Object-based unawareness: Axioms

This paper provides foundations for a model of unawareness, called object-based unawareness (OBU) structures, that can be used to distinguish between what an agent is unaware of and what she simply does not know. At an informal level, this distinction plays a key role in a number of papers such as Tirole (2009) and Chung & Fortnow (2016). In this paper, we give the model-theoretic description of OBU structures by showing how they assign truth conditions to every sentence of the formal language used. We then prove a model-theoretic sound and completeness theorem, which characterizes OBU structures in terms of a system of axioms. We then verify that agents in OBU structures do not violate any of the introspection axioms that are generally considered to be necessary conditions for a plausible notion of unawareness. Applications are provided in our companion paper.

Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC

We bring together in one place some of the main results and applications from our recent work on quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time to investigate the topic of distinguishability of sets of quantum states in quantum communication, with particular reference to the framework of one-way local quantum operations and classical communication (LOCC). We also derive a new graph-theoretic description of distinguishability in the case of a single-qubit sender.

De Rham–Witt sheaves via algebraic cycles

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

Leveraging Stochasticity for Open Loop and Model Predictive Control of Spatio-Temporal Systems

Stochastic spatio-temporal processes are prevalent across domains ranging from the modeling of plasma, turbulence in fluids to the wave function of quantum systems. This letter studies a measure-theoretic description of such systems by describing them as evolutionary processes on Hilbert spaces, and in doing so, derives a framework for spatio-temporal manipulation from fundamental thermodynamic principles. This approach yields a variational optimization framework for controlling stochastic fields. The resulting scheme is applicable to a wide class of spatio-temporal processes and can be used for optimizing parameterized control policies. Our simulated experiments explore the application of two forms of this approach on four stochastic spatio-temporal processes, with results that suggest new perspectives and directions for studying stochastic control problems for spatio-temporal systems.

Sandpile Models in the Large

This contribution is a review of the deep and powerful connection between the large-scale properties of critical systems and their description in terms of a field theory. Although largely applicable to many other models, the details of this connection are illustrated in the class of two-dimensional Abelian sandpile models. Bulk and boundary height variables, spanning tree–related observables, boundary conditions, and dissipation are all discussed in this context and found to have a proper match in the field theoretic description.

Reconstructing quantum theory from diagrammatic postulates

A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.

Group-Assign: Type Theoretic Framework for Human AI Orchestration

In today’s Information Age, we work under the constant drive to be more productive. Unsurprisingly, we progress towards being an AI-augmented workforce where we are augmented by AI assistants and collaborate with each other (and their AI assistants) at scale. In the context of humans, a human language suffices to describe and orchestrate our intents (and corresponding actions) with others. This, however, is clearly insufficient in the context of humans and machines. To achieve this, communication across a network of different humans and machines is crucial. With this objective, our research scope covers and presents a type theoretic framework and language built upon type theory (a branch of symbolic logic in mathematics), to enable the collaboration within a network of humans and AI assistants. While the idea of human-machine or human-computer collaboration is not new, to the best of our knowledge, we are one of the first to propose the use of type theory to orchestrate and describe human-machine collaboration. In our proposed work, we define a fundamental set of type theoretic rules and abstract functions Group and Assign to achieve the type theoretic description, composition and orchestration of intents and implementations for an AI-augmented workforce.

Introduction to Chern-Simons Theory

The 2 + 1 Yang-Mills theory allows for an interaction term called the Chern-Simons term. This topological term plays a useful role in understanding the field theoretic description of the excitation of the quantum hall system such as Anyons. While solving the non-Abelian Chern-simons theory is rather complicated, its knotty world allows for a framework for solving it. In the framework, the idea was to relate physical observables with the Jones polynomials. In this note, I will summarize the basic idea leading up to this framework.