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

What probabilities tell about quantum systems, with application to entropy and entanglement