Generic characterization of SBS pointer states under non-commuting system-environment dynamics

When the system Hamiltonian $H_\mathcal{S}$ commutes with the system-environment interaction $H_\mathcal{I}$, the standard notions of pointer states (common eigenstates of $H_\mathcal{S}$ and $H_\mathcal{I}$, states left pure by the evolution, and the system basis in a spectrum-broadcast-structure (SBS) decomposition) all coincide. Chisholm, Palma and Innocenti showed, in an explicit model of a single qubit coupled to an $n$-qubit environment, that objective, redundantly recorded states can still form when $[H_\mathcal{S},H_\mathcal{I}]\neq 0$, and proposed a relaxed, SBS-based definition of pointer states that is always well defined whenever information is redundant, but whose basis depends on the initial state and must be extracted post hoc from the evolved state, and provably disagrees with the predictability-sieve (least-decohering) selection. The open problem is the generic characterization of these SBS pointer states in the non-commuting regime: their universal structural features beyond the toy model, and the definitional choice between SBS-based and predictability-sieve pointer states. No resolution of these facets is known in the literature to date.

Background

Quantum Darwinism (QD) explains how a quantum system acquires objective, classical properties: the environment fragments into many independently accessible parts, each of which redundantly records the same information about the system, so that many observers can infer the system's state without disturbing it. Objectivity is quantified by the accessible mutual information $I_{\mathrm{acc}}(\mathcal{S}:\mathcal{F}_i)$ (the classical mutual information maximized over measurements on the system and an environment fraction) approaching the system entropy $S(\rho_\mathcal{S})$, and by the redundancy $\mathrm{Red}$, the number of environment fractions each carrying (within a threshold $\delta$) full information about the system. The states that the environment thereby records are the pointer states. In the traditional framework, where the system Hamiltonian $H_\mathcal{S}$ commutes with the interaction Hamiltonian $H_\mathcal{I}$, three characterizations of pointer states are equivalent: they are the simultaneous eigenstates of $H_\mathcal{S}$ and $H_\mathcal{I}$, they are the system states left pure and unperturbed by the joint evolution, and they are the orthogonal system states $\{|\psi_i\rangle\}$ appearing in the spectrum broadcast structure (SBS) decomposition of the joint state, $\rho_{\mathrm{SBS}}=\sum_i p_i|\psi_i\rangle\langle\psi_i|\bigotimes_j R_i^j$ with conditionally distinct environment states $R_i^j R_{i'}^j=0$ for $i\neq i'$. The SBS form is the state-level expression of objectivity: strong quantum Darwinism with strong independence is equivalent to SBS. Pointer states in this commuting setting can also be found by the predictability sieve, which selects the states suffering the least environment-induced decoherence (as in the coherent states of a damped oscillator, or the general conditions of Singh, Sawicki and Korbicz for the Born-Markov regime, the standard weak-coupling, memoryless approximation of open-system dynamics).

The commuting assumption $[H_\mathcal{S},H_\mathcal{I}]=0$ is, however, restrictive and generically violated in realistic measurement-like situations. Chisholm, Palma and Innocenti studied a model in which a single system qubit is coupled to $n$ environment qubits, with $H_\mathcal{S}=\omega(p\sigma_x+(1-p)\sigma_z)$ acting on the system and a dephasing-type interaction $H_\mathcal{I}=\sum_i\gamma\sigma_z^\mathcal{S}\otimes\sigma_x^{\mathcal{E}_i}$, so that the non-commutativity is tuned by $p$ and the timescale ratio by $\omega/\gamma$. They showed that non-commutativity monotonically hinders but does not prevent objectivity: redundancy degrades with $p$, and the degrading effects are minimized when the interaction is much faster than the intrinsic system dynamics ($\gamma\gg\omega$), the regime expected for macroscopic environments. Most importantly, whenever objectivity (redundant information) is present, the relaxed definition of pointer states as the basis $\{|\psi_i\rangle\}$ of the SBS decomposition of the evolved state remains well defined and coincides with the usual notion when commutativity is restored. In the non-commuting regime this SBS-pointer basis depends not only on $H_\mathcal{S}$ and $H_\mathcal{I}$ but also on the initial state of the system, cannot be determined a priori from the Hamiltonian, and provably does not coincide with the states selected by the predictability sieve; interpreting the environment as a measuring apparatus, the apparatus effectively measures in a basis different from the intended one, with high fidelity over a wide but unquantified parameter range. Related prior results treat adjacent aspects of commutativity: Ryan, Carolan, Campbell and Paternostro showed that a thermalizing environment that does not commute with the system-accessible interaction destroys objectivity, and Doucet and Deffner, building on Duruisseau, Touil and Deffner, classified the two-body Hamiltonians that support quantum Darwinism but only under commutation relations between the Hamiltonian's system operators and a pointer observable, a condition that excludes precisely the non-commuting regime considered here.

Problem Statement

Characterize the generic properties of the SBS-pointer states that emerge under non-commuting system-environment dynamics, $[H_\mathcal{S},H_\mathcal{I}]\neq 0$, beyond the proof-of-principle qubit model of Chisholm, Palma and Innocenti. This objective comprises two linked facets; each facet can be settled independently, and a complete resolution settles both. (a) Generic properties: characterize the SBS-pointer states (the orthogonal system basis of the spectrum-broadcast-structure decomposition of the evolved joint state, whenever redundant information is present) that emerge under non-commuting dynamics, and determine whether universal structural features exist for larger Hilbert spaces, many-body systems, or continuous-variable environments, and how they depend on the system's initial state, on the timescale ratio between $H_\mathcal{S}$ and $H_\mathcal{I}$, and on environment size and initial states, where the source paper's demonstrations were limited to 6-8 qubit environments with symmetric interactions and one threshold convention for redundancy. The three environment classes named above are alternative generality axes, so establishing the behaviour for any one of them, or arguing for them generically, discharges that part. Facet (a) then separates into two independently refutable parts. Throughout, a universal structural feature means a property of the SBS-pointer basis, or of how that basis emerges from the dynamics, that is not already entailed by the spectrum-broadcast-structure form itself (such as the orthogonality of the system basis, or the conditional orthogonality of the environment states) nor by the commuting-limit coincidences recalled in the Background, and that holds across the model families in scope rather than only at a single parameter value or a single initial state. Its existence part asks whether universal structural features, in this sense, exist beyond the toy model, and a demonstration that no such universal features exist resolves it on its own. Its dependence part, owed only if such features exist, asks how the features depend on the listed parameters, and is discharged per axis whenever that axis's dependence is settled, including by a definite statement that the dependence is fixed, absent, or unconstrained. (b) Definitional question: settle which definition of pointer states is appropriate in the non-commuting regime, the SBS-based one (states whose information proliferates into the environment) or the predictability-sieve one (states suffering least decoherence), which provably disagree there, including identifying regimes or limits in which they coincide.

References

1. Diana A. Chisholm, G. Massimo Palma, 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
2. Diana A. Chisholm, G. Massimo Palma, Luca Innocenti, 'Emergence of quantum Darwinism and pointer states for noncommuting evolutions', APS Open Science 1, 000024 (2026), DOI 10.1103/75jr-ltct, https://doi.org/10.1103/75jr-ltct
3. Harold Ollivier, David Poulin, Wojciech H. Zurek, 'Objective Properties from Subjective Quantum States: Environment as a Witness', Physical Review Letters 93, 220401 (2004), DOI 10.1103/PhysRevLett.93.220401, https://doi.org/10.1103/PhysRevLett.93.220401
4. Wojciech H. Zurek, Salman Habib, Juan Pablo Paz, 'Coherent states via decoherence', Physical Review Letters 70, 1187 (1993), DOI 10.1103/PhysRevLett.70.1187, https://doi.org/10.1103/PhysRevLett.70.1187
5. J. K. Korbicz, P. Horodecki, R. Horodecki, 'Objectivity in a noisy photonic environment through quantum state information broadcasting', Physical Review Letters 112, 120402 (2014), DOI 10.1103/PhysRevLett.112.120402, https://doi.org/10.1103/PhysRevLett.112.120402
6. R. Horodecki, J. K. Korbicz, P. Horodecki, 'Quantum origins of objectivity', Physical Review A 91, 032122 (2015), DOI 10.1103/PhysRevA.91.032122, https://doi.org/10.1103/PhysRevA.91.032122
7. Thao P. Le, Alexandra Olaya-Castro, 'Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure', Physical Review Letters 122, 010403 (2019), DOI 10.1103/PhysRevLett.122.010403, https://doi.org/10.1103/PhysRevLett.122.010403
8. J. K. Korbicz, 'Roads to objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong quantum Darwinism - a review', Quantum 5, 571 (2021), DOI 10.22331/q-2021-11-08-571, https://doi.org/10.22331/q-2021-11-08-571
9. E. Ryan, E. Carolan, S. Campbell, M. Paternostro, 'Commutativity and the emergence of classical objectivity', Journal of Physics Communications 6, 095005 (2022), DOI 10.1088/2399-6528/ac8f19, https://doi.org/10.1088/2399-6528/ac8f19 (arXiv:2206.08249)
10. Piotr Mironowicz, Paweł Horodecki, Ryszard Horodecki, 'Non-Perfect Propagation of Information to a Noisy Environment with Self-Evolution', Entropy 24, 467 (2022), DOI 10.3390/e24040467, https://doi.org/10.3390/e24040467
11. Paul Duruisseau, Akram Touil, Sebastian Deffner, 'Pointer States and Quantum Darwinism with Two-Body Interactions', Entropy 25, 1573 (2023), DOI 10.3390/e25121573, https://doi.org/10.3390/e25121573
12. Emery Doucet, Sebastian Deffner, 'Classifying Two-Body Hamiltonians for Quantum Darwinism', Physical Review X 14, 041064 (2024), DOI 10.1103/PhysRevX.14.041064, https://doi.org/10.1103/PhysRevX.14.041064
13. Uttam Singh, Adam Sawicki, Jarosław K. Korbicz, 'Pointer States in the Born-Markov Approximation', Physical Review Letters 132, 030203 (2024), DOI 10.1103/PhysRevLett.132.030203, https://doi.org/10.1103/PhysRevLett.132.030203 (arXiv:2212.09790)
14. Irene Valladares Duque, Philipp Strasberg, 'Toy Model Challenging Prevailing Definitions of Classicality', arXiv:2412.08444 (2024), https://arxiv.org/abs/2412.08444
15. Anthony Kiely, Diana A. Chisholm, Akram Touil, Sebastian Deffner, Gabriel Landi, Steve Campbell, 'Metrological approach to the emergence of classical objectivity', Physical Review A 113, 022403 (2026), DOI 10.1103/hn78-7xx3, https://doi.org/10.1103/hn78-7xx3 (arXiv:2510.12313)
16. Jie Gu, 'Hamiltonian Thresholds for Objective Records', Physical Review A 114, L020201 (2026), DOI 10.1103/fm9j-m9lh, https://doi.org/10.1103/fm9j-m9lh (arXiv:2608.00365)
17. Ravo Tokiniaina Ranaivoson, Angelo Mamitiana Ralaikoto, Diary Lova Ratsimbazafy, Rivo Herivola Manjakamanana Ravelonjato, Roland Raboanary, Raoelina Andriambololona, Nomenjanahary Tanjonirina Manampisoa, Fanamby Sahondraniandriana, 'Decoherence challenges in Nanoscience: A Quantum Phase Space perspective', arXiv:2512.22297 (2025), https://arxiv.org/abs/2512.22297