Relaxed objectivization in noisy, self-evolving environments: stability, retrievability, and determination of the generalized pointer (indicator) basis

When a quantum system broadcasts information into an environment that is imperfectly coupled to it, initially mixed (noisy), and subject to its own self-evolution, a spectrum-broadcast-like objectivity can still emerge - but with respect to a generalized pointer or indicator basis that is not selected by the interaction Hamiltonian, and with objective statistics that disclose dynamically generated parameters rather than parameters of the system's initial state, a phenomenon called relaxed objectivization. Mironowicz, P. Horodecki, and R. Horodecki (Entropy 24, 467 (2022)), who identified this phenomenon in small qubit models at the gate time $t=1$ and found the counterintuitive result that stronger environment self-dynamics or added initial mixedness can improve the quality of the formed broadcast structure, posed five interlinked open questions: whether relaxed objectivization can be stable in time and survive scaling to many independent environment fragments; whether information about the initial system state can still be retrieved from environment fragments under these unfriendly circumstances; whether the indicator basis can be exploited to read out parameters of the whole dynamics and determined from partial rather than full knowledge of the global dynamics; how the construction extends to strong quantum Darwinism and correlated macrofractions of the environment, where numerics become intractable; and what physical mechanism explains the observed non-monotonicities.

Background

Quantum Darwinism (QD) explains the emergence of objective classical facts by the redundant proliferation of information about a system $S$ into many fragments of its environment $E$, so that independent observers can each infer the state of $S$ by measuring a small fragment without disturbing it. The strongest dynamical form of this idea is the spectrum broadcast structure (SBS): a joint state $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i| \otimes \rho_i^{E_1} \otimes \cdots \otimes \rho_i^{E_N}$ in which the conditional states of each environmental fragment are mutually perfectly distinguishable ($\rho_i^{E_k}\rho_j^{E_k}=0$ for $i \neq j$), so that every fragment alone carries full information about the index $i$. Strong Quantum Darwinism (SQD), together with the strong-independence condition, was proven equivalent to SBS for fine-grained fragments (T. P. Le and A. Olaya-Castro, 2019); the source paper additionally describes a generalized SQD form in which local isometries acting on the environment parts may strip away correlations irrelevant to objectivity - a form expected when the fragments themselves interact. In the standard paradigm, the basis $\{|\psi_i\rangle\}$ - the pointer basis - is selected locally by the system-environment interaction Hamiltonian through einselection (the environment-induced suppression of superpositions between states that stably imprint records).

Mironowicz, P. Horodecki, and R. Horodecki (Entropy 24, 467 (2022)) asked what survives of this picture when its idealized assumptions fail. They studied a qubit system broadcasting information through an imperfect controlled-NOT gate (C-INOT, with gate-imperfection angle $\theta$) into environment qubits that (i) start in a noisy state $\varrho = p|0\rangle\langle 0| + (1-p)|1\rangle\langle 1|$ with $p\in[0,0.5]$ (mixedness $p$: pure at $p=0$ and maximally mixed at $p=0.5$; for vanishing system self-evolution and inter-environment coupling, $\varrho$ is a thermal state with $p = e^{-\alpha_2/\beta}/(e^{-\alpha_2/\beta}+e^{\alpha_2/\beta})$ at inverse temperature $\beta$, i.e. thermal noise), and (ii) self-evolve under $H_2 = \alpha_2\sum_i \sigma_Z^{(i)}$ (an external magnetic field acting on the environment) or, in one variant, interact with each other under $H_3 = \alpha_3\sum_i \sigma_Z^{(i)}\otimes\sigma_Z^{(i+1)}$. Because these Hamiltonians do not commute with the gate Hamiltonian, the pointer basis can no longer be read off from the interaction Hamiltonian; the basis that optimizes the distance of the evolved state to the nearest SBS state - which the authors call the generalized pointer or indicator basis - must be extracted from the full evolved state. Solving a three-qubit model analytically and bounding the SBS distance for an eight-qubit environment (via an algebraic upper bound on the distance to the SBS set), they found, at the gate time $t=1$ (the time at which the central interaction has fully occurred), that the quality of the formed SBS is non-monotonic both in the self-evolution strength $\alpha_2$ - a stronger magnetic field can turn a very vague measurement into a nearly ideal one - and in the initial mixedness $p$ - added noise can improve an otherwise poor objectivity. They interpret this as relaxed objectivization: the emergent objective statistics disclose new parameters generated by the dynamics rather than parameters of the initial system state. Both effects contradict standard intuition: for pure-decohering interactions in the good-decoherence regime, Zwolak, Quan, and Zurek showed that mixed (hazy) environments have a per-fragment storage capacity suppressed by their initial entropy, and non-commuting dynamics is generally expected to preclude stable pointer states.

Problem Statement

The problem is to determine whether relaxed objectivization - the SBS-like objectivity, with respect to a generalized pointer (indicator) basis, that emerges when an imperfect system-environment broadcast interaction competes with environment self-evolution and initial environmental mixedness - constitutes genuine, robust objectivity rather than a transient artifact of small environments, and to characterize the basis and the mechanisms that produce it.

Five linked sub-questions, posed in the conclusions of Mironowicz, P. Horodecki, and R. Horodecki (Entropy 24, 467 (2022)), frame this open problem:

1. Stability and scaling. Can the relaxed-objectivization effects observed at the gate time $t=1$ in environments of one observed plus one unobserved qubit (and eight-qubit environments accessed through bounds) persist stably in time and for a large number of environment fragments, as standard quantum Darwinism provides time-stable redundant records?

2. Retrieval of initial-state parameters. Despite the 'unfriendly circumstances' - environment dynamics, noise, and deviation from the ideal controlled-NOT interaction - can information about some parameters of the initial state of the system still be easily retrieved from the environment, and in which regimes?

3. The objectivized basis. Can the present objectivized (generalized pointer/indicator) basis be exploited to read out some well-defined parameters of the whole dynamics? And does one need to know all of the global dynamics to determine it, or are there cases in which knowledge of some parameters of the global dynamics, possibly combined with particular symmetries, is enough?

4. Strong Quantum Darwinism and macrofractions. How can generalized pointer bases and SBS/SQD structures be searched for when the parts of the environment interact and carry correlations - including macrofractions (large environmental fragments treated as single units) - where numerical optimization becomes intractable? The target SQD form is a state that local isometries on the environment parts transform into an SBS state after discarding objectivity-irrelevant correlated degrees of freedom, and new analytical methods are expected to be needed.

5. Mechanism of the non-monotonicity. What physical mechanism explains why and when increased speed of the environment's self-dynamics, or added initial mixedness of its states, improves the quality of the formed broadcast structure, and does this behavior generalize beyond the Hamiltonian families and gate imperfections studied by Mironowicz, P. Horodecki, and R. Horodecki?

These five sub-questions are the open directions the authors state in the paper's conclusions, "concerning the possibility of the emergence of objectivity close to the original quantum Darwinism paradigm, yet more relaxed, in some physical scenarios".

Each of these five sub-questions is independently resoluble: addressing any one of them is a self-contained research result, and the completion of the program does not require resolving all five together.

References

1. Piotr Mironowicz, Paweł Horodecki, Ryszard Horodecki. Non-perfect propagation of information to noisy environment with self-evolution. Entropy 24(4), 467 (2022). DOI: 10.3390/e24040467, arXiv:2201.11606. https://arxiv.org/abs/2201.11606
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, arXiv:2510.06867 (preprint title: On the emergence of quantum Darwinism and pointer states for non-commuting evolutions). https://arxiv.org/abs/2510.06867
3. Sophie Engineer, Tom Rivlin, Sabine Wollmann, Mehul Malik, Maximilian P. E. Lock. Equilibration of objective observables in a dynamical model of quantum measurements. Physical Review A 113, 032205 (2026). DOI: 10.1103/lkqb-jdmg (APS short DOI), arXiv:2403.18016. https://arxiv.org/abs/2403.18016
4. Irene Valladares Duque, Philipp Strasberg. Toy model challenging prevailing definitions of classicality. Europhysics Letters 152, 28001 (2025). DOI: 10.1209/0295-5075/ae1639, arXiv:2412.08444. https://arxiv.org/abs/2412.08444
5. Emery Doucet, Sebastian Deffner. Classifying two-body Hamiltonians for Quantum Darwinism. Physical Review X 14, 041064 (2024). DOI: 10.1103/PhysRevX.14.041064, arXiv:2405.00805. https://arxiv.org/abs/2405.00805
6. Michael Zwolak, H. T. Quan, Wojciech H. Zurek. Quantum Darwinism in a mixed environment. Physical Review Letters 103, 110402 (2009). DOI: 10.1103/PhysRevLett.103.110402, arXiv:0904.0418 (preprint title: Quantum Darwinism in a hazy environment). https://arxiv.org/abs/0904.0418
7. Michael Zwolak, H. T. Quan, Wojciech H. Zurek. Redundant imprinting of information in nonideal environments: objective reality via a noisy channel. Physical Review A 81, 062110 (2010). DOI: 10.1103/PhysRevA.81.062110, arXiv:0911.4307 (preprint title: Quantum Darwinism in non-ideal environments). https://arxiv.org/abs/0911.4307
8. 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
9. T. P. Le, A. 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, arXiv:1803.08936. https://arxiv.org/abs/1803.08936
10. 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, arXiv:2007.04276. https://arxiv.org/abs/2007.04276
11. Juha Torvinen, Esko Keski-Vakkuri, Nicola Pranzini. Quantum Darwinism and the quality of Petz recovery. arXiv:2605.06848 (2026). https://arxiv.org/abs/2605.06848
12. Katja Schneeweiss, Tom Rivlin, Maximilian P. E. Lock, Marcus Huber, Iva Březinová. Modelling quantum measurement dynamics: from decoherence to redundancy with site-hopping indistinguishable particles. arXiv:2608.11319 (2026). https://arxiv.org/abs/2608.11319
13. Arghya Maity, Kelvin Onggadinata, Teck Seng Koh. Exact Tradeoff Between Quantum Error Correction and Quantum Darwinism: An Information-Theoretic No-Go Theorem. arXiv:2608.03944 (2026). https://arxiv.org/abs/2608.03944
14. Marek Żukowski, Marcin Markiewicz. Against (unitary) interpretation (of quantum mechanics): removing the metaphysical load. arXiv:2409.17061 (2024). https://arxiv.org/abs/2409.17061
15. Piotr Mironowicz. Quantum security and theory of decoherence. New Journal of Physics 24, 113054 (2022). DOI: 10.1088/1367-2630/aca558, arXiv:2205.12927. https://arxiv.org/abs/2205.12927