Chapter 3 of S. Lloyd's 1988 Ph.D. thesis, `Black Holes, Demons, and the Loss of Coherence: How complex systems get information and what they do with it,' supervisor Heinz Pagels. Reformulates statistical mechanics in terms of pure states and shows that (a) quantum statistics of typical pure states are very close to the mechanics of statistical mechanical ensembles; (b) if a system is in a typical state with energy E, then the reduced density matrix of a subsystem is very close to a thermal state. (A similar result was derived using Levy's lemma some years later by S. Popescu, A.J. Short, A.Winter, Nature Physics 2, 754-758 (2006).) Pure state quantum statistical mechanics is applied to black holesto show that for typical states of matter insideand outside a black hole, the external state is likely to be thermal. Proposes novel interpretation of probabilities in quantum statistical mechanics. Full thesis available at http://meche.mit.edu/documents/slloyd_thesis.pdf. This chapter was submitted for publication to Physical Review in 1988 but rejected by one sentence referee report: `There is no physics in this paper.' You be the judge.
Chapter 3 of S. Lloyd's 1988 Ph.D. thesis, Black Holes, Demons, and the Loss of Coherence: How complex systems get information and what they do with it,' supervisor Heinz Pagels. Reformulates statistical mechanics in terms of pure states and shows that (a) quantum statistics of typical pure states are very close to the mechanics of statistical mechanical ensembles; (b) if a system is in a typical state with energy E, then the reduced density matrix of a subsystem is very close to a thermal state. (A similar result was derived using Levy's lemma some years later by S. Popescu, A.J. Short, A.Winter, Nature Physics 2, 754-758 (2006).) Pure state quantum statistical mechanics is applied to black holesto show that for typical states of matter insideand outside a black hole, the external state is likely to be thermal. Proposes novel interpretation of probabilities in quantum statistical mechanics. Full thesis available at http://meche.mit.edu/documents/slloyd_thesis.pdf. This chapter was submitted for publication to Physical Review in 1988 but rejected by one sentence referee report: There is no physics in this paper.' You be the judge.
The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way. This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
Quantum statistical mechanics can be fomulated based on a single pure quantum state.In this thesis, I establish this formulation, see the properties of such a pure quantum state, and apply the formulation to a practical application.Historically, it has been found that quantum states have robust "typicality", which is the property that almost all the realizable state give equilibrium values correctly for all "mechanical variables", such as energy and a correlation function.I generally call the pure quantum states which can represent the equilibrium state "thermal pure quantum (TPQ) state".I extend the typicality so that it is applicable to "genuine thermodynamic variables", such as entropy and temperature.I introduce many kinds of the TPQ states which include a "(grand)canonical TPQ state" , which is specified by intensive variables such as temperature, and a "microcanonical TPQ state", which is easy to generate.Owing to these results, I establish the new formulation of statistical mechanics based on the TPQ states.
Despite early developments on the foundations of quantum mechanics concern the wave function, quantum statistics has been developed with the density matrix formalism, leading to very important results in explaining molecular observations. Only recently, several authors argued a new interpretation by focusing on the wave function representing the quantum state of an isolated system, showing how a single wave function can exhibit statistical properties and generate the same results expected in the standard quantum statistical framework. Starting from these results, investigation on the foundations of quantum statistical mechanics has gained recently a renewed interest. As a matter of fact, the possibility of studying single molecule properties as well as the need of a better understanding of the effect of quantum dynamics, in order to develop new nanoscaled materials suitable to quantum computing tasks, have opened new intriguing questions leading to quantum statistical approaches far from being well understood and accepted. In this framework the behaviour of a single realization of quantum systems has gained a central role in the description of molecular systems. Furthermore, in recent years, an increasing number of studies has been presented on quantum dynamics through the numerical solution of the Schrödinger equation for systems of interacting components. These studies demonstrates that Quantum Dynamics Simulations could be a practicable route. In order to study phenomena such as dissipation, relaxation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation. An important issue concerns the rules to be employed for the choice of the initial quantum state for the simulation of isolated systems. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring that the simulation of the system is in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. A useful parametrization of the wave function will be presented in order to highlight the most appropriate variable for the statistical analysis. In particular some of them, called phases, retain all the dynamical information whereas the others, called populations, are the constants of motion. The latter, in particular are very important because they describe the equilibrium properties strictly related to the thermodynamic description. Since the dynamics of wave function does not supply any information about populations, the definition of a probability distribution on these variables is required. Different probability distributions on populations have been proposed, only on the bases of reasonable assumptions and they validation has been performed only with a posteriori considerations. In particular Fresch and Moro have demonstrated how the agreement with thermodynamics can be employed to discriminate different probability distributions on pure states. This had led the uniform ensembles to be, up to now, the most self-consistent models for quantum pure states. However, in this thesis I will highlight a drawback of the uniform distribution ensemble that can be described as follow: if we bring into contact two systems, even through a perturbative interaction, we are not able to describe the equilibrium properties after the interaction within the uniform distribution statistics, since the uniform character is lost. It represents a severe shortcoming of the statistical ensemble from a methodological point of view, since closed systems can be always considered as the result of interaction among previously isolated systems. On the other hand this drawback introduces a further requirement of a different nature that can be used for the definition of a new statistical ensemble. In this work I intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. The invariance of the thermal state in the coupling of identical systems will be used as a guideline in the definition af a new probability distribution on populations. Such an ensemble for pure states, called Thermalization Resilient Ensemble, provides a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamic properties is assured. In perspective it should be the privileged statistical ensemble to implement Quantum Dynamics Simulations. Once the average properties of the Thermalization Resilient Ensemble have been introduced, I will obtain a probability distribution on pure states with the use of a geometrical analysis on the Hilbert space. The surface elements of an ellipsoidal manifold will be related to the probability density on populations. As a matter of fact the explicit form of the probability distribution is a prerequisite in order to perform Quantum Dynamical Simulations. However the results obtained through the geometrical analysis cannot be easily extended to systems with unbounded energy spectrum and an alternative strategy has been developed. A scaling algorithm on the basis of the uniform statistical ensemble will be described and this allows a well defined sampling of a probability distribution with desired averages. In this framework I demonstrate the emergence of thermodynamic behavior in the limit of macroscopic systems. In the last part of the thesis I consider the dynamical features of the thermalization experiment. Two identical systems, initially at different temperature, will be brought in interaction and the analysis of the final equilibrium state will be performed for two different generic forms of the interaction Hamiltonian, highlighting how the statistical approach can be very useful in the definition of the equilibrium in complex quantum systems.
Abstract The fundamentals of the theory of density matrices are reviewed. The density matrix is introduced first in terms of state vectors for pure states over which a probability distribution is given. It is then shown that the applications of the density matrix do not require knowledge of the particular probability distribution over pure states that may have been used for its original determination. It is stressed that density matrices, like state vectors, describe ensembles rather than single systems, as determinism in quantum theory (except for some conservation laws) is of a statistical character only. The entropy of a quantum‐mechanical system is defined as the average per system of the entropy of an ensemble of systems described by the quantum‐mechanical (pure or mixed) state. For pure states this entropy is zero, in agreement with the fact that state vectors constitute the maximal information that can be given about a quantum‐mechanical system. As the entropy should be independent of the particular probability distribution over pure states that may have been used to form the mixed state, it is calculated from statistical considerations about an imaginary ensemble in which the eigenvalues of the density matrix are probabilities over pure states given by the eigenfunctions of the density matrix. The entropy is calculated for a mixture of two nonorthogonal states. The time evolution of the density matrix, in general, is not a unitary transformation, as observable systems are not closed and therefore in the Hamiltonian contain unknown external parameters that are not the same for the entire mixture. Therefore transitions between pure states and mixed states take place. Those from pure to mixed states take place spontaneously and irreversibly under increase of entropy. Transitions from mixed states to pure states can be fabricated; an example is given. As Wigner's thought experiment of mixing coherently the two beams from a Stern‐Gerlach apparatus according to estimates of Bohm should be impossible by loss of coherence in the gradient magnetic field, it is indicated how Wigner's ideas may be realized by using, instead, polarized light beams from birefringent crystals. After a brief review of the quantum theory of measurements and the various arguments usually given for the reducibility of state after the measurement, Wigner's argument against blind use of these arguments is interpreted as a limit of validity of the familiar probability rule of quantum theory, which should not be applied after an unsuccessful measurement. By splitting light beams coherently by glass plates, one can by the same apparatus have measurements both of the relative intensities of two coherent beams, and of interference effects between the two beams. This shows that complementarity applies to the individual systems in the beam, as they reach the glass plate where a choice must be made between two complementary possibilities of future measurement. Finally, by describing Schrödinger's cat by a density matrix and asking about its time dependence, it is shown that, if it were possible to measure the interference effects between living and dead cats used by some people as a justification for writing down a state vector for the cat, then this measurement could be used to revive some dead cats.