All my publications can also be found on the arXiv, on ORCID, and some statistics is available via Google Scholar and my ResearcherID in case you care about such numbers...

#### Sample complexity of device-independently certified "quantum supremacy"

##### Dominik Hangleiter, Martin Kliesch, Jens Eisert, Christian Gogolin

Results on the hardness of approximate sampling are seen as important stepping stones towards a convincing demonstration of the superior computational power of quantum devices. The most prominent suggestions for such experiments include boson sampling, IQP circuit sampling, and universal random circuit sampling. A key challenge for any such demonstration is to certify the correct implementation. For all these examples, and in fact for all sufficiently flat distributions, we show that any non-interactive certification from classical samples and a description of the target distribution requires exponentially many uses of the device. Our proofs rely on the same property that is a central ingredient for the approximate hardness results: namely, that the sampling distributions, as random variables depending on the random unitaries defining the problem instances, have small second moments.

arXiv:1812.01023 [quant-ph] |

#### Evaluating analytic gradients on quantum hardware

##### Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran

An important application for near-term quantum computing lies in optimization tasks, with applications ranging from quantum chemistry and drug discovery to machine learning. In many settings --- most prominently in so-called parametrized or variational algorithms --- the objective function is a result of hybrid quantum-classical processing. To optimize the objective, it is useful to have access to exact gradients of quantum circuits with respect to gate parameters. This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same, architecture that executes the original circuit. It generalizes previous results for qubit-based platforms, and proposes recipes for the computation of gradients of continuous-variable circuits. Interestingly, in many important instances it is sufficient to run the original quantum circuit twice while shifting a single gate parameter to obtain the corresponding component of the gradient. More general cases can be solved by conditioning a single gate on an ancilla.

arXiv:1811.11184 [quant-ph] |

#### PennyLane: Automatic differentiation of hybrid quantum-classical computations

##### Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, and Nathan Killoran

PennyLane is a Python 3 software framework for optimization and machine learning of quantum and hybrid quantum-classical computations. The library provides a unified architecture for near-term quantum computing devices, supporting both qubit and continuous-variable paradigms. PennyLane's core feature is the ability to compute gradients of variational quantum circuits in a way that is compatible with classical techniques such as backpropagation. PennyLane thus extends the automatic differentiation algorithms common in optimization and machine learning to include quantum and hybrid computations. A plugin system makes the framework compatible with any gate-based quantum simulator or hardware. We provide plugins for StrawberryFields and ProjectQ (including a IBMQE device interface). PennyLane can be used for the optimization of variational quantum eigensolvers, quantum approximate optimization, quantum machine learning models, and many other applications.

arXiv:1811.04968 [quant-ph] |

#### Verification of Quantum Optimizers

##### Flavio Baccari, Christian Gogolin, Peter Wittek, and Antonio Acín

Methods for finding ground states of classical spin models are of great importance in optimization and are gaining additional relevance now for verifying the results quantum optimizers. We combine the state-of-the-art branch and bound method for solving such optimization problems via converging upper- and lower-bounds with ideas from polynomial optimization and semidefinite programming (SDP). The resulting chordal branch and bound (CBB) algorithm can exploit the locality and resulting sparsity in relevant Ising spin models in a systematic way. This yields certified solutions for many of the problems that are being used to benchmark quantum annealing devices more efficiently and for larger system sizes. We are able to verify the output of a D-Wave 2000Q device for the largest triangular lattice that can be embedded in the hardware and provide exact ground states for cases in which the quantum annealer returns a configuration with almost minimal energy but markedly different spin pattern. The method always yields provable polynomial time upper and lower bounds on the ground state energy. We benchmark our method against further planar and non-planar graphs and show that these bounds often converge after a small number of steps, even though the NP-hardness of general Ising models implies that exponentially many steps are required in the worst case. This new tool is a flexible and scalable solution for the verification and benchmarking of the next generation of quantum optimization devices.

arXiv:1808.01275 [quant-ph,math.OC] |

#### Automated discovery of characteristic features of phase transitions in many-body localization

##### Patrick Huembeli, Alexandre Dauphin, Peter Wittek, and Christian Gogolin

We identify a new "order parameter" for the disorder driven many-body localization (MBL) transition by leveraging artificial intelligence. This allows us to pin down the transition, as the point at which the physics changes qualitatively, from vastly fewer disorder realizations and in an objective and cleaner way than is possible with the existing zoo of quantities. Contrary to previous studies, our method is almost entirely unsupervised. A game theoretic process between neural networks defines an adversarial setup with conflicting objectives to identify what characteristic features to base efficient predictions on. This reduces the numerical effort for mapping out the phase diagram by a factor of ~100x. This approach of automated discovery is applicable specifically to poorly understood phase transitions and exemplifies the potential of machine learning assisted research in physics.

arXiv:1806.00419 [quant-ph,cond-mat.dis-nn] |

#### What it takes to avoid equilibration

##### R. Gallego, H. Wilming, J. Eisert, and C. Gogolin

Numerous works have shown that under mild assumptions, unitary dynamics inevitably leads to equilibration of physical expectation values if many energy eigenstates contribute to the initial state. Here, we consider systems driven by arbitrary time-dependent Hamiltonians as a protocol to prepare systems that do not equilibrate. We introduce a measure of the resilience against equilibration of such states, and we show, under natural assumptions, that in order to increase the resilience against equilibration of a given system, one needs to possess a resource system that itself has a large resilience. In this way, we establish a link between the theory of equilibration and resource theories by quantifying the resilience against equilibration and the resources that are needed to produce it. We connect these findings with insights into local quantum quenches, and we investigate the (im)possibility of formulating a second law of equilibration by studying how resilience can be either only redistributed among subsystems, if these remain completely uncorrelated, or in turn created in a catalytic process if subsystems are allowed to build up some correlations.

arXiv:1711.09832 [quant-ph,cond-mat.stat-mech] |

Phys. Rev. A 98, 022135 (2018) |

#### Eigenstate Thermalization for Degenerate Observables

##### Fabio Anza, Christian Gogolin, and Marcus Huber

Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of the ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive, or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian. We show that the relative overlaps are unbiased for highly degenerate observables and demonstrate that unless relative phases conspire to cumulative effects, this makes such observables verify the ETH. Through this we elucidate potential pathways towards proofs of thermalization.

arXiv:1708.02881 [cond-mat.stat-mech,quant-ph] |

Phys. Rev. Lett. 120, 150603 (2018) |

#### Decay of correlations in systems of fermions with long-range interactions at non-zero temperature

##### Senaida Hernández-Santana, Christian Gogolin, J. Ignacio Cirac, and Antonio Acín

We study correlations in fermionic systems wit h long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators based on long-range Lieb-Robinson type bounds. Our result shows that correlations between such operators in fermionic long-range systems of spatial dimension D with at most two-site interactions decaying algebraically with the distance with an exponent α≥2D, decay at least algebraically with an exponent arbitrarily close to α. Our bound is asymptotically tight, which we demonstrate by numerically analyzing density-density correlations in a 1D quadratic (free, exactly solvable) model, the Kitaev chain with long-range interactions. Away from the quantum critical point correlations in this model are found to decay asymptotically as slowly as our bound permits.

arXiv:1702.00371 [quant-ph,cond-mat.stat-mech] |

Phys. Rev. Lett. 119, 110601 (2018) |

#### Optimal quantum error correcting codes from absolutely maximally entangled states

##### Zahra Raissi, Christian Gogolin, Arnau Riera, and Antonio Acín

Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q >= n-1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q >= n-1 prime, we show how to construct stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [[n,1,n/2]]_q for n even with the highest distance allowed by the quantum Singleton bound.

arXiv:1701.03359 [quant-ph] |

J. Phys. A: Math. Theor. 51 075301 (2018) |

#### Quantum Enhanced Inference in Markov Logic Networks

##### Peter Wittek and Christian Gogolin

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

arXiv:1611.08104 [stat.ML,cs.AI,quant-ph] |

Scientific Reports 7, Article number: 45672 (2017) |

#### Total correlations of the diagonal ensemble as a generic indicator for ergodicity breaking in quantum systems

##### Francesca Pietracaprina, Christian Gogolin, and John Goold

The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general α-Renyi entropies and focus on the the cases α=1 and α=2 with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a sub-extensive behaviour when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many body localized phase in systems with disorder or quasi-periodicity.

arXiv:1611.02046 [cond-mat.dis-nn,quant-ph] |

Phys. Rev. B 95, 125118 (2017) |

#### Random bosonic states for robust quantum metrology

##### Michał Oszmaniec, Remigiusz Augusiak, Christian Gogolin, Janek Kołodyński, A. Acín, and Maciej Lewenstein

We study how useful random states are for quantum metrology, i.e., surpass the classical limits imposed on precision in the canonical phase estimation scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to super-classical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for states of arbitrarily low purity and preserved under finite particle losses. Moreover, we prove that for such states a standard photon-counting interferometric measurement suffices to typically achieve the Heisenberg scaling of precision for all possible values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam-splitters and a non-linear (Kerr-like) transformation.

arXiv:1602.05407 [quant-ph] |

Phys. Rev. X 6, 041044 (2016) |

#### Equilibration via Gaussification in fermionic lattice systems

##### M. Gluza, C. Krumnow, M. Friesdorf, C. Gogolin, and J. Eisert

The perspective of probing quantum many-body systems out of equilibrium under well controlled conditions is attracting enormous attention in recent years, a perspective that extends to the study of fermionic systems. In this work, we present an argument that precisely captures the dynamics causing equilibration and Gaussification under quadratic non-interacting fermionic Hamiltonians. Specifically, based on two basic assumptions - the initial clustering of correlations and the Hamiltonian exhibiting delocalizing transport - we prove that systems become locally indistinguishable from fermionic Gaussian states on precisely controlled time scales. The argument gives rise to rigorous instances of a convergence to a generalized Gibbs ensemble. This argument is general enough to allow for arbitrary pure and mixed initial states, including thermal and ground states of interacting models, and large classes of systems, including high-dimensional lattice and classes of spin systems. Our results allow to develop an intuition of equilibration that is expected to be generally valid, and at the same time relates to current experiments of cold atoms in optical lattices.

arXiv:1601.00671 [cond-mat.quant-gas,quant-ph] |

Phys. Rev. Lett. 117, 190602 (2016) |

#### Quantum annealing for the number-partitioning problem using a tunable spin glass of ions

##### T. Graß, D. Raventós, B. Juliá-Díaz, C. Gogolin, and M. Lewenstein

Exploiting quantum properties to outperform classical ways of information-processing is an outstanding goal of modern physics. A promising route is quantum simulation which aims at implementing relevant and computationally hard problems in controllable quantum systems. Here we consider trapped ions which have proven very flexible for realizing the physics of interacting spins. We demonstrate concretely that, with present day technology, a spin model of the Mattis type can be obtained, that exhibits spin glass phases. Remarkably, our method produces the glassy behaviour without the need of any disorder potential, just by controlling the detuning of the spin-phonon coupling. Applying a transverse field, the system can be used to benchmark quantum annealing strategies which aim at reaching the ground state of the spin glass starting from the paramagnetic phase. In the vicinity of a phonon resonance, the problem maps onto number partitioning, and instances which are difficult to address classically can be implemented.

arXiv:1507.07863 [cond-mat.quant-gas] |

Nature Comm. 7, 11524 (2016) |

ICFO press release |

UB press release |

articulo en La Vanguardia |

#### Total correlations of the diagonal ensemble herald the many-body localization transition

##### J. Goold, C. Gogolin, S. R. Clark, J. Eisert, A. Scardicchio, and A. Silva

The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work, we introduce the total correlations, a concept from quantum information theory capturing multi-partite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sub-linear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field.

arXiv:1504.06872 [cond-mat.dis-nn] |

Phys. Rev. B 92, 180202(R) |

#### 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.

arXiv:1503.07538 [quant-ph, cond-mat.stat-mech, math-ph] |

Rep. Prog. Phys. 79, 056001 (2016) |

#### 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.

arXiv:1408.5148 [quant-ph, cond-mat.quant-gas, cond-mat.stat-mech] |

Nature Physics 11, 124–130 (2015) |

ICFO news web abstract |

MPQ press release |

#### 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.

arXiv:1407.4817 [quant-ph] |

Nature Comm. 6, 8498 (2015) |

MPQ press release |

ICFO press release |

FU Berlin press release |

#### Equilibration and thermalization in quantum systems

##### Christian Gogolin

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.

doi: 10.13140/2.1.1636.4485 |

direct download (pdf) |

urn:nbn:de:kobv:188-fudissthesis000000097097-7 |

#### 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 this 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.

arXiv:1311.0169 [quant-ph,cond-mat.stat-mech] |

Phys. Rev. Lett. 112, 130403 (2014) |

#### 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.

arXiv:1309.0816 [quant-ph,cond-mat.stat-mech,math-ph,math] |

Phys. Rev. X 4, 031019 (2014) |

#### 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.

arXiv:1306.3995 [quant-ph] |

#### 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.

arXiv:1306.0716 [quant-ph] |

Mathematical Physics Studies 2014, pp 301-318 |

#### 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.

arXiv:1111.3965 [quant-ph] |

Phys. Rev. Lett. 108, 260501 (2012) |

Article in Phys.org |

Article in Компьютерное Обозрение |

FQXi article |

#### 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.

arXiv:1105.3986 [quant-ph,math-phys,cond-mat.stat-mech] |

Phys. Rev. Lett. 107, 120501 (2011) |

Viewpoint in Physics |

#### 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.

arXiv:1102.2389 [quant-ph,stat-mech,math-ph] |

Phys. Rev. Lett. 108, 080402 (2012) |

#### 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.

arXiv:1102.0103 [stat-mech] |

J. Phys.: Conf. Ser. 297 012001 |

#### 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 non-locally 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.

New Journal of Physics 13 (2011) 063024 |

arXiv:1012.1215 [quant-ph] |

#### 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.

Phys. Rev. Lett. 106, 040401 (2011) |

arXiv:1009.2493 [quant-ph,cond-mat.stat-mech] |

#### Pure State Quantum Statistical Mechanics

##### Christian Gogolin

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.

arXiv:1003.5058 [quant-ph,cond-mat.stat-mech] |

urn:nbn:de:bvb:20-opus-106065 |

#### Einselection without pointer states

##### Christian Gogolin

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.

Phys. Rev. E 81, 051127 (2010) |

arXiv:0908.2921 [quant-ph,cond-mat.stat-mech] |

#### 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 non-wet 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.

Phys. Rev. E 79, 041111 (2009) |

arXiv:0809.2542 [cond-mat.stat-mech] |