## Three ways to classicalize (nearly) any probabilistic theory

Alexander Wilce Susquehanna University

It is commonplace that quantum theory can be viewed as a non-classical" probability calculus. This observation has inspired the study of more general non-classical probabilistic theories modeled on QM, the so-called generalized probabilistic theories or GPTs. However, the boundary between these putatively non-classical probabilistic theories and classical probability theory is somewhat blurry, and perhaps even conventional. This is because, as is well known, any probabilistic model can be understood in classical terms if we are willing to embrace some form of contextuality.

## Quantum axiomatics à la carte

Alexander Wilce Susquehanna University
The past decade or so has produced a handful of derivations, or reconstructions, of finite-dimensional quantum mechanics from various packages of operational and/or information-theoretic principles. I will present a selection of these principles --- including symmetry postulates, dilational assumptions, and versions of Hardy's subspace axiom --- in a common framework, and indicate several ways, some familiar and some new, in which these can be combined to yield either standard complex QM (with or without SSRs) or broader theories embracing formally real Jordan algebras.

## Symmetry, Self-Duality and the Jordan Structure of Quantum Theory

Alexander Wilce Susquehanna University
This talk reviews recent and on-going work, much of it joint with Howard Barnum, on the origins of the Jordan-algebraic structure of finite-dimensional quantum theory. I begin by describing a simple recipe for constructing highly symmetrical probabilistic models, and discuss the ordered linear spaces generated by such models. I then consider the situation of a probabilistic theory consisting of a symmetric monoidal *-category of finite-dimensional such models: in this context, the state and effect cones are self-dual.

## Foundations and Interpretation of Quantum Theory - Lecture 11

Alexander Wilce Susquehanna University
After a review of the axiomatic formulation of quantum theory, the generalized operational structure of the theory will be introduced (including POVM measurements, sequential measurements, and CP maps). There will be an introduction to the orthodox (sometimes called Copenhagen) interpretation of quantum mechanics and the historical problems/issues/debates regarding that interpretation, in particular, the measurement problem and the EPR paradox, and a discussion of contemporary views on these topics.

## Foundations and Interpretation of Quantum Theory - Lecture 10

Alexander Wilce Susquehanna University
After a review of the axiomatic formulation of quantum theory, the generalized operational structure of the theory will be introduced (including POVM measurements, sequential measurements, and CP maps). There will be an introduction to the orthodox (sometimes called Copenhagen) interpretation of quantum mechanics and the historical problems/issues/debates regarding that interpretation, in particular, the measurement problem and the EPR paradox, and a discussion of contemporary views on these topics.

## Four and a Half Axioms for Quantum Mechanics

Alexander Wilce Susquehanna University
I will discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. These require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent up to the action of a compact group of symmetries, and that every state be the marginal of a bipartite state perfectly correlating two measurements.

## Entanglement and measurement in general probabilistic theories

Alexander Wilce Susquehanna University
Quantum mechanics is a non-classical probability theory, but hardly the most general one imaginable: any compact convex set can serve as the state space for an abstract probabilistic model (classical models corresponding to simplices). From this altitude, one sees that many phenomena commonly regarded as characteristically quantum' are in fact generically non-classical'. In this talk, I'll show that almost any non-classical probabilistic theory shares with quantum mechanics a notion of entanglement and, with this, a version of the so-called measurement problem.

## Probability theory --classical, quantum and otherwise

Alexander Wilce Susquehanna University
Quantum mechanics is a non-classical probability calculus -- but hardly the most general one imaginable. In this talk, I'll discuss some familiar non-classical properties of quantum-probabilistic models that turn out to be features of {em all} non-classical models. These include a generic no-cloning theorem obtained in recent work with Howard Barnum, Jon Barrett and Matt Leifer.

## Symmetry and quantum logic

Alexander Wilce Susquehanna University

## Introduction to and Historical Overview of Quantum Logics

Alexander Wilce Susquehanna University
Introductory lecture summary: 1. Finite dimensional hilbert spaces and (complemented) modular lattices; infinite-dimensional hilbert spaces and orthomodularity.