Some 2-Categorical Aspects in Physics

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description of the Gelfand-Naimark-Segal construction as a canonical procedure from states on C*-algebras to representations of C*-algebras equipped with pure state

Reversing information flow: retrodiction in semicartesian categories

In statistical inference, retrodiction is the act of inferring potential causes in the past based on knowledge of the effects in the present and the dynamics leading to the present. Retrodiction is applicable even when the dynamics is not reversible, and it agrees with the reverse dynamics when it exists, so that retrodiction may be viewed as an extension of inversion, i.e., time-reversal. Recently, an axiomatic definition of retrodiction has been made in a way that is applicable to both classical and quantum probability using ideas from category theory. Almost simultaneously, a framework for information flow in in terms of semicartesian categories has been proposed in the setting of categorical probability theory. Here, we formulate a general definition of retrodiction to add to the information flow axioms in semicartesian categories, thus providing an abstract framework for retrodiction beyond classical and quantum probability theory. More precisely, we extend Bayesian inference, and more generally Jeffrey's probability kinematics, to arbitrary semicartesian categories.Comment: 20.5 pages + references, some diagram

Gauge invariant surface holonomy and monopoles

There are few known computable examples of non-abelian surface holonomy. In this paper, we give several examples whose structure 2-groups are covering 2-groups and show that the surface holonomies can be computed via a simple formula in terms of paths of 1-dimensional holonomies inspired by earlier work of Chan Hong-Mo and Tsou Sheung Tsun on magnetic monopoles. As a consequence of our work and that of Schreiber and Waldorf, this formula gives a rigorous meaning to non-abelian magnetic flux for magnetic monopoles. In the process, we discuss gauge covariance of surface holonomies for spheres for any 2-group, therefore generalizing the notion of the reduced group introduced by Schreiber and Waldorf. Using these ideas, we also prove that magnetic monopoles have an abelian group structure.Comment: 99 pages, 31 figures (2 are new), v2 is published version, updates include: several points clarified, added Defn 2.33 and 3.37 for markings, statement of smoothness removed from Thm 2.39 and 3.41, proof of Thm 4.13 corrected, proof of Lem 3.46 has been enhanced, appendix on 2-categories removed, index of notation adde

From time-reversal symmetry to quantum Bayes' rules

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.Comment: Some adjustments and organizational changes, typos fixed, new tables added; 24 pages tota

On dynamical measures of quantum information

In this work, we use the theory of quantum states over time to define an entropy $S(\rho,\mathcal{E})$ associated with quantum processes $(\rho,\mathcal{E})$, where $\rho$ is a state and $\mathcal{E}$ is a quantum channel responsible for the dynamical evolution of $\rho$. The entropy $S(\rho,\mathcal{E})$ is a generalization of the von Neumann entropy in the sense that $S(\rho,\mathrm{id})=S(\rho)$ (where $\mathrm{id}$ denotes the identity channel), and is a dynamical analogue of the quantum joint entropy for bipartite states. Such an entropy is then used to define dynamical formulations of the quantum conditional entropy and quantum mutual information, and we show such information measures satisfy many desirable properties, such as a quantum entropic Bayes' rule. We also use our entropy function to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes.Comment: Comments welcome