Related papers: State transitions as morphisms for complete lattic…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…
We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply…
We introduce the concept of deck transformations within the category of developable complexes of groups. Drawing inspiration from classical covering theory for topological spaces, we propose an alternative construction of the universal…
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we…
We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that…
In this work, we present several aspects of the interplay between classical and quantum theories. After reviewing the equivalence between positivity and complete positivity in the commutative setting, we introduce and analyze intermediate…
We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy…
We study the entanglement spectrum of a translationally-invariant lattice system under a random partition, implemented by choosing each site to be in one subsystem with probability $p\in[0, 1]$. We apply this random partitioning to a…
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a…
We consider an ``integral'' extension of the classical notion of affine connection providing a correspondence between paths in the manifold and diffeomorphisms of the manifold. These path-diffeomorphisms are a generalization of parallel…
We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of…
A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we demonstrate that the ground state of a topological phase itself encodes critical properties of its transition to a…
Using the signed laplacian matrix, and weighted and hybrid graphs, we present additional ways to interpret graphs as grid states. Hybrid graphs offer the most general interpretation. Existing graphical methods that characterize entanglement…
Conformal field theory has turned out to be a powerful tool to derive two-dimensional lattice models displaying fractional quantum Hall physics. So far most of the work has been for lattices with open boundary conditions in at least one of…
Morphisms are homomorphisms under the concatenation operation of the set of words over a finite set. Changing the elements of the finite set does not essentially change the morphism. We propose a way to select a unique representing member…
We propose a class of generalizations of the geometric entanglement for pure states by exploiting the matrix product state formalism. This generalization is completely divested from the notion of separability and can be freely tuned as a…
The use of the quantizer-dequantizer formalism to describe the evolution of a quantum system is reconsidered. We show that it is possible to embed a manifold in the space of quantum states of a given auxiliary system by means of an…
Lax extensions of set functors play a key role in various areas including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between…