The relationship between classical and quantum mechanics is usually understood via the limit ħ→0. This is the underlying idea behind the quantization of classical objects. The apparent incompatibility of general relativity with quantum mechanics and quantum field theory has challenged for many decades this basic idea. We recently showed (Bru and de Siqueira Pedra, 0000; Bru and de Siqueira Pedra, 2021 [46,47]) the emergence of classical dynamics for very general quantum lattice systems with mean-field interactions, without (complete) suppression of its quantum features, in the infinite volume limit. This leads to a theoretical framework in which the classical and quantum worlds are entangled. Such an entanglement is noteworthy and is a consequence of the highly non-local character of mean-field interactions. Therefore, this phenomenon should not be restricted to systems with mean-field interactions only, but should also appear in presence of interactions that are sufficiently long-range, yielding effective, classical background fields, in the spirit of the Higgs mechanism of quantum field theory. In order to present the result in a less abstract way than in its original version, here we apply it to a concrete, physically relevant, example and discuss, by this means, various important aspects of our general approach. The model we consider is not exactly solvable and the particular results obtained are new.
The relationship between classical and quantum mechanics is usually understood via the limit ħ→0. This is the underlying idea behind the quantization of classical objects. The apparent incompatibility of general relativity with quantum mechanics and quantum field theory has challenged for many decades this basic idea. We recently showed (Bru and de Siqueira Pedra, 0000; Bru and de Siqueira Pedra, 2021 [46,47]) the emergence of classical dynamics for very general quantum lattice systems with mean-field interactions, without (complete) suppression of its quantum features, in the infinite volume limit. This leads to a theoretical framework in which the classical and quantum worlds are entangled. Such an entanglement is noteworthy and is a consequence of the highly non-local character of mean-field interactions. Therefore, this phenomenon should not be restricted to systems with mean-field interactions only, but should also appear in presence of interactions that are sufficiently long-range, yielding effective, classical background fields, in the spirit of the Higgs mechanism of quantum field theory. In order to present the result in a less abstract way than in its original version, here we apply it to a concrete, physically relevant, example and discuss, by this means, various important aspects of our general approach. The model we consider is not exactly solvable and the particular results obtained are new.
Experimental progress in atomic, molecular, and optical platforms in the last decade has stimulated strong and broad interest in the quantum coherent dynamics of many long-range interacting particles. The prominent collective character of these systems enables novel non-equilibrium phenomena with no counterpart in conventional quantum systems with local interactions. Much of the theory work in this area either focussed on the impact of variable-range interaction tails on the physics of local interactions or relied on mean-field-like descriptions based on the opposite limit of all-to-all infinite-range interactions. In this Report, we present a systematic and organic review of recent advances in the field. Working with prototypical interacting quantum spin lattices without disorder, our presentation hinges upon a versatile theoretical formalism that interpolates between the few-body mean-field physics and the many-body physics of quasi-local interactions. Such a formalism allows us to connect these two regimes, providing both a formal quantitative tool and basic physical intuition. We leverage this unifying framework to review several findings of the last decade, including the peculiar non-ballistic spreading of quantum correlations, counter-intuitive slowdown of entanglement dynamics, suppression of thermalization and equilibration, anomalous scaling of defects upon traversing criticality, dynamical phase transitions, and genuinely non-equilibrium phases stabilized by periodic driving. The style of this Report is on the pedagogical side, which makes it accessible to readers without previous experience in the subject matter.
Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac limit. If both attractive and repulsive long-range forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the "free" part of the model.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
Abstract. 75 years after the term "entanglement " was coined to a peculiar feature inherent to quantum systems, the connection between quantum and classical mechanics remains an open problem. Drawing on recent results obtained in semiclassical systems, we discuss here the fate of entanglement in a closed system as Planck’s constant becomes vanishingly small. In that case the generation of entanglement in a quantum system is perfectly reproduced by properly defined correlations of the corresponding classical system. We speculate on what these results could imply regarding the status of entanglement and of the ensuing quantum correlations.
This expository review reformulates the BCS analyses of Haag, Emch--Guenin, and Thirring--Wehrl, together with subsequent operator-algebraic mean-field results of Bóna, Raggio--Werner, and Bru--de Siqueira Pedra, in the language of quasi-local C∗-algebras and state decompositions. The quasi-spin model is realized on the UHF algebra of type 2∞, with product sectors described by von Neumann's incomplete infinite tensor products. This framework distinguishes strong limits of intensive observables, domain-restricted limits of extensive observables, and sectorwise Bogoliubov--Haag limits, including their operator criterion, gap equation, and limiting dynamics. In the degenerate model, the ground-state and thermal gauge averages admit central direct-integral decompositions over the gauge circle, and the thermal decomposition converges to the ground-state decomposition at zero temperature. Adjoining the gauge phase as a central classical variable combines the phase-dependent Bogoliubov dynamics into one automorphism group on C(S1,A).
75 years after the term "entanglement" was coined to a peculiar feature inherent to quantum systems, the connection between quantum and classical mechanics remains an open problem. Drawing on recent results obtained in semiclassical systems, we discuss here the fate of entanglement in a closed system as Planck's constant becomes vanishingly small. In that case the generation of entanglement in a quantum system is perfectly reproduced by properly defined correlations of the corresponding classical system. We speculate on what these results could imply regarding the status of entanglement and of the ensuing quantum correlations.