This note studies how quantum information is distributed when an entangled environment decoheres a system. It brings together three previous results: information stored nonlocally in environmental correlations, the separation between coherence loss and locally readable leakage, and the partial-information curve describing how much evidence can be recovered from a fraction of the environment. The paper proves that these effects are different aspects of one exact structure: a duality law for distributed quantum evidence. The readable-evidence functions of any environment fraction and its complementary fraction are exact reciprocals, with the product of the two evidence exponents remaining constant across all partitions. The self-dual point occurs when half of the environment is available, where the accessible evidence reaches the vacuum-level value. The extreme limits reproduce the previously identified eavesdropper deficit, showing that hidden environmental correlations control how much information becomes locally accessible. The paper also computes exact mutual-information curves in a hybrid qubit–Gaussian model. It shows that mutual information and readable which-path evidence are not the same quantity. A fragment may contain correlations with the system while carrying zero locally readable information. The difference represents quantum correlations that cannot be converted into an observable record by local measurements. The results reveal a fundamental distinction between information existing in correlations and information becoming an objective classical fact. Broadcast environments create redundant public records, while entangled environments can store information in a hidden, nonlocal form. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
This note studies how quantum information is distributed when an entangled environment decoheres a system. It brings together three previous results: information stored nonlocally in environmental correlations, the separation between coherence loss and locally readable leakage, and the partial-information curve describing how much evidence can be recovered from a fraction of the environment. The paper proves that these effects are different aspects of one exact structure: a duality law for distributed quantum evidence. The readable-evidence functions of any environment fraction and its complementary fraction are exact reciprocals, with the product of the two evidence exponents remaining constant across all partitions. The self-dual point occurs when half of the environment is available, where the accessible evidence reaches the vacuum-level value. The extreme limits reproduce the previously identified eavesdropper deficit, showing that hidden environmental correlations control how much information becomes locally accessible. The paper also computes exact mutual-information curves in a hybrid qubit–Gaussian model. It shows that mutual information and readable which-path evidence are not the same quantity. A fragment may contain correlations with the system while carrying zero locally readable information. The difference represents quantum correlations that cannot be converted into an observable record by local measurements. The results reveal a fundamental distinction between information existing in correlations and information becoming an objective classical fact. Broadcast environments create redundant public records, while entangled environments can store information in a hidden, nonlocal form. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
This note extends the evidence duality of distributed quantum records from simple environment partitions to general Gaussian environments. It studies how quantum information is distributed when a record-carrying mode is spread across multiple environmental modes through different network structures. The paper proves that the evidence duality is universal for passive Gaussian environments: regardless of how the environment is distributed through passive networks, the accessible evidence depends only on the fraction of the partner information captured by an observer. Complementary observers always obey an exact reciprocal relation, showing that only the captured fraction matters, not the way the information is physically distributed. The active boundary is then explored. Generic active networks involving squeezing break the duality, while special configurations can preserve it exactly. This identifies the precise domain where the universal duality holds and opens the question of classifying the surviving active cases. The paper also discovers a second form of complementarity: pattern duality in entangled record carriers. Independent environmental carriers combine additively, while entangled carriers can hide or amplify evidence depending on the correlation pattern. Correlated records suppress accessible information, while anti-correlated records enhance it, with an exact reciprocal relation between the two cases. Together, the results show that distributed quantum evidence obeys complementary laws both across environmental fractions and across correlation patterns. The structure of the environment determines whether information is hidden, amplified, or made accessible. All results are derived in exactly solvable Gaussian models, verified through independent analytical and numerical methods, and accompanied by a reproducibility package.
Quantum Darwinism explains classical objectivity through the redundant spread of information across fragments of the environment. This note studies the opposite regime: an environment whose record is distributed through entangled correlations rather than independent fragments. The model considers a mediator carrying a which-branch displacement record while being purified by an unobserved environment. The central question is: how much of the original quantum information can an observer recover when only part of the environment is accessible? The paper proves that any fragment excluding the record-bearing mode contains exactly zero which-branch information. No individual fragment carries a readable record, and redundancy disappears completely. A closed-form partial-information curve is derived. The amount of accessible evidence depends only on the captured fraction of the environment purification and is governed by a single rational function. The curve connects two limits: the minimal information available to an external observer and the full information recovered when the complete environment is captured. A precise threshold is obtained: strong environmental entanglement requires capturing almost the entire environment before objectivity emerges. The results show that decoherence and objectivity are not equivalent. The same loss of coherence can either create a public, redundantly observable fact or leave information hidden entirely in nonlocal correlations. The structure of the environment determines whether quantum information becomes an objective classical fact or remains inaccessible. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
Decoherence is usually viewed as the loss of local coherence, but it also writes information into the environment. Here we show that the environment stores this information in two distinct forms. One is a recoverable quantum record: after a transverse qubit measurement, the bath is projected onto a Schrödinger-cat-like state in a mode-matched physical collective coordinate. The other is a redundant classical which-path record distributed across physical frequency-band fragments. Using tensor-network simulations of a spin--boson reservoir, we reconstruct the conditional bath Wigner function and find visible negativity. In the chain representation used for the simulations, one natural orbital carries more than 95% of the bath one-body occupation associated with the record. A parity symmetry gives an exact nonperturbative identity between the remaining qubit coherence and the overlap of the two environmental branches, while the classical pointer information forms a Darwinian record across fragments. At finite temperature the quantum record is thermally smoothed. In the pure-dephasing limit it becomes an exact mixture of displaced cat states, and negativity remains visible over the simulated range. These results connect decoherence to phase-space tomography and outline how both records can be observed by qubit readout and collective-mode Wigner tomography.
This note studies mediated quantum entanglement when the mediator is not only heated but also entangled with an unobserved environment. It investigates how hidden correlations affect the relationship between decoherence, information leakage, and the ability to recover which-path information. The first result is an unbreakable revival window: at specific revival times, the mediator interaction closes exactly, restoring the full entanglement between the target systems. The reduced state becomes identical to the ideal mediator case, with no remaining which-path information anywhere, even when the mediator is correlated with an external environment. The second result is the eavesdropper’s deficit: coherence loss and locally accessible information are not the same quantity when the environment is entangled. The paper derives an exact relation showing that the coherence lost can exceed the information available to any local observer by a factor of (2n̄+1)². The missing information is not destroyed; it is stored nonlocally in correlations between the mediator and its environment. Local measurements cannot recover this hidden record, while the complete joint system preserves the full information balance. These results demonstrate that entangled environments can separate decoherence from observable information leakage: a system may lose coherence strongly while leaving only limited locally accessible evidence about the cause. All results are derived in exactly solvable Gaussian models, verified through analytical and numerical checks, and accompanied by a reproducibility package.
This paper quantifies a structural exclusion principle in quantum decoherence dynamics: when environment fragments carry redundant classical records of a pointer observable, information about any conjugate observable (the frame variable) is forced into nonlocal, high-order correlations. The work fills a gap in the quantum Darwinism program (Zurek 2003, 2009), which has thoroughly characterized how pointer information is redundantly broadcast to environment fragments but has not provided a quantitative treatment—with explicit constants and tight bounds—of the complementary question: what happens to conjugate (phase/coherence) information as pointer redundancy increases. The central result is an exact complementarity law across every bipartition, demonstrating that pointer accessibility in the unobserved complement is precisely the resource that suppresses frame accessibility to a verifier. This identity is fully consistent with the no-hiding theorem (Braunstein and Pati 2007), which guarantees that frame information remains in principle recoverable from the environment; the contribution here is to quantify how the required coalition size scales with pointer redundancy according to an explicit exponential curve, establishing the resource cost of recovery that the no-hiding theorem leaves open. Key results include a quantitative redundancy–leakage tradeoff with explicit constants, a universal record-to-conjugate exclusion theorem valid for general pointer-preserving Stinespring isometries without product-branching or conditional-independence assumptions, and a formal cross-domain equivalence establishing that physical branching dynamics induce a ramp secret-sharing and data-hiding access structure for the frame bit. The secret-sharing structure connects to the quantum data hiding literature (DiVincenzo, Leung, and Terhal 2002; Eggeling and Werner 2002; Lami, Palazuelos, and Winter 2018), with the distinction that the hiding here is not artificially constructed but arises naturally from decoherence with redundant pointer records, parameterized by a single physically measurable quantity—the decoherence factor of the unobserved complement. The effective dephasing channel induced by fragment erasure has a structural analogy to the erasure channels studied in fault-tolerant quantum error correction, with pointer redundancy playing the role of an effective code distance for conjugate observables; the key difference from the QEC threshold setting is that the encoding is imposed by nature rather than designed, and the access structure may not be known a priori. The paper also complements information-theoretic equilibration approaches (Short and Farrelly 2012) and discord-based irreversibility arguments (Zurek 2018) by providing explicit coalition-size thresholds tied to physical parameters rather than generic typicality bounds or discord measures. The work formalizes the transition from quantum coherence to classical objectivity by elevating complementary information bounds to exact, model-independent structural identities, while acknowledging that the branching-family starting point inherits the system–environment partition assumed by the quantum Darwinism framework.
The sum of the Holevo quantity (that bounds the capacity of quantum channels to transmit classical information about an observable) and the quantum discord (a measure of the quantumness of correlations of that observable) yields an observable-independent total given by the quantum mutual information. This split naturally delineates information about quantum systems accessible to observers – information that is redundantly transmitted by the environment – while showing that it is maximized for the quasi-classical pointer observable. Other observables are accessible only via correlations with the pointer observable. We also prove an anti-symmetry property relating accessible information and discord. It shows that information becomes objective – accessible to many observers – only as quantum information is relegated to correlations with the global environment, and, therefore, locally inaccessible. The resulting complementarity explains why, in a quantum Universe, we perceive objective classical reality while flagrantly quantum superpositions are out of reach.