  As described quantum mechanically, an experimental trial parses into "a
preparation" expressed by a density operator and "a measurement" expressed by a
set of detection operators, one for each measurable event. A density operator
and a detection operator combine via the "trace rule" to generate the
probability of a measurable event. As used to describe experiments, both
density operators and detection operators depend on parameters expressing
experimental choices, so the probabilities they generate also depend on these
parameters. The trace rule answers the question: "what parametrized
probabilities are generated by a given parametrized density operator and given
parametrized detection operator?"
  Recognizing that the accessibility of operators to experimental tests is only
indirect, via probabilities, we ask what probabilities tell about operators,
or, put more precisely: "what combinations of a parametrized density operator
and parametrized detection operators generate any given set of parametrized
probabilities?"
  We show that any parametrized probability measure can be explained by many
inequivalent models expressed by density operators and detection operators. so
that a parametrized probability measure, detached from any of the (infinitely
many) parametrized operators that generate it, becomes an interesting object in
its own right. By detaching a parametrized probability measure from the
operators that may have led us to it, we (1) strengthen Holevo's bound on a
quantum communication channel and (2) clarify a role for multiple levels of
modeling in an example based on quantum key distribution. We then inquire into
some parametrized probability measures generated by entangled states and into
the topology of the associated parameter spaces; in particular we display some
previously overlooked topological features of level sets of these probability
measures.
