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

Abstract

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.