arXiv subjects

Jochen Rau

Publications and source records attributed to Jochen Rau.

At least 19 recordsLinked to original sources

arXiv2015

Reconstruction of Markovian dynamics from untimed data

I develop a theoretical framework for inferring nonequilibrium equations of motion from incomplete experimental data. I focus on genuinely irreversible, Markovian processes, for which the incomplete data are given in the form of snapshots of the macrostate at different instances of the evolution, yet without any information about the timing of these snapshots. A reconstruction of the equation of motion must therefore be preceded by a reconstruction of time.

cond-mat.stat-mech↗

arXiv2014

Appearance of Gibbs states in quantum-state tomography

I investigate the extent to which the description of quantum systems by Gibbs states can be justified purely on the basis of tomographic data, without recourse to theoretical concepts such as infinite ensembles, environments, information, or to the systems' dynamics. I show that the use of Gibbs states amounts to a relevance hypothesis, which I spell out in detail. This hypothesis can be subjected to statistical hypothesis testing and hence assessed on the basis of the experimental data.

quant-ph↗

arXiv2012

Direct estimation of decoherence rates

The decoherence rate is a nonlinear channel parameter that describes quantitatively the decay of the off-diagonal elements of a density operator in the decoherence basis. We address the question of how to experimentally access such a nonlinear parameter directly without the need of complete process tomography. In particular, we design a simple experiment working with two copies of the channel, in which the registered mean value of a two-valued measurement directly determines the value of the average decoherence rate. No prior knowledge of the decoherence basis is required.

quant-ph↗

arXiv2011

Reconstructing the relaxation dynamics induced by an unknown heat bath

In quantum state tomography, one potential source of error is uncontrolled contact of the system with a heat bath whose detailed properties are not known, and whose impact on the system moreover varies between different runs of the experiment. Precisely these variations provide a handle for reconstructing the system's effective relaxation dynamics. I propose a pertinent estimation scheme which is based on a steepest-descent ansatz and maximum likelihood. After reconstructing the relaxation dynamics, the original quantum state of the system can be constrained to a curve in state space.

quant-ph↗

arXiv2011

Assessing thermalization and estimating the Hamiltonian with output data only

I consider the generic situation where a finite number of identical test systems in varying (possibly unknown) initial states are subjected independently to the same unknown process. I show how one can infer from the output data alone whether or not the process in question induces thermalization, and if so, which constants of the motion characterize the final equilibrium states. In case thermalization does occur and there is no evidence for constants of the motion other than energy, I further show how the same output data can be used to estimate the test systems' effective Hamiltonian. For both inference tasks I devise a statistical framework inspired by the generic techniques of factor and principal component analysis. I illustrate its use in the simple example of qubits.

quant-ph↗

arXiv2011

Inferring the Gibbs state of a small quantum system

Gibbs states are familiar from statistical mechanics, yet their use is not limited to that domain. For instance, they also feature in the maximum entropy reconstruction of quantum states from incomplete measurement data. Outside the macroscopic realm, however, estimating a Gibbs state is a nontrivial inference task, due to two complicating factors: the proper set of relevant observables might not be evident a priori; and whenever data are gathered from a small sample only, the best estimate for the Lagrange parameters is invariably affected by the experimenter's prior bias. I show how the two issues can be tackled with the help of Bayesian model selection and Bayesian interpolation, respectively, and illustrate the use of these Bayesian techniques with a number of simple examples.

quant-ph↗

arXiv2010

On the metric structure of space-time

I present an analysis of the physical assumptions needed to obtain the metric structure of space-time. For this purpose I combine the axiomatic approach pioneered by Robb with ideas drawn from works on Weyl's "Raumproblem". The concept of a Lorentzian manifold is replaced by the weaker concept of an "event manifold", defined in terms of volume element, causal structure and affine connection(s). Exploiting properties of its structure group, I show that distinguishing Lorentzian manifolds from other classes of event manifolds requires the key idea of general relativity: namely that the manifold's physical structure, rather than being fixed, is itself a variable.

gr-qc↗

arXiv2010

Evidence procedure for efficient quantum state tomography

I show that in tomographic experiments measurement of a small set of observables suffices to confirm or incrementally amend prior expectations with a high degree of confidence. To this end I adapt the evidence procedure, an estimation technique used in classical image reconstruction, to use in quantum state tomography.

quant-ph↗

arXiv2010

Convergence of macrostates under reproducible processes

I show that whenever a system undergoes a reproducible macroscopic process the mutual distinguishability of macrostates, as measured by their relative entropy, diminishes. This extends the second law which regards only ordinary entropies, and hence only the distinguishability between macrostates and one specific reference state (equidistribution). The new result holds regardless of whether the process is linear or nonlinear. Its proof hinges on the monotonicity of quantum relative entropy under arbitrary coarse grainings, even those that cannot be represented by completely positive maps.

quant-ph↗

arXiv2010

Measurement-based quantum foundations

I show that quantum theory is the only probabilistic framework that permits arbitrary processes to be emulated by sequences of local measurements. This supports the view that, contrary to conventional wisdom, measurement should not be regarded as a complex phenomenon in need of a dynamical explanation but rather as a primitive -- and perhaps the only primitive -- operation of the theory.

quant-ph↗

arXiv2009

On quantum vs. classical probability

Quantum theory shares with classical probability theory many important properties. I show that this common core regards at least the following six areas, and I provide details on each of these: the logic of propositions, symmetry, probabilities, composition of systems, state preparation and reductionism. The essential distinction between classical and quantum theory, on the other hand, is shown to be joint decidability versus smoothness; for the latter in particular I supply ample explanation and motivation. Finally, I argue that beyond quantum theory there are no other generalisations of classical probability theory that are relevant to physics.

quant-ph↗

arXiv2008

Reasoning about quantum systems at the macroscopic level

In this didactical note I review in depth the rationale for using generalised canonical distributions in quantum statistics. Particular attention is paid to the proper definitions of quantum entropy and quantum relative entropy, as well as to quantum state reconstruction on the basis of incomplete data. There are two appendices in which I outline how generalised canonical distributions link to the conventional formulation of statistical mechanics, and how classical probability calculus emerges at the macroscopic level.

quant-ph↗

arXiv2007

Consistent reasoning about a continuum of hypotheses on the basis of finite evidence

In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however, that--at least in principle--the amount of evidence that an experimenter can accumulate always matches the size of the hypothesis space. I investigate how the framework for consistent reasoning must be modified in non-classical situations where hypotheses form a continuum, yet the maximum evidence accessible through experiment is not allowed to exceed some finite upper bound. Invoking basic consistency requirements pertaining to the preparation and composition of systems, as well as to the continuity of probabilities, I show that the modified theory must have an internal symmetry isomorphic to the unitary group. It thus appears that the only consistent algorithm for plausible reasoning about a continuum of hypotheses on the basis of finite evidence is furnished by quantum theory in complex Hilbert space.

quant-ph↗

arXiv1998

Statistical Mechanics in a Nutshell

I give a concise introduction to some essential concepts of statistical mechanics: 1. Probability theory (constrained distributions, concentration theorem, frequency estimation, hypothesis testing); 2. Macroscopic systems in equilibrium (macrostate, thermodynamic variables, entropy, first law, thermodynamic potentials, correlations); 3. Linear response (Kubo formula).

physics.ed-ph↗

arXiv1997

Geometric magnetism in classical transport theory

The effective dynamics of a slow classical system coupled to a fast chaotic environment is described by means of a Master equation. We show how this approach permits a very simple derivation of geometric magnetism.

cond-mat.stat-mech↗

arXiv1997

Transport theory yields renormalization group equations

We show that dissipative transport and renormalization can be described in a single theoretical framework. The appropriate mathematical tool is the Nakajima-Zwanzig projection technique. We illustrate our result in the case of interacting quantum gases, where we use the Nakajima-Zwanzig approach to investigate the renormalization group flow of the effective two-body interaction.

cond-mat.stat-mech↗

arXiv1996

Generalized Bloch equations for a strongly driven tunneling system

Using the Robertson projection operator formalism, we derive generalized Bloch equations which describe the dynamics of a biased two-level tunneling system strongly driven by an external field and weakly coupled to a super-Ohmic heat bath. The generalized Bloch equations constitute a set of coupled nonlinear integro-differential equations. With their help we investigate the influence of phonons on the phenomenon of dynamical localization.

cond-mat↗

arXiv1996

Algebraic approach to renormalization

In close analogy to the Bloch-Feshbach formalism known from the theory of nuclear dynamics, I develop a mathematical framework that allows one to understand renormalization in terms of purely algebraic operations (projections, dilatations) in Hilbert space. This algebraic approach is put to the test in the study of the low-energy dynamics of interacting quantum gases, and proves to be efficient in deriving such diverse results as the renormalization group equation for an interacting Bose gas, the $β$ function of $ϕ^4$ theory, the screening of fermion-fermion interactions or the BCS instability.

cond-mat↗