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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).

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  • Not sure - it's a bit difficult to parse. However I think this question should go into the physics stack exchange :-)
    – Frank
    Jan 13, 2023 at 4:06
  • QM has lots of connections and inspirations from SM where ensemble is a key concept introduced by Gibbs as a frequentist interpretation for probability: an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system... Jan 13, 2023 at 4:11
  • @DoubleKnot That's well taken; that's the sense in which I thought it was being used (as e.g. Reif does in his Fundamentals of Statistical Mechanics). That answers Q1 of mine, thank you.
    – EE18
    Jan 13, 2023 at 4:43
  • @Frank I did put it there, but I thought Q2 was well-inclined towards philosophy SE, insofar as it asks about the distinction between two concepts which seem to the layperson (like me) to be the same thing (one being -- I think -- a redundant recapitulation of the other).
    – EE18
    Jan 13, 2023 at 4:45
  • 1
    For your Q2 even for a non-free quantum particle in a (infinite) potential well, to probe its observables such as its position you need to apply their corresponding Hermitian operator to its complex-valued (usually position-based) wavefunction whose product with its complex conjugate is nothing but the probability density function of its state (position), the observables are the guaranteed real-valued eigenvalues of the said operator and form a discrete spectrum in such scenario like an electron near a hydrogen atom. Thus ensemble is useful to conceive quantum state of probabilistic nature... Jan 13, 2023 at 5:53

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Comment (but too long for comments):
    Ballentine's whole conception of "state" is based on information acquirable by reproducible experiments. And that information-based interpretation is pretty much the standard Bohr interpretation. But, as I'm sure you already know, there are many various interpretations of quantum mechanics. And Ballentine's 2nd edition is from 1998, way before the often-cited (908 times as per https://scholar.google.com/citations?user=XeqvafYAAAAJ) Pusey-Barrett 2012 paper, https://arxiv.org/abs/1111.3328 (and also see, e.g., https://en.wikipedia.org/wiki/PBR_theorem), where...
    "we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory".

So I think maybe your "(2)the key/more meaningful question" is just elaborating some of your own confusions about Ballentine's epistemic state=information interpretation. But that interpretation can't be completely correct, i.e., can't be everything about states, which necessarily possess an ontic reality beyond experimentally measurable information. So maybe just broaden your scope beyond Ballentine, whereby you'll just get even more confused. But now your confusion will be in a broader context: nobody has yet come up with a completely satisfactory notion of "state".

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