Information in statistical physics

We review with a tutorial scope the information theory foundations of quantum statistical physics. Only a small proportion of the variables that characterize a system at the microscopic scale can be controlled, for both practical and theoretical reasons, and a probabilistic description involving the observers is required. The criterion of maximum von Neumann entropy is then used for making reasonable inferences. It means that no spurious information is introduced besides the known data. Its outcomes can be given a direct justification based on the principle of indifference of Laplace. We introduce the concept of relevant entropy associated with some set of relevant variables; it characterizes the information that is missing at the microscopic level when only these variables are known. For equilibrium problems, the relevant variables are the conserved ones, and the Second Law is recovered as a second step of the inference process. For non-equilibrium problems, the increase of the relevant entropy expresses an irretrievable loss of information from the relevant variables towards the irrelevant ones. Two examples illustrate the flexibility of the choice of relevant variables and the multiplicity of the associated entropies: the thermodynamic entropy (satisfying the Clausius-Duhem inequality) and the Boltzmann entropy (satisfying the H-theorem). The identification of entropy with missing information is also supported by the paradox of Maxwell's demon. Spin-echo experiments show that irreversibility itself is not an absolute concept: use of hidden information may overcome the arrow of time.Comment: latex InfoStatPhys-unix.tex, 3 files, 2 figures, 32 pages http://www-spht.cea.fr/articles/T04/18

Entanglement in BF theory I: Essential topological entanglement

We study the entanglement structure of Abelian topological order described by $p$-form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement. This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.Comment: updated references, corrected minor typos; 38 pages, 3 figure

Information Theoretic Graph Kernels

This thesis addresses the problems that arise in state-of-the-art structural learning methods for (hyper)graph classification or clustering, particularly focusing on developing novel information theoretic kernels for graphs. To this end, we commence in Chapter 3 by defining a family of Jensen-Shannon diffusion kernels, i.e., the information theoretic kernels, for (un)attributed graphs. We show that our kernels overcome the shortcomings of inefficiency (for the unattributed diffusion kernel) and discarding un-isomorphic substructures (for the attributed diffusion kernel) that arise in the R-convolution kernels. In Chapter 4, we present a novel framework of computing depth-based complexity traces rooted at the centroid vertices for graphs, which can be efficiently computed for graphs with large sizes. We show that our methods can characterize a graph in a higher dimensional complexity feature space than state-of-the-art complexity measures. In Chapter 5, we develop a novel unattributed graph kernel by matching the depth-based substructures in graphs, based on the contribution in Chapter 4. Unlike most existing graph kernels in the literature which merely enumerate similar substructure pairs of limited sizes, our method incorporates explicit local substructure correspondence into the process of kernelization. The new kernel thus overcomes the shortcoming of neglecting structural correspondence that arises in most state-of-the-art graph kernels. The novel methods developed in Chapters 3, 4, and 5 are only restricted to graphs. However, real-world data usually tends to be represented by higher order relationships (i.e., hypergraphs). To overcome the shortcoming, in Chapter 6 we present a new hypergraph kernel using substructure isomorphism tests. We show that our kernel limits tottering that arises in the existing walk and subtree based (hyper)graph kernels. In Chapter 7, we summarize the contributions of this thesis. Furthermore, we analyze the proposed methods. Finally, we give some suggestions for the future work

Entanglement through interfaces and toy models of holography

This thesis is dedicated to the analysis of correlation structure in two quite different ex-amples. In the first one we investigate the entanglement entropy (EE) through conformal inter-faces in two-dimensional conformal field theories. The EE is a measure for the strength of entanglement between two subsystems of a system in a pure state. Interfaces are one-dimensional objects on a two-dimensional surface along which two conformal field theories are glued together. They are called conformal if their gluing conditions is invariant under conformal transformations that preserve the shape of the interface. Special interfaces are topological ones. They are called topological because the partition function does not change when the interface is continuously deformed. We in particular show that in a vast class of conformal field theories the presence of a topological interface at the boundary between two subregions makes the EE get dressed with an additional sub-leading but universal contribution that solely depends on the interface data. This sub-leading contribution can be interpreted as a relative entropy measuring the relative loss of entanglement compared to no interface insertion. We also compute the EE through general conformal interfaces in the critical Ising model and find that its leading order is affected by the interfaces if and only if it is not topological. Any conformal defect can be characterized by its transmissivity. As physically expected, the EE though conformal interfaces in the Ising model decreases for lower transmissivities. The second example is what we call classical holographic codes. They are classical probabilistic codes defined by a network on a uniform tiling of a constant time slice of AdS 3 -spacetime. They share some remarkable properties with certain quantum error correcting codes that are designed to mimic particular properties of the AdS/CFT correspondence. Under these features are the Ryu-Takayanagi formula and bulk reconstruction properties that both reflect deep connections between the correlation structure of a theory and the ge- ometry of its dual description. Our classical codes can be seen as toy models for holography that show that the latter features do not necessarily originate from a quantum description.In dieser Arbeit analysieren wir Korrelationsstrukturen in zwei unterschiedlichen Beispielen. Im ersten behandeln wir die “Entanglement Entropie” (EE) durch Defekte in zweidimensionalen konformen Feldtheorien. Die EE is ein Maß für die Quantenverschränkung zwischen zwei Untersystemen eines Systems, das durch einen reinen Zustand beschrieben wird. Defekte sind eindimensionale Objekte auf einer zweidimensionalen Fläche entlang derer konforme Feldtheorien verklebt werden können. Sie werden konform genannt, wennihre Eigenschaften invariant unter jenen konformen Transformationen sind, die die Form des Defektes nicht verändern. Topologische Defekte bilden eine besondere Klasse. Sie sind dadurch definiert, dass kontinuierlichen Deformationen keinen Effekt auf die Zustandssumme des Systems haben. Wir zeigen in dieser Arbeit, dass in einer großen Klasse von konformen Feldtheorien die EE durch topologischen Defekt das Ergebnis ohne Defekt um einen universellen Term nicht führender Ordnung ergänzt. Dieser hängt nur von den Defektdaten ab und kann als relative Entropie interpretiert werden, die den Verlust an Ver- schränkung relativ zur Situation ohne Defekt misst. Wir berechnen zudem die EE durch beliebige konforme Defekte des kritischen Isingmodells. Diese können insbesondere durch ihre Transmissivität charakterisiert werden. Wir können zeigen, dass nicht-topologische Defekte auch die führende Ordnung der EE beeinflussen und dass sie, wie es physikalisch zu erwarten ist, für niedrigere Werte der Transmissivität sinkt. Beim zweiten Beispiel handelt es sich um spezielle Systeme, die wir klassische holografische Codes nennen. Man kann sie als probabilistische klassische Codes realisieren, die durch ein Netzwerk auf einer uniformen Abdeckung eines Schnitts entlang konstanter Zeit durch einen AdS 3 -Raum definiert sind. Erstaunlicherweise teilen sie einige Eigenschaften mit speziellen Quantencodes zur Fehlerkorrektur, die wiederum spezielle Eigenschaften holographischer Theorien und im besonderen der AdS/CFT Korrespondenz mimen. Unsere klassischen Codes zeigen, dass Eigenschaften wie die Ryu-Takayanagi-Formel und Rekonstruktioneigenschaften, die beide eine Verbindung zwischen der Korrelationsstruktur einer Theorie und der Geometrie ihrer dualen Beschreibung aufzeigen, nicht notwendigerweise von einer Quantenbeschreibung herrühren

Entropic order parameters for the phases of QFT

We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT's. We do so by an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the 't Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by the dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.Comment: 60 pages + appendice

Entanglement in tripartitions of topological orders: a diagrammatic approach

Recent studies have demonstrated that measures of tripartite entanglement can probe data characterizing topologically ordered phases to which bipartite entanglement is insensitive. Motivated by these observations, we compute the reflected entropy and logarithmic negativity, a mixed state entanglement measure, in tripartitions of bosonic topological orders using the anyon diagrammatic formalism. We consider tripartitions in which three subregions meet at trijunctions and tetrajunctions. In the former case, we find a contribution to the negativity which distinguishes between Abelian and non-Abelian order while in the latter, we find a distinct universal contribution to the reflected entropy. Finally, we demonstrate that the negativity and reflected entropy are sensitive to the $F$-symbols for configurations in which we insert an anyon trimer, for which the Markov gap, defined as the difference between the reflected entropy and mutual information, is also found to be non-vanishing.Comment: v2: updated references; v3: updated references, corrected typos and notatio