Using the framework of decoherent histories, we study which past events leave detectable records in isolated quantum systems under the realistic assumption that decoherence is approximate and not perfect. In the first part we establish -- asymptotically for a large class of (pseudo-)random histories -- that the number of reliable records can be much smaller than the number of possible events, depending on the degree of decoherence. In the second part we reveal a clear decoherence structure for long histories based on a numerically exact solution of a random matrix model that, as we argue, captures generic aspects of decoherence. We observe recoherence between histories with a small Hamming distance, for localized histories admitting a high purity Petz recovery state, and for maverick histories that are statistical outliers with respect to Born's rule. From the perspective of the Many Worlds Interpretation, the first part -- which views the self-location problem as a coherent version of quantum state discrimination -- reveals a "branch selection problem", and the second part sheds light on the emergence of Born's rule and the theory confirmation problem.
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Main text: 25 pages with 16 figures split into two related but independently readable parts. Supplement: 13 pages with 9 figures. Comments welcome
Using the framework of decoherent histories, we study which past events leave detectable records in isolated quantum systems under the realistic assumption that decoherence is approximate and not perfect. In the first part we establish -- asymptotically for a large class of (pseudo-)random histories -- that the number of reliable records can be much smaller than the number of possible events, depending on the degree of decoherence. In the second part we reveal a clear decoherence structure for long histories based on a numerically exact solution of a random matrix model that, as we argue, captures generic aspects of decoherence. We observe recoherence between histories with a small Hamming distance, for localized histories admitting a high purity Petz recovery state, and for maverick histories that are statistical outliers with respect to Born's rule. From the perspective of the Many Worlds Interpretation, the first part -- which views the self-location problem as a coherent version of quantum state discrimination -- reveals a "branch selection problem", and the second part sheds light on the emergence of Born's rule and the theory confirmation problem.
Within the histories formalism the decoherence functional is a formal tool to investigate the emergence of classicality in isolated quantum systems, yet an explicit evaluation of it from first principles has not been reported. We provide such an evaluation for up to five-time histories based on exact numerical diagonalization of the Schrödinger equation. We find a robust emergence of decoherence for slow and coarse observables of a generic random matrix model and extract a finite-size scaling law by varying the Hilbert space dimension over 4 orders of magnitude. Specifically, we conjecture and observe an exponential suppression of coherent effects as a function of the particle number of the system. This suggests a solution to the preferred basis problem of the many-worlds interpretation (or the set selection problem of the histories formalism) within a minimal theoretical framework without relying on environmentally induced decoherence, quantum Darwinism, Markov approximations, low-entropy initial states, or ensemble averages. Published by the American Physical Society 2024
There exist two different paradigms for records in isolated quantum systems: decoherent histories (which determines which past events leave formal records) and quantum Darwinism (which determines which current events get redundantly recorded). Neither one implies the other, and connections between the two are scarcely investigated. We unify quantum Darwinism and decoherent histories based on the insight that only redundant records about past events matter. The resulting framework of past redundancy encompasses quantum Darwinism when applied to events at the current time, and it implies the probability sum rule or decoherence of histories. The Bhattacharyya coefficient (or fidelity) plays an important role to establish this connection. Importantly, various examples show that past redundancy is a more widespread phenomenon than the fine-tuned models necessary for quantum Darwinism suggest. We thus arrive at a richer picture in which redundancy can be caused by mechanisms different from quantum Darwinism (such as macroevents or cascade structures), which emphasizes that redundancy is necessary but not sufficient for objectivity, and which, overall, allows to realistically investigate the question: how generic is redundancy in nature?
Motivated by the advances of quantum Darwinism and recognizing the role played by redundancy in identifying the small subset of quantum states with resilience characteristic of objective classical reality, we explore the implications of redundant records for consistent histories. The consistent histories formalism is a tool for describing sequences of events taking place in an evolving closed quantum system. A set of histories is consistent when one can reason about them using Boolean logic, i.e., when probabilities of sequences of events that define histories are additive. However, the vast majority of the sets of histories that are merely consistent are flagrantly nonclassical in other respects. This embarras de richesses (known as the set selection problem) suggests that one must go beyond consistency to identify how the classical past arises in our quantum universe. The key intuition we follow is that the records of events that define the familiar objective past are inscribed in many distinct systems, e.g., subsystems of the environment, and are accessible locally in space and time to observers. We identify histories that are not just consistent but redundantly consistent using the partial-trace condition introduced by Finkelstein as a bridge between histories and decoherence. The existence of redundant records is a sufficient condition for redundant consistency. It selects, from the multitude of the alternative sets of consistent histories, a small subset endowed with redundant records characteristic of the objective classical past. The information about an objective history of the past is then simultaneously within reach of many, who can independently reconstruct it and arrive at compatible conclusions in the present.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
The consistent histories formalism can be used to describe histories comprised of events across many systems, times, and places, plausibly rich enough to describe our experiences of the classical world; however, many consistent history sets are nonclassical and thus not obviously relevant to our experiences. Meanwhile, the program of environmentally induced decoherence identifies dynamically privileged classical degrees of freedom, but provides no general account of when or how many such degrees of freedom consistently combine to form histories. This work shows that the strengths of these two approaches can be combined by adopting a dynamics-first perspective on decoherence. Inspired by quantum causal models and quantum Darwinism, we define the process of decoherence in terms of the causal influences through unitary dynamics required for the proliferation of information about observables. We characterise decoherence as a property of the unitary dynamics, without presupposing the existence of any quantum state. Instead, we show that the state emerges from dual decoherence, related to decoherence by time-reversal of the unitary dynamics. Indeed, for any set of systems in an arbitrary unitary circuit, decoherence and its dual single out a privileged consistent history set -- and we demonstrate through examples that states emerge from dual decoherence while outcomes emerge from decoherence. Hence the idea that quantum states emerge from the process of decoherence turns out to be the key missing ingredient for unifying environmentally induced decoherence and consistent histories. Taking this idea ontologically seriously leads to a recently proposed causal interpretation of quantum theory or a dynamics-first version of the Everett interpretation. The causal approach also sheds light on the suppression of off-diagonal terms, time asymmetry, and robustness of the pointer basis.
Two recent results suggest Hamming distance organizes quantum coherence dynamics, but no unified framework connects them. We establish the Hamming Persistence Principle: quantum correlations at Hamming distance persist with amplitude scaling as e−κd, where κ depends on context but the geometric structure is universal. The principle unies two independent discoveries: (1) In open systems under local noise, density matrix elements decay as ρij(t) = ρij(0)e−γH(i,j)ta theorem following from locality of interactions. (2) In isolated systems, Strasberg et al. observed numerically that o-diagonal elements of the decoherence functional correlate negatively with Hamming distance, but noted they had no elaborate theoretical explanation. We provide that explanation: both phenomena arise because physical interactions are local, making Ham-ming distancethe minimum number of single-bit operations connecting two congurationsthe dynamically privileged metric. Numerical verication conrms the framework: (i) Coherence magnitude decreases with Hamming distance(r =−0.26, p < 10−70), reproducing Strasberg's Figure 12. (ii) The decay coecient scales as α ≈0.5 ln D(r= 0.965), supporting the locality derivation. (iii) Maverick histories that deviate from Born statistics cluster at smaller Hamming distance (3/4 tests p < 0.05) and are 2×more coherent with each otherexplaining why they recohere together. The synthesis resolves why entanglement fails to predict decoherence rates: entanglement ignores Hamming geometry. It also suggests a geometric origin for branch selection: histories spread across Hamming space (large D∗H) are stable, while clustered histories (small D∗H) recohere away. The classical world may emerge not from environmental monitoring but from the intrinsic geometry of conguration space.