A quantitative theory of the redundancy-consensus gap in quantum objectivity for nonlocally encoded information

Chisholm, Innocenti and Palma (Quantum 7, 1074 (2023)) gave operative definitions distinguishing redundancy (how many environmental fractions in the best partition of the environment each carry nearly all information about a system) from consensus (how many independent observers, given a random partition of the environment, can actually retrieve that information), and showed that for nonlocally encoded states, such as GHZ- and W-type encodings, the two can differ starkly, breaking the established hierarchy between spectrum broadcast structure (SBS) and quantum Darwinism (QD). The open problem is to turn this operative framework into a rigorous, predictive theory, along the directions the paper itself formulates. This includes determining how generic large-redundancy/low-consensus states are across broad state classes; how physically motivated biased sampling of environmental fractions changes consensus; whether a measurable fragility parameter, possibly grounded in quantum error correction, predicts the gap; how the redundancy and consensus values depend on the explicit thresholds epsilon and delta needed to make the definitions rigorous; and under what precise conditions SBS can be guaranteed to imply QD for nonlocally encoded states.

Background

Decoherence is the process by which a quantum system that interacts with an environment loses the phase relations between its components: the environment effectively 'measures' the system, suppressing superpositions of the states most robust to monitoring (the pointer states selected by 'einselection'). This explains the disappearance of quantum coherence, but not why many independent observers agree on the properties of the system. Quantum Darwinism (QD) [Zurek, Nat. Phys. 5, 181 (2009)] addresses this second step: the environment is claimed to redundantly record information about the pointer observable, so that many observers, each accessing only a small fraction of the environment, infer the same system state. The standard QD diagnostic plots the quantum mutual information between the system and a fraction of the environment against the fraction size and looks for a plateau at the system entropy, which indicates redundancy. Spectrum broadcast structure (SBS) is the other main framework: a state has SBS when it decomposes as a mixture over pointer values with each environmental fragment holding a perfectly distinguishable conditional state, which was proven by Horodecki, Korbicz and Horodecki [Phys. Rev. A 91, 032122 (2015)] to imply the QD mutual-information condition, so that SBS was understood to sit at the top of an objectivity hierarchy. Le and Olaya-Castro [Phys. Rev. Lett. 122, 010403 (2019)] further showed that strong quantum Darwinism (where the full quantum mutual information equals the classically accessible part) plus a strong-independence condition between fragments (the fragments are mutually uncorrelated once the system's pointer value is fixed) is equivalent to SBS; Feller, Roussel, Frérot and Degiovanni [Phys. Rev. Lett. 126, 188901 (2021)] then showed by counterexample that the pairwise strong-independence condition as originally formulated is necessary but not sufficient for this equivalence, and gave a strengthened formulation. A key subtlety is that the quantum mutual information can include correlations that no local measurement can reveal; the gap between it and the accessible mutual information $I_{\rm acc}$, defined as the classical mutual information of the outcome distributions maximized over local measurements, is the symmetric discord, and ignoring it can produce false positives of objectivity. Chisholm, Innocenti and Palma (Quantum 7, 1074 (2023)) built on this distinction to separate two notions usually conflated: the redundancy $\mathrm{Red}(\rho)$ of a system-environment state $\rho$, defined as the largest $n$ for which there exists a partition of the environment into $n$ fractions with $I_{\rm acc}(\mathcal{S}:\mathcal{E}_i)\simeq S(\rho_{\mathcal{S}})$ for every fraction (with $S(\rho_{\mathcal{S}})$ the von Neumann entropy of the system), and the consensus, defined as the largest $n$ such that the probability $P(\rho,n)$ that a uniformly random partition into $n$ equal-size fractions has all fractions $I_{\rm acc}$-correlated to the system is close to one. Consensus can never exceed redundancy, but it can be much smaller: the two notions coincide for locally encoded information, while for states where the information about the system is stored nonlocally in correlations inside groups of environment qubits (GHZ states are the standard multipartite entangled states of the form $(|0\cdots0\rangle+|1\cdots1\rangle)/\sqrt2$, and W states the single-excitation superpositions), a carefully chosen partition reveals high redundancy while a random one typically leaves each observer with almost no accessible information. In such scenarios the SBS property with respect to a well-chosen partition no longer guarantees the QD condition built on the averaged accessible information, breaking the hierarchy between the two main frameworks for objectivity. The same authors also showed that using a non-averaged mutual information as an objectivity quantifier can be severely misleading whenever the encoding is not homogeneous across the environment [Phys. Rev. A 110, 012218 (2024)]. The paper states that large environments probabilistically compensate the nonlocal encoding, but leaves open the systematic theory of when and how much the gap persists.

Problem Statement

Build the quantitative, threshold-rigorous theory of the gap between redundancy and consensus for system-environment states with nonlocally encoded information, within the operative framework of Chisholm, Innocenti and Palma: redundancy $\mathrm{Red}(\rho)$ as the largest number $n$ of environmental fractions in some partition of the environment each carrying accessible mutual information $I_{\rm acc}(\mathcal{S}:\mathcal{E}_i)$ close to the system entropy $S(\rho_{\mathcal{S}})$, and consensus as the largest $n$ such that the probability $P(\rho,n)$ over uniformly random partitions into $n$ equal-size fractions that every fraction is $I_{\rm acc}$-correlated to the system is close to one. The theory should address the following interrelated components: (1) Genericity: a systematic characterization, over broad classes of system-environment states beyond GHZ- and W-type tensor encodings (including dynamical families and random state ensembles), of when $\mathrm{Red}(\rho)$ can be large while consensus remains small, determining whether the redundancy-consensus gap is generic in the quantum-to-classical transition or confined to finely structured encodings. (2) Sampling: replace the uniform-equiprobable sampling of equal-size environmental fractions with physically motivated biased distributions (for example favoring spatially contiguous or connected fractions) and characterize how the degree of consensus changes. (3) Fragility: define a measurable fragility parameter, potentially formalized through quantum error correction notions quantifying how rapidly accessible information degrades upon loss of environmental constituents, that predicts the size of the redundancy-consensus gap for arbitrary states, with the GHZ and W cases recovered as calibration points. (4) Rigor: give the operative definitions a mathematical formulation with explicit thresholds $I_{\rm acc}\ge S(\rho_{\mathcal{S}})-\epsilon$ and $P(\rho,n)\ge 1-\delta$, and characterize how the resulting redundancy and consensus values depend on $\epsilon$ and $\delta$. (5) Hierarchy restoration: identify and prove precise conditions, for example requiring SBS to hold for all environmental partitionings, or on average over them, of a certain minimum size, under which SBS implies the QD condition even for nonlocally encoded states, restoring the relation between the two main frameworks for quantum objectivity outside the local-encoding regime. Each of the five components is an independently sufficient sub-result: a rigorous, self-contained treatment of any single component resolves the task, and the remaining components are optional extensions rather than requirements. The overarching aim is an integrated theory: from the structure of a system-environment state alone, via the thresholds, the sampling model, and the fragility parameter, it should be possible to predict whether independent observers sampling the environment will agree on the system's state, and whether the SBS and QD diagnoses coincide.

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