Abstract Quantum statistics and non-locality are deeply rooted in quantum mechanics and go beyond our intuition reflected in classical physics. Quantum statistics can be derived using statistical methods for indistinguishable particles - particles of quantum mechanics. Violation of strong locality - colloquially called the ghostly action at a distance - is one of the most amazing properties of nature derived from quantum mechanics. An intriguing question is whether the non-local evolution of indistinguishable particles is needed to reach the equilibrium state given by quantum statistics. Motivated by the above and similar questions, we developed a simple framework that allows us to follow space-time evolution of assembly of particles. It is based on a discrete-time Markov chain on countable space for indistinguishable particles. We summarise well-known and introduced new constraints on the transition matrix that grant space-time symmetries, locality of particle-transport, strong locality, and equilibrium state. Then, within the framework, several important cases are considered. First, we show that the simplest transition matrix leads to equilibrium but violates particle transport and strong localities. Furthermore, we construct a simple matrix that leads to equilibrium obeying particle-transport locality and violating strong locality. This resembles the properties of quantum mechanics. Finally, we demonstrate that it is also possible to reach equilibrium by obeying both particle-transport and strong localities. Thus, within this framework, the violation of a strong locality is not needed to reach the equilibrium of indistinguishable particles. However, to obey strong locality, a complex structure of the transition matrix is needed. In addition, we comment on distinguishable particles and, in particular, show that their evolution seen by an observer blind to particle differences may look like the evolution of indistinguishable particles with the properties of quantum mechanics. We hope that this work may help to study the relation between symmetries, localities and the evolution to equilibrium for indistinguishable and distinguishable particles.
Abstract Quantum statistics and non-locality are deeply rooted in quantum mechanics and go beyond our intuition reflected in classical physics. Quantum statistics can be derived using statistical methods for indistinguishable particles - particles of quantum mechanics. Violation of strong locality - colloquially called the ghostly action at a distance - is one of the most amazing properties of nature derived from quantum mechanics. An intriguing question is whether the non-local evolution of indistinguishable particles is needed to reach the equilibrium state given by quantum statistics. Motivated by the above and similar questions, we developed a simple framework that allows us to follow space-time evolution of assembly of particles. It is based on a discrete-time Markov chain on countable space for indistinguishable particles. We summarise well-known and introduced new constraints on the transition matrix that grant space-time symmetries, locality of particle-transport, strong locality, and equilibrium state. Then, within the framework, several important cases are considered. First, we show that the simplest transition matrix leads to equilibrium but violates particle transport and strong localities. Furthermore, we construct a simple matrix that leads to equilibrium obeying particle-transport locality and violating strong locality. This resembles the properties of quantum mechanics. Finally, we demonstrate that it is also possible to reach equilibrium by obeying both particle-transport and strong localities. Thus, within this framework, the violation of a strong locality is not needed to reach the equilibrium of indistinguishable particles. However, to obey strong locality, a complex structure of the transition matrix is needed. In addition, we comment on distinguishable particles and, in particular, show that their evolution seen by an observer blind to particle differences may look like the evolution of indistinguishable particles with the properties of quantum mechanics. We hope that this work may help to study the relation between symmetries, localities and the evolution to equilibrium for indistinguishable and distinguishable particles.
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.
The surprising aspects of quantum information are due to two distinctly non-classical features of the quantum world: first, different quantum states need not be orthogonal and, second, quantum states may be entangled. Non-orthogonality leads to the blurring of classical distinctions. On the other hand, entanglement leads via non-locality to teleportation and other ``entanglement-assisted'' forms of communication that go beyond what is classically possible. In this article we attempt to understand these new possibilities via an analysis of the significance of entanglement for the basic physical concepts of a ``particle'' and a ``localized physical system''. Classical particles can be individuated on the basis of qualitative differences in their sets of properties. But in entangled states the ``particle labels'' of the quantum formalism usually do not pick out such sets of individuating particle properties. It is sometimes nevertheless possible to think in terms of individual particles, which may be localized; but we argue that in general the structure of quantum mechanics is at odds with such a particle interpretation. This finally leads us to the conclusion that quantum mechanics is best seen as not belonging to the category of space-time theories, in which physical quantities are functions on space-time points. The resulting picture of the quantum world is relevant for our understanding of the way in which quantum theory is non-local, and it sheds light on the novel aspects of quantum information.
Classical statistical mechanics is based on the notion of a particle (i.e. its x and p phase space) as is Newtonian mechanics which is based on x(t). Quantum mechanics, on the other hand, contains particle-wave duality and is linked to Newtonian mechanics (in that one considers bound state, motion etc), but is also considered to be statistical. In this note we consider the statistical features of quantum mechanics by examining a reflection/refraction of light problem with incident light moving along the positive x direction. If considered from the point of view of a particle (i.e. photon) (in accordance with Newtonian mechanics and classical statistical mechanics) there is no equilibrium. One may imagine one photon strike the medium at a time. This photon retains its energy, but has a certain probability “a” to reflect at x=0 and 1-a to refract. Thus energy and photon number are conserved, but “a” is unknown and the energy and particle number to the left and right of x are not the same, even if one considers a steady state stream. In other words, there is a discontinuity at x=0 because of the difference in energy and photon numbers, yet a steady state picture seems to imply a kind of equilibrium which is not present in the particle picture. In other words, there should be invariance along the x axis with continuity at x=0, but this cannot be achieved by only considering particles (i.e. photons). Nevertheless, photons are physical and whatever solution is found must be consistent with photon energy and number conservation. This seems to suggest a new physical property needed to establish the equilibrium which somehow coexists with the particle picture. We suggest this property is wavelength with a photon being described by exp(ipx) (the square root of flux) and that it is this scenario which allows one to see the reflection/refraction problem as an equilibrium one. In other words, one does not need a particle point of view to establish equilibrium, yet particle number/energy may still be conserved in this picture.
The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results.
Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.
Collapse models possibly suggest the need for a better understanding of the structure of space-time. We argue that physical space, and space-time, are emergent features of the Universe, which arise as a result of dynamical collapse of the wave function. The starting point for this argument is the observation that classical time is external to quantum theory, and there ought to exist an equivalent reformulation which does not refer to classical time. We propose such a reformulation, based on a non-commutative special relativity. In the spirit of Trace Dynamics, the reformulation is arrived at, as a statistical thermodynamics of an underlying classical dynamics in which matter and non-commuting space-time degrees of freedom are matrices obeying arbitrary commutation relations. Inevitable statistical fluctuations around equilibrium can explain the emergence of classical matter fields and classical space-time, in the limit in which the universe is dominated by macroscopic objects. The underlying non-commutative structure of space-time also helps understand better the peculiar nature of quantum non-locality, where the effect of wave-function collapse in entangled systems is felt across space-like separations.