The R-algebra of Quasiknowledge and Convex Optimization

TL;DR

This paper introduces a convex algebraic framework for representing and analyzing the states of knowledge of classical and quantum agents, generalizing semidefinite programs and offering new insights into information evolution and experimental design.

Contribution

It develops a convex description of knowledge states within a commutative R-algebra, generalizing quantum semidefinite programs to classical and faulty-quantum settings, and proposes a differential equation model for knowledge evolution.

Findings

Generalizes quantum semidefinite programs to classical and faulty-quantum contexts

Provides a formal differential equation for knowledge state evolution

Offers a new perspective on states of knowledge as convex subsets of R-algebras

Abstract

This article develops a convex description of a classical or quantum learner's or agent's state of knowledge about its environment, presented as a convex subset of a commutative R-algebra. With caveats, this leads to a generalization of certain semidefinite programs in quantum information (such as those describing the universal query algorithm dual to the quantum adversary bound, related to optimal learning or control of the environment) to the classical and faulty-quantum setting, which would not be possible with a naive description via joint probability distributions over environment and internal memory. More philosophically, it also makes an interpretation of the set of reduced density matrices as "states of knowledge" of an observer of its environment, related to these techniques, more explicit. As another example, I describe and solve a formal differential equation of states of knowledge in that algebra, where an agent obtains experimental data in a Poissonian process, and its state of knowledge evolves as an exponential power series. However, this framework currently lacks impressive applications, and I post it in part to solicit feedback and collaboration on those. In particular, it may be possible to develop it into a new framework for the design of experiments, e.g. the problem of finding maximally informative questions to ask human labelers or the environment in machine-learning problems. The parts of the article not related to quantum information don't assume knowledge of it.