Early in Chapter 2 of Ballentine's Quantum Mechanics, he gives what I will call Statement 1:
The empirical content of a probability statement is revealed only in the relative frequencies in a sequence of events that result from the same (or an equivalent) state preparation procedure. Thus, although the primary definition of a state is the abstract set of probabilities for the various observables, it is also possible to associate a state with an ensemble of similarly prepared systems. However, it is important to remember that this ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space.
One page later, he gives what I will call Statement 2:
The quantum state description may be taken to refer to an ensemble of similarly prepared systems.
My questions are as follows:
(1) What exactly is Ballentine's conception of an ensemble? From Statement 1, it seems that an ensemble is the conceptual and unbounded set of systems prepared by equivalent state preparation procedures, where earlier in Ballentine it was noted that a state preparation procedure is "any repeatable process that yields well-defined probabilities for all observables". Also, why does Ballentine insist on using words like "similarly" and "may potentially result" if (if my understanding of state preparation procedure is correct) a state preparation procedure results in the exact same distribution of probabilities for every conceivable observable for the given (in general conceptual) system?
(2) This is the key/more meaningful question I think. If my understanding of an ensemble in (1) is correct, then what advantage is there in considering a state as an ensemble (per Statement 2) -- which is an unbounded set of systems each with the same probability distributions for measurement of a given observable -- over a state as simply the set of probability distributions for measurement of a given observable (this is like a set vs. a set of sets). The "ensemble" interpretation seems only to hint at not being able to say things about an individual particle, but at best it seems to be a redundant recapitulation of the latter notion (of a state as simply the set of probability distributions for measurement of a given observable).