Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first give a sufficient condition for canonical states to typically appear in subsystems of phase-random states, which reveals a trade-off relation between the initial state in the bounded energy subspace and the energy eigenstates that define that subspace. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that, starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix-product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.
Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first give a sufficient condition for canonical states to typically appear in subsystems of phase-random states, which reveals a trade-off relation between the initial state in the bounded energy subspace and the energy eigenstates that define that subspace. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that, starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix-product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.
In order to study generic properties of time-evolving states by time-independent Hamiltonian dynamics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We compute the average amount of entanglement of phase-random states analytically, and show that the average can be extremely large when the amplitudes are equal and the basis is separable. We also study implications on Hamiltonian dynamics, in particular the realization of the canonical state in a subsystem.
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)].
Most states in the Hilbert space are maximally entangled. This fact has proven useful to investigate--among other things--the foundations of statistical mechanics. Unfortunately, most states in the Hilbert space of a quantum many-body system are not physically accessible. We define physical ensembles of states acting on random factorized states by a circuit of length k of random and independent unitaries with local support. We study the typicality of entanglement by means of the purity of the reduced state. We find that for a time k=O(1), the typical purity obeys the area law. Thus, the upper bounds for area law are actually saturated, on average, with a variance that goes to zero for large systems. Similarly, we prove that by means of local evolution a subsystem of linear dimensions L is typically entangled with a volume law when the time scales with the size of the subsystem. Moreover, we show that for large values of k the reduced state becomes very close to the completely mixed state.
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.
We propose a method to calculate finite-temperature properties of a quantum many-body system for a microcanonical ensemble by introducing a pure quantum state named here an energy-filtered random-phase state, which is also a potentially promising application of near-term quantum computers. In our formalism, a microcanonical ensemble is specified by two parameters, i.e., the energy of the system and its associated energy window. Accordingly, the density of states is expressed as a sum of Gaussians centered at the target energy with its spread corresponding to the width of the energy window. We then show that the thermodynamic quantities such as entropy and temperature are calculated by evaluating the trace of the time-evolution operator and the trace of the time-evolution operator multiplied by the Hamiltonian of the system. We also describe how these traces can be evaluated using random diagonal-unitary circuits appropriate to quantum computation. The pure quantum state representing our microcanonical ensemble is related to a state of the form introduced by Wall and Neuhauser for the filter diagonalization method [M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011 (1995)], and therefore we refer to it as an energy-filtered random-phase state. The energy-filtered random-phase state is essentially a Fourier transform of a time-evolved state whose initial state is prepared as a random-phase state, and the cut-off time in the time-integral for the Fourier transform sets the inverse of the width of the energy window. The proposed method is demonstrated numerically by calculating thermodynamic quantities for the one-dimensional spin-1/2 Heisenberg model on small clusters up to 28 qubits, showing that the method is most efficient for the target energy around which the dense distribution of energy eigenstates is found.
Mixed state ensembles such as the Bures-Hall and Hilbert-Schmidt measure are probability distributions that characterise the statistical properties of random density matrices and can be used to determine the typical features of mixed quantum states. Here we extend this framework by considering the properties of random states with fixed average energy, and the ensemble-averaged density matrix is derived under this additional physical constraint. This gives rise to a type of microcanonical ensemble for random mixed states and we connect its properties to a statistical mechanical entropy and temperature. Our results are illustrated using a variety of simple spin systems, and we find that they can exhibit exotic features such as phase transitions in the absence of energetic interactions and finite relative energy fluctuations in the thermodynamic limit.