Characterizing quantum state-space with a single quantum measurement

Can the state-space of $d$-dimensional quantum theory be derived from studying the behavior of a single ``reference'' measuring device? The answer is yes, if the measuring device corresponds to a complex-projective 3-design. In this privileged case, not only does each quantum state correspond to a probability distribution over the outcomes of a single measurement, but also the probability distributions which correspond to quantum states can be elegantly characterized as those which respect a generalized uncertainty principle. The latter takes the form of a lower-bound on the variance of a natural class of observables as measured by the reference. We give simple equations which pure-state probability distributions must satisfy, and contextualize these results by showing how 3-designs allow the structure-coefficients of the Jordan algebra of observables to be extracted from the probabilities which characterize the reference measurement itself. This lends credence to the view that quantum theory ought to be primarily understood as a set of normative constraints on probability assignments which reflect nature's lack of hidden variables and further cements the significance of 3-designs in quantum information science.