  The Jordan structure of finite-dimensional quantum theory is derived, in a
conspicuously easy way, from a few simple postulates concerning abstract
probabilistic models (each defined by a set of basic measurements and a convex
set of states). The key assumption is that each system A can be paired with an
isomorphic $\textit{conjugate}$ system, $\overline{A}$, by means of a
non-signaling bipartite state $\eta_A$ perfectly and uniformly correlating each
basic measurement on A with its counterpart on $\overline{A}$. In the case of a
quantum-mechanical system associated with a complex Hilbert space $\mathcal H$,
the conjugate system is that associated with the conjugate Hilbert space
$\overline{\mathcal H}$, and $\eta_A$ corresponds to the standard maximally
entangled EPR state on ${\mathcal H} \otimes \overline{\mathcal H}$. A second
ingredient is the notion of a $\textit{reversible filter}$, that is, a
probabilistically reversible process that independently attenuates the
sensitivity of detectors associated with a measurement. In addition to offering
more flexibility than most existing reconstructions of finite-dimensional
quantum theory, the approach taken here has the advantage of not relying on any
form of the "no restriction" hypothesis. That is, it is not assumed that
arbitrary effects are physically measurable, nor that arbitrary families of
physically measurable effects summing to the unit effect, represent physically
accessible observables. An appendix shows how a version of Hardy's "subspace
axiom" can replace several assumptions native to this paper, although at the
cost of disallowing superselection rules.
