  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.
