Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
C. Gogolin and J. Eisert
We review selected advances in the theoretical understanding of complex
quantum many-body systems with regard to emergent notions of quantum statistical
mechanics. We cover topics such as equilibration and thermalisation in pure state
statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of
ensembles, non-equilibration dynamics following global and local quenches as well
as ramps. We also address initial state independence, absence of thermalisation,
and many-body localisation. We elucidate the role played by key concepts for
these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality
arguments, quantum maximum entropy principles and the generalised Gibbs
ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches
and present the most important results in a unified language.
Quantum many-body systems out of equilibrium
J. Eisert, M. Friesdorf, and C. Gogolin
How do closed quantum many-body systems driven out of equilibrium eventually achieve equilibration? And how do these systems thermalize, given that they comprise so many degrees of freedom? Progress in answering these—and related—questions has accelerated in recent years—a trend that can be partially attributed to success with experiments performing quantum simulations using ultracold atoms and trapped ions. Here we provide an overview of this progress, specifically in studies probing dynamical equilibration and thermalization of systems driven out of equilibrium by quenches, ramps and periodic driving. In doing so, we also address topics such as the eigenstate thermalization hypothesis, typicality, transport, many-body localization and universality near phase transitions, as well as future prospects for quantum simulation.
Reliable quantum certification for photonic quantum technologies
L. Aolita, C. Gogolin, M. Kliesch, and J. Eisert
Photonic devices involving many optical modes promise major advances in quantum technologies, with applications ranging from quantum metrology over quantum computing to quantum simulations. A significant current roadblock for the development of such devices, however, is the lack of practical reliable certification tools. Here, we present one such tool. We start by carefully defining different notions of quantum-state certification tests. Then, we introduce an experimentally friendly, yet mathematically rigorous, certification test for experimental preparations of arbitrary m-mode pure Gaussian states as well as a class of pure non-Gaussian states common in linear-optical experiments, including those given by a Gaussian unitary acting on Fock basis states with n bosons. The protocol is efficient for all Gaussian states and all mentioned non-Gaussian states with constant n. We follow the formal mindset of an untrusted prover, who prepares the state, and a skeptic certifier, equipped only with classical computing and single-mode measurement capabilities. No assumptions are made on the type of quantum noise or experimental capabilities of the prover. We build upon an extremality property that leads to a practical fidelity lower bound, interesting in its own right. Experimentally, our technique relies on single-mode homodyne detection. With this method, the efficient and reliable certification of large-scale photonic networks, with a constant number of input photons, as those used for photonic quantum simulations, boson samplers, and quantum metrology is within reach.
Equilibration and thermalization in quantum systems
This thesis fathoms out the capabilities of the theory of quantum mechanics to explain thermodynamic behavior.
It covers in particular equilibration and thermalization in closed quantum systems, typicality, time scales for equilibration, quantum integrability and its connection to thermalization, decoherence, and a maximum entropy principle.
Together, the presented results form the body of the theory of pure state quantum statistical mechanics.
With almost 300 references, ranging from the groundbreaking works of the early 20th century to the most recent discoveries (up to 2013), this work arguably constitutes the most comprehensive review of the literature on equilibration and thermalization in closed quantum systems.
All results are presented in a unified notation and many are slightly strengthened or generalized.
Pushing the limits of the eigenstate thermalization hypothesis towards mesoscopic quantum systems
Robin Steinigeweg, Abdelah Khodja, Hendrik Niemeyer, Christian Gogolin, and Jochen Gemmer
In the ongoing discussion on thermalization in closed quantum many-body systems, the eigenstate thermalization hypothesis (ETH) has recently been proposed as a universal concept which attracted considerable attention. So far this concept is, as the name states, hypothetical. The majority of attempts to overcome sthis hypothetical character is based on exact diagonalization which implies for, e.g., spin systems a limitation to roughly 15 spins. In this Letter we present an approach which pushes this limit up to system sizes of roughly 35 spins, thereby going significantly beyond what is possible with exact diagonalization. A concrete application to a Heisenberg spin-ladder which yields conclusive results is demonstrated.
Locality of temperature
M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert
This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems with short range interactions. We provide results leading to a local definition of temperature, thereby extending the notion of "intensivity of temperature" to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized covariance. For this covariance, we prove exponential clustering of correlations above a universal critical temperature that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above the critical temperature, thermal states are stable against distant Hamiltonian perturbations. Moreover, our results imply that above the critical temperature, local expectation values can be approximated efficiently in the error and the system size.
Boson-Sampling in the light of sample complexity
C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert
Boson-Sampling is a classically computationally hard problem that can — in principle — be efficiently solved with linear quantum optical networks.
Very recently, a rush of experimental activity has ignited with the aim of developing such devices as feasible instances of quantum simulators.
Even approximate Boson-Sampling is believed to be hard with high probability if the unitary describing the optical network is drawn from the Haar measure.
In this work we show that in this setup, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform one from fewer than exponentially many samples.
This means that the two distributions are operationally indistinguishable without detailed a priori knowledge.
We carefully discuss the prospects of efficiently using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this.
We conclude that due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach.
Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems
Martin Kliesch, Christian Gogolin, and Jens Eisert
This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound. Finally we discuss a number of important consequences in quantum many-body theory.
To appear as a chapter in the book "Many-Electron Approaches in Physics, Chemistry and Mathematics" to be published in "Mathematical Physics Studies" by Springer.
Quantum measurement occurrence is undecidable
J. Eisert, M. P. Mueller, C. Gogolin
A famous result by Alan Turing dating back to 1936 is that a general algorithm solving the halting problem on a Turing machine for all possible inputs and programs cannot exist — the halting problem is undecidable. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. In this work, we show that surprisingly, very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability appears as a genuine quantum property. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable.
A dissipative quantum Church-Turing theorem
M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert
We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.
Thermalization in nature and on a quantum computer
Arnau Riera, Christian Gogolin, and Jens Eisert
In this work, we put several questions related to the emergence of Gibbs states in quantum physics to rest. We show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, by completing the program of dynamical typicality and by introducing a novel general perturbation theorem that is robust under the thermodynamic limit, rigorously capturing the intuition of a meaningful weak coupling limit. We discuss the physics of thermal states occurring and identify the precise conditions under which this happens. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime, including full error estimates, complementing quantum Metropolis algorithms which are expected to be efficient but have no known runtime estimate.
Non-equilibrium Dynamics, Thermalization and Entropy Production
Haye Hinrichsen, Christian Gogolin, and Peter Janotta
This paper addresses fundamental aspects of statistical mechanics such as the motivation of a classical state space with spontaneous transitions, the meaning of non-equilibrium in the context of thermalization, and the justification of these concepts from the quantum-mechanical point of view. After an introductory part we focus on the problem of entropy production in non-equilibrium systems. In particular, the generally accepted formula for entropy production in the environment is analyzed from a critical perspective. It is shown that this formula is only valid in the limit of separated time scales of the system's and the environmental degrees of freedom. Finally, we present an alternative simple proof of the fluctuation theorem.
Limits on nonlocal correlations from the structure of the local state space
Peter Janotta, Christian Gogolin, Jonathan Barrett, and Nicolas Brunner
The outcomes of measurements on entangled quantum systems can
be nonlocally correlated. However, while it is easy to write down toy theories
allowing arbitrary nonlocal correlations, those allowed in quantum mechanics
are limited. Quantum correlations cannot, for example, violate a principle known
as macroscopic locality, which implies that they cannot violate Tsirelson’s
bound. This paper shows that there is a connection between the strength of
nonlocal correlations in a physical theory and the structure of the state spaces
of individual systems. This is illustrated by a family of models in which local
state spaces are regular polygons, where a natural analogue of a maximally
entangled state of two systems exists. We characterize the nonlocal correlations
obtainable from such states. The family allows us to study the transition between
classical, quantum and super-quantum correlations by varying only the local
state space. We show that the strength of nonlocal correlations — in particular
whether the maximally entangled state violates Tsirelson’s bound or not —
depends crucially on a simple geometric property of the local state space, known
as strong self-duality. This result is seen to be a special case of a general
theorem, which states that a broad class of entangled states in probabilistic
theories — including, by extension, all bipartite classical and quantum states — cannot violate macroscopic locality. Finally, our results show that models exist
that are locally almost indistinguishable from quantum mechanics, but can
nevertheless generate maximally nonlocal correlations.
Absence of thermalization in non-integrable systems
Christian Gogolin, Markus P. Müller, and Jens Eisert
We present rigorous results establishing a link between unitary relaxation dynamics after a quench in closed many-body systems in non-equilibrium and the entanglement in the energy eigenbasis.
We find that even if reduced states equilibrate, and appear perfectly relaxed, they can still have memory on the initial conditions even in models that are far from integrable, thereby giving rise to "equilibration without thermalization".
We show that in such situations the equilibrium states are however still described by a Jaynes maximum entropy or generalized Gibbs ensemble and, moreover, that this is always the case if equilibration happens, regardless of whether a model is integrable or not.
In addition, we discuss individual aspects of thermalization processes separately, comment on the role of Anderson localization, and collect and compare different notions of integrability.
In May 2011 I received the Leibniz publication award for young academics for this publication.
Pure State Quantum Statistical Mechanics
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.
Einselection without pointer states
We show that the existence of a basis of pointer states is not necessary for environment-induced super selection.
This is achieved by using recent results on equilibration of small subsystems of large, closed quantum systems evolving according to the von Neumann equation.
Without making any special assumptions on the form of the interaction we prove that, for almost all initial states and almost all times, the off-diagonal elements of the density matrix of the subsystem in the eigenbasis of its local Hamiltonian must be small whenever the energies of the corresponding eigenstates differ by more than the interaction energy.
Dynamic wetting with two competing adsorbates
Christian Gogolin, Christian Meltzer, Marvin Willers, and Haye Hinrichsen
We study the dynamic properties of a model for wetting with two competing adsorbates on a planar substrate. The two species of particles have identical properties and repel each other. Starting with a flat interface one observes the formation of homogeneous droplets of the respective type separated by nonwet regions where the interface remains pinned. The wet phase is characterized by slow coarsening of competing droplets. Moreover, in 2+1 dimensions an additional line of continuous phase transition emerges in the bound phase, which separates an unordered phase from an ordered one. The symmetry under interchange of the particle types is spontaneously broken in this region and finite systems exhibit two metastable states, each dominated by one of the species. The critical properties of this transition are analyzed by numeric simulations.