We consider an alternative approach to the foundations of statistical mechanics, in which subjective randomness, ensemble-averaging or time-averaging are not required. Instead, the universe (i.e. the system together with a sufficiently large environment) is in a quantum pure state subject to a global constraint, and thermalisation results from entanglement between system and environment. We formulate and prove a "General Canonical Principle", which states that the system will be thermalised for almost all pure states of the universe, and provide rigorous quantitative bounds using Levy's Lemma.

The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics. Despite considerable progress, it remains an open problem. Motivated by this issue, we address the more general question of equilibration. We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems: Almost any subsystem in interaction with a large enough bath will reach an equilibrium state and remain close to it for almost all times. We also prove several general results about other aspects of thermalisation besides equilibration, for example, that the equilibrium state does not depend on the detailed micro-state of the bath.Leave your hot cup of coffee or cold beer alone for a while and they soon lose their appeal -the coffee cools down and the beer warms up and they both reach room temperature. And it is not only coffee and beer -reaching thermal equilibrium is a ubiquitous phenomenon: everything does it. Thermalization is one of the most fundamental facts of nature.But how exactly does thermalization occur? How can one derive the existence of this phenomenon from the basic dynamical laws of nature (such as Newton's or Schrödinger's equations)? These have been open questions since the very beginning of statistical mechanics more than a century and a half ago.One -but by no means the only -stumbling block has been the fact that the basic postulates of statistical mechanics rely on subjective lack of knowledge and ensemble averages, which is very controversial as a physical principle. Recently however there has been significant progress: it was realized that ensemble averages and subjective ignorance are not needed, because individual quantum states of systems can exhibit statistical properties. This is a purely quantum phenomenon, and the key is entanglement, which leads to objective lack of knowledge. Namely, in quantum mechanics even when we have complete knowledge of the state of a system, i.e. it is in a pure state and has zero entropy, the state of a subsystem may be mixed and have non-zero entropy. In this situation we cannot describe the subsystem by any particular pure state, and for all purposes it behaves as if we have a lack of knowledge about what its pure state is (i.e. it behaves as a probability distribution over pure states). This is in stark contrast to classical physics where complete knowledge of the state of the whole system implies complete knowledge of the state of any subsystem, and hence probabilities can only arise as purely subjective lack of knowledge (i.e. the subsystem has a well-defined state only we don't know what that is).This approach has become a very fruitful direction of research in recent years [1, 2,3], see also important earlier work [4,5,6,7] and numerical studies [8]. Most notably, it was shown in [1, 2,3] that almost all (pure) states of a large system are such that any small subsystem is in a canonical stat...

Thermodynamics is traditionally concerned with systems comprised of a large number of particles. Here we present a framework for extending thermodynamics to individual quantum systems, including explicitly a thermal bath and work-storage device (essentially a 'weight' that can be raised or lowered). We prove that the second law of thermodynamics holds in our framework, and gives a simple protocol to extract the optimal amount of work from the system, equal to its change in free energy. Our results apply to any quantum system in an arbitrary initial state, in particular including non-equilibrium situations. The optimal protocol is essentially reversible, similar to classical Carnot cycles, and indeed, we show that it can be used to construct a quantum Carnot engine.

We consider one copy of a quantum system prepared in one of two orthogonal pure states, entangled or otherwise, and distributed between any number of parties. We demonstrate that it is possible to identify which of these two states the system is in by means of local operations and classical communication alone. The protocol we outline is both completely reliable and completely general; it will correctly distinguish any two orthogonal states 100% of the time.

It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has nondegenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate. 9 Acknowledgments 10 Appendix A. Proof of equation (13) 10 Appendix B. Proof of theorem 2 11 Appendix C. Proof of theorem 3 111 For a precise definition of these criteria, see the next section. 2 If desired, we could define d E to be the number of distinct energies on which ρ(t) has support. This gives a slight improvement in the results, but has the disadvantage of making d E and N (ε) depend on the state, rather than the Hamiltonian alone.