, pp. 36–47. 2016arXivDOI
  4. Lyons, Russell; Peres, Yuval. Probability on Trees and Networks. Cambridge University Press, vol. 42. 2016DOI
  5. Schönemann, Peter H. A Generalized Solution of the Orthogonal Procrustes Problem. Psychometrika, vol. 31, no. 1, pp. 1–10. 1966DOI
  6. Bhatia, Rajendra; Jain, Tanvi; Lim, Yongdo. On the Bures–Wasserstein Distance between Positive Definite Matrices. Expositiones Mathematicae, vol. 37, no. 2, pp. 165–191. 2019arXivDOI
  7. Gell-Mann, Murray; Hartle, James B. Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology. Physical Origins of Time Asymmetry, pp. 311–345. 1994arXiv
  8. Chefles, Anthony; Jozsa, Richard; Winter, Andreas. On the Existence of Physical Transformations between Sets of Quantum States. International Journal of Quantum Information, vol. 2, no. 1, pp. 11–21. 2004arXivDOI
  9. Napoli, Carmine; Bromley, Thomas R.; Cianciaruso, Marco; Piani, Marco; Johnston, Nathaniel; Adesso, Gerardo. Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence. Physical Review Letters, vol. 116, no. 15, pp. 150502. 2016arXivDOI
  10. Piani, Marco; Cianciaruso, Marco; Bromley, Thomas R.; Napoli, Carmine; Johnston, Nathaniel; Adesso, Gerardo. Robustness of Asymmetry and Coherence of Quantum States. Physical Review A, vol. 93, no. 4, pp. 042107. 2016arXivDOI
  11. Goldstein, Sheldon; Page, Don N. Linearly Positive Histories: Probabilities for a Robust Family of Sequences of Quantum Events. Physical Review Letters, vol. 74, no. 19, pp. 3715–3719. 1995arXivDOI
  12. Halliwell, J. J.; Yearsley, J. M. Negative Probabilities, Fine's Theorem, and Linear Positivity. Physical Review A, vol. 87, no. 2, pp. 022114. 2013DOI
  13. Kent, Adrian. Consistent Sets Yield Contrary Inferences in Quantum Theory. Physical Review Letters, vol. 78, no. 15, pp. 2874–2877. 1997arXivDOI
  14. Wallden, Petros. Contrary Inferences in Consistent Histories and a Set Selection Criterion. Foundations of Physics, vol. 44, pp. 1195–1215. 2014arXivDOI
  15. Strassen, Volker. The Existence of Probability Measures with Given Marginals. The Annals of Mathematical Statistics, vol. 36, no. 2, pp. 423–439. 1965DOI
  16. Abramsky, Samson; Barbosa, Rui Soares; Mansfield, Shane. Contextual Fraction as a Measure of Contextuality. Physical Review Letters, vol. 119, no. 5, pp. 050504. 2017DOI
  17. Robertson, Sawyer Jack; Kohli, Dhruv; Mishne, Gal; Cloninger, Alexander. On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs. SIAM Journal on Scientific Computing, vol. 47, no. 5, pp. A2774–A2800. 2025arXivDOI
  18. Childs, Andrew M.; Wiebe, Nathan. Hamiltonian Simulation Using Linear Combinations of Unitary Operations. Quantum Information and Computation, vol. 12, no. 11–12, pp. 901–924. 2012arXivDOI
  19. Leggett, A. J.; Garg, Anupam. Quantum Mechanics versus Macroscopic Realism: Is the Flux There when Nobody Looks?. Physical Review Letters, vol. 54, no. 9, pp. 857–860. 1985DOI
  20. Halliwell, J. J.; Mawby, C. Fine's Theorem for Leggett–Garg Tests with an Arbitrary Number of Measurement Times. Physical Review A, vol. 100, no. 4, pp. 042103. 2019arXivDOI
  21. Vorob'ev, N. N. Consistent Families of Measures and Their Extensions. Theory of Probability and Its Applications, vol. 7, no. 2, pp. 147–163. 1962DOI
  22. Atserias, Albert; Kolaitis, Phokion G. Consistency, Acyclicity, and Positive Semirings. Samson Abramsky on Logic and Structure in Computer Science and Beyond, vol. 25, pp. 623–668. 2023arXivDOI

Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentNov 04, 2026