Operational completion of the compatibility criterion for emergent classical objectivity: statistically rigorous Kirkwood-Dirac tests, approximate criteria with timescales, and emergence conditions for Quantum Darwinism

Doucet and Deffner (Phys. Rev. A 111, 042217 (2025); arXiv:2411.10862) proved an exact algebraic characterization: for a multipartite quantum system under unitary dynamics, the Kirkwood-Dirac quasiprobability (KDQ) distributions of all collections of measurements on disjoint subsystems are classical for every initial state and measurement schedule if and only if the Hamiltonian generates no effective interactions between those subsystems, a condition expressed by the vanishing of an infinite hierarchy of nested commutators, and this is exactly the Hamiltonian structure required to support Quantum Darwinism. The theorem is exact but not yet an operational theory; the paper itself flags the first two of the following three linked issues as open and notes the third without carrying it out, which together define this problem: (1) experimentally feasible, statistically robust protocols or derived observable tests that detect KDQ non-classicality between environment fragments with quantifiable confidence scaling — an operational falsifier of Quantum Darwinism, given that full KDQ reconstruction is as demanding as state tomography; (2) approximate or perturbative versions of the compatibility criterion, characterizing the timescales over which small-but-nonzero effective interactions produce detectable KDQ non-classicality; and (3) integration of the mild additional restrictions on the Hamiltonian and initial system-environment state that the paper notes are needed for unidirectional information transfer and the mutual-information plateau with the algebraic commutator criteria — an integration the paper mentions but does not carry out — toward operational conditions for the emergence (not merely the impossibility) of objectivity. A 2026 follow-up by the same authors demonstrated KDQ measurement on noisy intermediate-scale quantum (NISQ) hardware but found device noise dominates and left statistical certification open.

Background

When a quantum system is monitored by its environment, decoherence singles out preferred 'pointer' states that are robust against monitoring, and information about which pointer state the system occupies spreads into many disjoint fragments of the environment (Zurek, Rev. Mod. Phys. 75, 715 (2003)). Quantum Darwinism (Ollivier, Poulin, and Zurek, Phys. Rev. Lett. 93, 220401 (2004)) proposes that classical objectivity — many observers independently agreeing on a system's properties — arises when this information is encoded redundantly: each observer measures a different environment fragment and reconstructs the same pointer value, visible as a plateau of the mutual information between the system and environment fragments. The states supporting objectivity have a 'spectrum broadcast structure' (Horodecki, Korbicz, and Horodecki, Phys. Rev. A 91, 032122 (2015)): the system is classically correlated with each fragment and the fragments are uncorrelated with one another, a form shown equivalent to 'singly-branching' states (Touil, Anza, Deffner, and Crutchfield, Quantum 8, 1494 (2024); comparisons in Le and Olaya-Castro, Phys. Rev. A 98, 032103 (2018)).

A separate thread concerns joint measurement statistics. In a classical system, any set of measurements by different observers can be captured by a single joint probability distribution. In quantum mechanics, measurements may be incompatible (noncommuting), and the natural object describing the sequential statistics of two observables with outcome projectors $A_i$ and $B_j$ on a state $\rho$ is the Kirkwood-Dirac quasiprobability (KDQ) distribution, $q_{ij}=\mathrm{tr}[B_j A_i\rho]$. A two-point measurement (TPM) protocol, in which each projector is measured in sequence, instead yields $p_{ij}=\mathrm{tr}[B_j A_i\rho A_i]$; the KDQ decomposes into this TPM probability plus quantum modification terms that vanish exactly when the relevant operators commute. The KDQ is a genuine joint probability precisely when all its entries are real and non-negative; negative or imaginary entries are standard witnesses of non-classicality and contextuality (reviewed in Arvidsson-Shukur et al., New J. Phys. 26, 121201 (2024)). A key subtlety is that noncommutation is necessary but not sufficient for KDQ non-classicality (Arvidsson-Shukur, Drori, and Yunger Halpern, J. Phys. A 54, 284001 (2021)).

Doucet and Deffner (Phys. Rev. A 111, 042217 (2025)) connected the two threads. Consider a multipartite quantum system evolving under a time-independent Hamiltonian, with several observers who each measure, at freely chosen times, arbitrary observables on their own disjoint collections of subsystems. They proved that the KDQ distributions of all such collections of measurements are classical — they coincide with TPM joint probabilities — for all initial states and measurement schedules, if and only if the Hamiltonian generates no effective interactions between the observed subsystems. Concretely, decomposing the interaction Hamiltonian terms between each observed region and the inaccessible remainder as sums of $V^{\mathfrak{A}}_a\otimes S^{\mathfrak{A}}_a$ and $V^{\mathfrak{B}}_b\otimes S^{\mathfrak{B}}_b$ (the $S$ operators acting on the shared inaccessible region), the condition is that every nested commutator of an $S^{\mathfrak{A}}_a$ with an $S^{\mathfrak{B}}_b$, with arbitrary interleaved powers of the Hamiltonian on the inaccessible region, vanishes. Strikingly, this coincides with the Hamiltonian structure previously found to be required for a model to support Quantum Darwinism in the two-body classification program (Duruisseau, Touil, and Deffner, Entropy 25, 1573 (2023); Doucet and Deffner, Phys. Rev. X 14, 041064 (2024)): environment fragments accessed by different observers must not interact, even effectively through higher-order processes. A single non-classical quasiprobability between two fragments therefore certifies that the model cannot exhibit Quantum Darwinism, suggesting a falsification test that avoids the discord-optimization procedures used in earlier objectivity witnesses.

The theorem, however, is exact and purely algebraic, and the paper itself flags what is missing to make it an operational theory. First, complete KDQ reconstruction is as demanding as reduced-state tomography, and the details of a practical hypothesis test — which quasiprobabilities or derived quantities to measure, and how confidence scales with the number of tests — are left unspecified. Second, the criterion is all-or-nothing: any nonzero effective interaction eventually produces KDQ non-classicality, but if the interaction is small the manifestation can be delayed to long times, and no approximate or perturbative version of the criterion is given. Third, Quantum Darwinism requires, in addition to the Hamiltonian structure, mild restrictions on initial states and on unidirectional (system-to-environment) information transfer for the mutual-information plateau to form, and the paper states that the Hamiltonian structures coincide without carrying out this integration.

Problem Statement

The problem is the operational and quantitative completion of the Doucet-Deffner characterization: turn the exact algebraic equivalence — 'KDQ-classicality of all disjoint-measurement statistics for all states and schedules, no effective Hamiltonian interactions between the observed subsystems, and Quantum-Darwinism-supporting Hamiltonian structure' — into a testable and approximate theory, along the three directions the source paper identifies as unresolved or leaves incomplete.

1. Statistical detection. Construct experimentally feasible protocols — sampling a few quasiprobabilities, derived observable witnesses, or moment-based tests, such as those implementable through interferometric or cycle-test circuits — together with a rigorous decision procedure that detects KDQ non-classicality between two or more environment fragments with quantified confidence: error rates and sample complexity as functions of the model parameters and of the chosen measurement settings, including how the conclusion aggregates over multiple measurement settings and times. Since a single non-classical quasiprobability certifies that the model cannot exhibit Quantum Darwinism, such a protocol is an operational falsifier of Quantum Darwinism; conversely, a precise statement is required of what can and cannot be concluded from finitely many settings that show no non-classicality.

2. Approximate criteria and timescales. Define a quantitative notion of approximate classical compatibility — for example, bounds on the norms of the nested commutators entering the exact criterion, or on the total non-classicality of the KDQ distributions at a given time — and characterize how the magnitude of residual effective interactions controls the time and strength at which KDQ non-classicality becomes detectable. This includes perturbative criteria that reduce to the exact nested-commutator conditions in the limit of vanishing interactions, and quantitative bounds or estimates on the detection timescale in terms of the commutator norms and inter-measurement delays.

3. Emergence conditions. Combine the Hamiltonian commutator criteria with the state-level restrictions (formation of singly-branching / spectrum-broadcast states from a specified class of initial states) and the requirement of unidirectional information transfer — the 'mild additional restrictions on the Hamiltonian and initial system-environment state' that the source paper notes are needed so that the mutual information between the system and environment fragments eventually develops a classical plateau — so as to give operational conditions for the emergence (not merely the impossibility) of classical objectivity: conditions under which the mutual-information plateau forms, and how its formation relates to the commutator criteria. The source paper states that the Hamiltonian structures coincide without carrying out this integration.

The three facets jointly constitute the operational completion of one theorem, and they are scientifically linked: the statistical protocol of facet 1 is meaningful precisely because of the exact equivalence, the approximate criteria of facet 2 quantify how the protocol's detection horizon scales with residual interactions, and the emergence conditions of facet 3 delimit what a negative test outcome can imply. An answer may resolve one or more facets, provided it matches the rigor and the generality of the source theorem (generic finite-dimensional multipartite systems, arbitrary measurements on disjoint subsystems, arbitrary initial states and measurement schedules, or explicitly delimited weaker generality) and states precisely which facets it addresses.

References

1. Emery Doucet and Sebastian Deffner, 'Compatibility of quantum measurements and the emergence of classical objectivity', Phys. Rev. A 111, 042217 (2025), DOI 10.1103/PhysRevA.111.042217, arXiv:2411.10862, https://arxiv.org/abs/2411.10862 (source paper)
2. Emery Doucet and Sebastian Deffner, 'Classifying Two-Body Hamiltonians for Quantum Darwinism', Phys. Rev. X 14, 041064 (2024), DOI 10.1103/PhysRevX.14.041064, arXiv:2405.00805, https://arxiv.org/abs/2405.00805
3. Paul Duruisseau, Akram Touil, and Sebastian Deffner, 'Pointer States and Quantum Darwinism with Two-Body Interactions', Entropy 25, 1573 (2023), DOI 10.3390/e25121573, arXiv:2309.03299, https://arxiv.org/abs/2309.03299
4. Akram Touil, Fabio Anza, Sebastian Deffner, and James P. Crutchfield, 'Branching States as The Emergent Structure of a Quantum Universe', Quantum 8, 1494 (2024), DOI 10.22331/q-2024-10-10-1494, arXiv:2208.05497, https://arxiv.org/abs/2208.05497
5. Emery Doucet and Sebastian Deffner, 'From compatibility of measurements to exploring Quantum Darwinism on NISQ', Quantum Sci. Technol. 11, 035008 (2026), DOI 10.1088/2058-9565/ae753f, arXiv:2601.05350, https://arxiv.org/abs/2601.05350
6. Anthony Kiely, Diana A. Chisholm, Akram Touil, Sebastian Deffner, Gabriel Landi, and Steve Campbell, 'Metrological approach to the emergence of classical objectivity', Phys. Rev. A 113, 022403 (2026), DOI 10.1103/hn78-7xx3, arXiv:2510.12313, https://arxiv.org/abs/2510.12313
7. Shou-I Tang, Emery Doucet, Akram Touil, Sebastian Deffner, and Akira Sone, 'Information processing in quantum thermodynamic systems: An autonomous Hamiltonian approach', Phys. Rev. E 114, 024144 (2026), DOI 10.1103/rwkz-dc5z, arXiv:2511.08858, https://arxiv.org/abs/2511.08858
8. Alessandro Coppo, Nicola Pranzini, and Paola Verrucchi, 'Quantum model for black holes and clocks', arXiv:2601.07437 (2026), https://arxiv.org/abs/2601.07437
9. Abdelkader El Makouri, Abdallah Slaoui, and Rachid Ahl Laamara, 'Fluctuations in measurement-based quantum engines and quantum batteries derived from the Kirkwood-Dirac quasiprobability distribution', APS Open Science 1, 000020 (2026), DOI 10.1103/fls6-mxgq, https://doi.org/10.1103/fls6-mxgq
10. Sudip Chakrabarty, 'Quantifying Margenau-Hill Nonclassicality', arXiv:2609.07869 (2026), https://arxiv.org/abs/2609.07869
11. Zitian Zhu, Kiera Salice, Akram Touil, Zehang Bao, Zixuan Song, Pengfei Zhang, Hekang Li, Zhen Wang, Chao Song, Qiujiang Guo, H. Wang, and Rubem Mondaini, 'Observation of Quantum Darwinism and the Origin of Classicality with Superconducting Circuits', arXiv:2504.00781 (2025), https://arxiv.org/abs/2504.00781
12. Jędrzej Burkat and Sergii Strelchuk, 'Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions', arXiv:2502.11784 (2025), https://arxiv.org/abs/2502.11784
13. Lijun Liu and Shuming Cheng, 'The boundary of Kirkwood-Dirac quasiprobability', arXiv:2504.09238 (2025), https://arxiv.org/abs/2504.09238
14. Sagar Silva Pratapsi, Sebastian Deffner, and Stefano Gherardini, 'Quantum speed limit for Kirkwood-Dirac quasiprobabilities', Quantum Sci. Technol. 10, 035019 (2025), DOI 10.1088/2058-9565/add55d, https://doi.org/10.1088/2058-9565/add55d, arXiv:2402.07582, https://arxiv.org/abs/2402.07582
15. Diana A. Chisholm, G. Massimo Palma, and Luca Innocenti, 'On the emergence of quantum Darwinism and pointer states for non-commuting evolutions', arXiv:2510.06867 (2025), https://arxiv.org/abs/2510.06867
16. Jie Gu, 'Hamiltonian Thresholds for Objective Records', Phys. Rev. A 114, L020201 (2026), DOI 10.1103/fm9j-m9lh, https://doi.org/10.1103/fm9j-m9lh, arXiv:2608.00365, https://arxiv.org/abs/2608.00365
17. Rafael Wagner, Zohar Schwartzman-Nowik, Ismael L. Paiva, Amit Te'eni, Antonio Ruiz-Molero, Rui Soares Barbosa, Eliahu Cohen, and Ernesto F. Galvão, 'Quantum circuits for measuring weak values, Kirkwood-Dirac quasiprobability distributions, and state spectra', Quantum Sci. Technol. 9, 015030 (2024), DOI 10.1088/2058-9565/ad124c, arXiv:2302.00705, https://arxiv.org/abs/2302.00705
18. Wojciech Hubert Zurek, 'Decoherence, einselection, and the quantum origins of the classical', Rev. Mod. Phys. 75, 715 (2003), DOI 10.1103/RevModPhys.75.715, https://doi.org/10.1103/RevModPhys.75.715
19. 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
20. R. Horodecki, J. K. Korbicz, and P. Horodecki, 'Quantum origins of objectivity', Phys. Rev. A 91, 032122 (2015), DOI 10.1103/PhysRevA.91.032122, https://doi.org/10.1103/PhysRevA.91.032122
21. Thao P. Le and Alexandra Olaya-Castro, 'Objectivity (or lack thereof): Comparison between predictions of quantum Darwinism and spectrum broadcast structure', Phys. Rev. A 98, 032103 (2018), DOI 10.1103/PhysRevA.98.032103, https://doi.org/10.1103/PhysRevA.98.032103
22. D. R. M. Arvidsson-Shukur, W. F. Braasch Jr, S. De Bièvre, J. Dressel, A. N. Jordan, C. Langrenez, M. Lostaglio, J. S. Lundeen, and N. Y. Halpern, 'Properties and applications of the Kirkwood-Dirac distribution', New J. Phys. 26, 121201 (2024), DOI 10.1088/1367-2630/ada05d, https://doi.org/10.1088/1367-2630/ada05d
23. D. R. M. Arvidsson-Shukur, Jacob Chevalier Drori, and Nicole Yunger Halpern, 'Conditions tighter than noncommutation needed for nonclassicality', J. Phys. A: Math. Theor. 54, 284001 (2021), DOI 10.1088/1751-8121/ac0289, https://doi.org/10.1088/1751-8121/ac0289