In direct answer to the question. As I said in my answer of 24th December which seems to have been deleted.
As a quantum system is prepared [by way of the apparatus configuration] to make a prepared state, its density operator evolves deterministically, resulting in a density operator determined by its preparation history. So, prior to measurement, the density operator is deterministic.
But critically — for mixed states — the density operator [so prepepared, deterministic definite] does not have a unique history. Hence, when a measurement asks the question: what history caused this density matrix? the answer is an ambiguous one.
When I say this density operator does not have a unique history, I mean that a different history would have resulted in the same definite density operator. So this definite density operator can't tell us which of these histories it originated from. In the case of photon polarisation experiments, those different histories are different configurations of polarisers.
ADDENDA:
Physics is about discovering rules that the Natural World obeys. So it's about the relationship between rules and what Nature does.
To gain insight into Quantum Indeterminacy it's instructive to consider the relationship between rules and what we witness in the Motion of the Planets. We might say the planets move according to rules layed down by Newton; or maybe, according to rules of General Relativity. The crucial point to notice is that these rules not only prescribe the unique path taken by a planet; they also deny every other path. We might say that the motion of a planet is caused by these rules, AND that every other motion is prevented by them.
But this is not how the Quantum World is. The rule followed by a quantum system as it evolves is the density matirx, or density operator. This conveys information about the full set of complimentary variables as they evolve. For instance, in the free particle, the density matrix conveys information about both position and momentum. Unlike the rules obeyed by planets, there are "paths" available to the quantum system which are not caused by the density matrix, but neither are they prevented.
And so in considering physical processes we must account for all of the following: "caused paths", "prevented paths", and "paths which are neither caused nor prevented". Mathematically these correspond to logical consequences [comprising proofs and disproofs] deriving from Principles, Postulates or Axioms; AND or PLUS, logical consistencies which are not in contradiction with those same Principles, Postulates or Axioms.
And so, looking at physics generally from this 'rules' point of view, there exist both constraints and freedoms.
Notice that this explanation is not in some new Physical Law, Principle or Postulate; the rules remain the same. No; the explanation is in the processesing of those rules.
It's not realistic to expect an anwer here that fully answers the question, And the answer is a mathematical one because the processes in Nature entail the conveyance of mathematical information.
I cannot do better than to refer readers to the experiments done in Vienna, of Tomasz Paterek et al, published under the title: Logical Independence and Quantum Randomness. And then my own book which follows up in detail on the meaning of the Paterek results. [I do not include here, its title or a link to it's website, because that would attract the deletion of this entry.]
Quote:
In the early 20th century Physics suffered two crises: first, Special Relativity and then Quantum Mechanics. Then, from 1930 onward, Mathematics suffered its own shattering crisis, after [Kurt] Gödel announced his First Incompleteness Theorem. Its consequence today is that there are statements in Applied Mathematics that are true but not provable. One such statement concerns existence of the square root of minus one. Knowing precisely what drives necessity for this number's presence in Quantum Theory resolves the question of quantum indeterminacy.
