Motivated by a recent proof of free choices in linking equations to the experiments they describe, I clarify some relations among purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and operator-valued measures), thereby allowing applications of these entities to the modeling of a wider variety of physical situations. I relate conditional probabilities associated with projection-valued measures to conditional density operators identical, in some cases but not in others, to the usual reduced density operators. While a fatal obstacle precludes associating conditional density operators with general non-projective measures, tensor products of general positive operator-valued measures (POVMs) are associated with conditional density operators. This association together with the free choice of probe particles allows a postulate of state reductions to be replaced by a theorem. An application shows an equivalence between one form of quantum key distribution and another with respect to certain eavesdropping attacks.
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Myers, J.M. Conditional Probabilities and Density Operators in Quantum Modeling. Found Phys 36, 1012–1035 (2006). https://doi.org/10.1007/s10701-006-9053-0
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DOI: https://doi.org/10.1007/s10701-006-9053-0