University of New Mexico
Talks by Howard Barnum
Much progress has recently been made on the fine-grained thermodynamics and statistical mechanics of microscopic physical systems, by conceiving of thermodynamics as a resource theory: one which governs which transitions between states are possible using specified "thermodynamic" (e.g. adiabatic or isothermal) means. In this talk we lay some groundwork for investigating thermodynamics in generalized probabilistic theories.
The talk will focus primarily on recent work with Alexander Wilce in which we show that any locally tomographic composite of a qubit with any finite-dimensional homogeneous self-dual (equivalently Jordan-algebraic) system must be a standard finite-dimensional quantum (i.e. $C^*$-algebraic) system. I may touch on work in progress with collaborators on composites of arbitrary homogeneous self-dual systems.
I will consider various attempts to derive the quantum probabilities from the HIlbert space formalism within the many-worlds interpretation, and argue that they either fail, or depend on tacit probabilistic assumptions. The main problem with the project is that it is difficult to understand what the state of system X is psi even *means* without already supposing some probabilistic link to definite observed or observable phenomena involving X.
The normalized-state spaces of finite-dimensional Jordan algebras constitute a relatively narrow class of convex sets that includes the finite-dimensional quantum mechanical and classical state spaces. Several beautiful mathematical characterizations of Jordan statespaces exist, notably Koecher's characterization as the bases of homogeneous self-dual cones, and Alfsen and Shultz's characterization based on the notion of spectral convex sets plus additional axioms.
The question whether SICs exist can be viewed as a question about the structure of the convex set of quantum measurements, or turned into one about quantum states, asserting that they must have a high degree of symmetry. I\'ll address Chris Fuchs\' contrast of a \'probability first\' view of the issue with a \'generalized probabilistic theories\' view of it. I\'ll review some of what\'s known about the structure of convex state and measurement spaces with symmetries of a similar flavor, including the quantum one, and speculate on connections to recent SIC triple product results.
Information Processing in Convex Operational Theories: Toward a characterization of quantum mechanics
The rise of quantum information science has been paralleled by the development of a vigorous research program aimed at obtaining an informational characterization or reconstruction of the quantum formalism, in a broad framework for stochastic theories that encompasses quantum and classical theory, but also a wide variety of other theories that can serve as foils to them.