On the foundations and extremal structure of the holographic entropy cone

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. Some interesting open problems are stated throughout.Comment: 32 pages, 2 figures, 3 tables. Revised to expand the description of the connection to quantum information theory, including additional reference

Constraints on Multipartite Quantum Entropies

The von Neumann entropy plays a vital role in quantum information theory. As the Shannon entropydoes in classical information theory, the von Neumann entropy determines the capacities of quan-tum channels. Quantum entropies of composite quantum systems are important for future quantumnetwork communication their characterization is related to the so calledquantum marginal problem.Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropiesof multipartite quantum systems is the main object of interest. The problem of characterizing this setis not new – however, progress has been sparse, indicating that the problem may be considered hardand that new methods might be needed. Here, a variety of different and complementary aprroachesare taken.First, I look at global properties. It is known that the von Neumann entropy region – just likeits classical counterpart – forms aconvex cone. I describe the symmetries of this cone and highlightgeometric similarities and differences to the classical entropy cone.In a different approach, I utilize thelocalgeometric properties ofextremal raysof a cone. I showthat quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have avery simple structure.As the set of all quantum states is very complicated, I look at a simple subset calledstabilizerstates. I improve on previously known results by showing that under a technical condition on the localdimension, entropies of stabilizer states respect an additional class of information inequalities that isvalid for random variables from linear codes.In a last approach I find a representation-theoretic formulation of the classical marginal problemsimplifying the comparison with its quantum mechanical counterpart. This novel correspondenceyields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf.Theory, 48(7):1992–1995, 2002) in purely combinatorial terms

On the Fundamental Limits and Symmetric Designs for Distributed Information Systems

Many multi-terminal communication networks, content delivery networks, cache networks, and distributed storage systems can be modeled as a broadcast network. An explicit characterization of the capacity region of the general network coding problem is one of the best known open problems in network information theory. A simple set of bounds that are often used in the literature to show that certain rate tuples are infeasible are based on the graph-theoretic notion of cut. The standard cut-set bounds, however, are known to be loose in general when there are multiple messages to be communicated in the network. This dissertation focuses on broadcast networks, for which the standard cut-set bounds are closely related to union as a specific set operation to combine different simple cuts of the network. A new set of explicit network coding bounds, which combine different simple cuts of the network via a variety of set operations (not just the union), are established via their connections to extremal inequalities for submodular functions. The tightness of these bounds are demonstrated via applications to combination networks. The tightness of generalized cut-set bounds has been further explored by studying the problem of “latency capacity region” for a broadcast channel. An implicit characterization of this region has been proved by Tian, where a rate splitting based scheme was shown to be optimal. However, the explicit characterization of this region was only available when the number of receivers are less than three. In this dissertation, a precise polyhedral description of this region for a symmetric broadcast channel with complete message set and arbitrary number of users has been established. It has been shown that a set of generalized cut-set bounds, characterizes the entire symmetrical multicast region. The achievability part is proved by showing that every maximum rate vector is feasible by using a successive encoding scheme. The framework for achievability strongly relies on polyhedral combinatorics and it can be useful in network information theory problems when a polyhedral description of a region is needed. Moreover, it is known that there is a direct relationship between network coding solution and characterization of entropy region. This dissertation, also studies the symmetric structures in network coding problems and their relation with symmetrical projections of entropy region and introduces new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. Finally, due to the relationship between linear entropic vectors and representability of integer polymatroids, construction of such vector has been discussed. Specifically, It is shown that representability of the particularly constructed matroid is a sufficient condition for integer polymatroids to be linearly representable over real numbers. Furthermore, it has been shown that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid

Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems

We prove a conjecture of J. Palis: any diffeomorphism of a compact manifold can be C1-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism having a transverse homoclinic intersection. ----- Cr'eation d'intersection homoclines : un mod`ele pour la dynamique centrale des syst`emes partiellement hyperboliques. Nous montrons une conjecture de J. Palis : tout diff'eomorphisme d'une vari'et'e compacte peut ^etre approch'e en topologie C1 par un diff'eomorphisme Morse-Smale ou par un diff'eomorphisme ayant une intersection homocline transverse

The holographic entropy arrangement

We develop a convenient framework for characterizing multipartite entanglement in composite systems, based on relations between entropies of various subsystems. This continues the program initiated in arXiv:1808.07871, of using holography to effectively recast the geometric problem into an algebraic one. We prove that, for an arbitrary number of parties, our procedure identifies a finite set of entropic information quantities that we conveniently represent geometrically in the form of an arrangement of hyperplanes. This leads us to define the holographic entropy arrangement, whose algebraic and combinatorial aspects we explore in detail. Using the framework, we derive three new information quantities for four parties, as well as a new infinite family for any number of parties. A natural construct from the arrangement is the holographic entropy polyhedron which captures holographic entropy inequalities describing the physically allowed region of entropy space. We illustrate how to obtain the polyhedron by winnowing down the arrangement through a sieve to pick out candidate sign-definite information quantities. Comparing the polyhedron with the holographic entropy cone, we find perfect agreement for 4 parties and corroborating evidence for the conjectured 5-party entropy cone. We work with explicit configurations in arbitrary (time-dependent) states leading to both simple derivations and an intuitive picture of the entanglement pattern.Comment: 100+epsilon page

Random Metric Spaces and Universality

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S_{\infty}-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so called anarchical uniformly distributed sequences. We give examples of other categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE