Bratteli diagrams where random orders are imperfect

For the simple Bratteli diagrams B where there is a single edge connecting any two vertices in consecutive levels, we show that a random order has uncountably many infinite paths if and only if the growth rate of the level-n vertex sets is super-linear. This gives us the dichotomy: a random order on a slowly growing Bratteli diagram admits a homeomorphism, while a random order on a quickly growing Bratteli diagram does not. We also show that for a large family of infinite rank Bratteli diagrams B, a random order on B does not admit a continuous Vershik map

Perfect orderings on finite rank Bratteli diagrams

Given a Bratteli diagram B, we study the set OB of all possible orderings on B and its subset PB consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set OB with product measure µ and prove that there is some 1 ≤ j ≤ k such that µalmost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then µ-almost all orderings are imperfect

First-order corrections to the mean-field limit and quantum walks with non-orthogonal position states

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Progress in Group Field Theory and Related Quantum Gravity Formalisms

Following the fundamental insights from quantum mechanics and general relativity, geometry itself should have a quantum description; the search for a complete understanding of this description is what drives the field of quantum gravity. Group field theory is an ambitious framework in which theories of quantum geometry are formulated, incorporating successful ideas from the fields of matrix models, ten-sor models, spin foam models and loop quantum gravity, as well as from the broader areas of quantum field theory and mathematical physics. This special issue collects recent work in group field theory and these related approaches, as well as other neighbouring fields (e.g., cosmology, quantum information and quantum foundations, statistical physics) to the extent that these are directly relevant to quantum gravity research

RECOVERY AND RECONSTRUCTION IN QUANTUM SYSTEMS

Quantum systems are prone to noises. Accordingly, many techniques are developed tocancel the action of a quantum operation, or to protect the quantum information against the noises. In this dissertation, I discuss two such schemes, namely the recovery channel and the quantum error correction, and various scenarios in which they are applied. The first scenario is the perfect recovery in the Gaussian fermionic systems. When therelative entropy between two states remains unchanged under a channel, the perfect recovery can be achieved. It is realized by the Petz recovery map. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. This scenario is based on the work [1]. The second scenario is the approximate recovery in the context of quantum field theory.When perfect recovery is not achievable, the existence of a universal approximate recovery channel is proven. The approximation is in the sense that the fidelity between the recovered state and the original state is lower bounded by the change of the relative entropy under the quantum channel. This result is a generalization of previous results that applied to type-I von Neumann algebras in [2]. To deal with quantum field theory, the type of the von Neumann algebras is not restrained here. This induces qualitatively new features and requires extra proving techniques. This results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda Lp norms. This part is based on the work [3]. The third scenario is applying quantum error correction codes on tensor networks on hyperbolicplanes. This kind of model is proposed to be toy models of the AdS/CFT duality, thus also dubbed holographic tensor networks. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility, we provide guidelines for constructing networks consisting of multiple types of tensors which are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. An explicit construction of such networks is also provided. It approximates the holographic HaPPY pentagon code when variational parameters are taken to be small. This part is based on the work [4]. Supplementary materials and technical details are collected in the appendices

Dynamical systems via domains:Toward a unified foundation of symbolic and non-symbolic computation

Non-symbolic computation (as, e.g., in biological and artificial neural networks) is astonishingly good at learning and processing noisy real-world data. However, it lacks the kind of understanding we have of symbolic computation (as, e.g., specified by programming languages). Just like symbolic computation, also non-symbolic computation needs a semantics—or behavior description—to achieve structural understanding. Domain theory has provided this for symbolic computation, and this thesis is about extending it to non-symbolic computation. Symbolic and non-symbolic computation can be described in a unified framework as state-discrete and state-continuous dynamical systems, respectively. So we need a semantics for dynamical systems: assigning to a dynamical system a domain—i.e., a certain mathematical structure—describing the system’s behavior. In part 1 of the thesis, we provide this domain-theoretic semantics for the ‘symbolic’ state-discrete systems (i.e., labeled transition systems). And in part 2, we do this for the ‘non-symbolic’ state-continuous systems (known from ergodic theory). This is a proper semantics in that the constructions form functors (in the sense of category theory) and, once appropriately formulated, even adjunctions and, stronger yet, equivalences. In part 3, we explore how this semantics relates the two types of computation. It suggests that non-symbolic computation is the limit of symbolic computation (in the ‘profinite’ sense). Conversely, if the system’s behavior is fairly stable, it may be described as realizing symbolic computation (here the concepts of ergodicity and algorithmic randomness are useful). However, the underlying concept of stability is limited by a no-go result due to a novel interpretation of Fitch’s paradox. This also has implications for AI-safety and, more generally, suggests fruitful applications of philosophical tools in the non-symbolic computation of modern AI