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your words “You may be asking whether in place of one (or more) of the properties at a particular time one can use a pre-probability rho, and my immediate response is that I don’t see the physical motivation for it, why it would make sense. The classical analog would be that in some stochastic process in place of a state j at some time, to which the process will assign a probability, you want instead to employ a probability distribution” properly describe my question.
There are a few reasons backing my question.
First, as you say in your reply, mixed states (ensembles) are often regarded on the equal footing as the pure states, even for single systems (single elements of an ensemble). This attitude seems unavoidable in Quantum Information/Computation.
Second, mixed states (density matrices) are simply a part of the formalism e.g. in the open systems theory and at least formally should be considered — i understand your position and reluctance, but mathematical completeness has always been important to me.
Finally, there is a physical motivation that you may not be interested in — it can be found in ‘Other topics’ of this forum. To this end, it may be better, IF you find it relevant, to continue our discussion via email; if so, please let me know.