146 From Classical Mechanics to Quantum Field Theory. A Tutorial
(b) Measuring instruments are commonly employed toprepare a system in
a certain pure state. Theoretically-speaking the preparation of apurestate is
carried out like this. A finite collection of compatiblepropositionsP 1 ,...,Pn
is chosen so that the projection subspace of P 1 ∧···∧Pn =P 1 ···Pnis one-
dimensional.InotherwordsP 1 ···Pn=〈ψ, 〉ψfor some vector with||ψ||=1.
The existence of such propositions is seen in practically all quantum systems used
in experiments. (From a theoretical point of view these areatomicpropositions)
Then propositions Piare simultaneously measured on several identical copies of
the physical system of concern (e.g., electrons), whose initial states, though, are
unknown. If for one system the measurements of all propositions are successful,
the post-measurement state is determined by the vectorψ, and the system was
preparedin that particular pure state.
Normally each projectorPibelongs to the PVMP(A)of an observableAiwhose
spectrum is made of isolated points (thus a pure point spectrum) andPi=P{(λAi)}
withλi∈σp(Ai).
(c)Let us finally explain how to practically obtain non-pure states from pure
ones. Considerq 1 identical copies of systemSprepared in the pure state associated
toψ 1 ,q 2 copies ofSprepared in the pure state associated toψ 2 and so on, up toψn.
If we mix these states each one will be in the non-pure state:ρ=
∑n
i=1pi〈ψi,〉ψi,
where pi:=qi/
∑n
i=1qi. In general,〈ψi,ψj〉is not zero ifi=j, so the above
expression forρis not the decomposition with respect to an eigenvector basis for
ρ. This procedure hints at the existence of two different types of probability, one
intrinsic and due to the quantum nature of stateψi, the other epistemic, and
encoded in the probabilitypi. But this is not true: once a non-pure state has been
created, as above, there is no way, within QM, to distinguish the states forming
the mixture. For example, the sameρcould have been obtained mixing other pure
states than those determined by theψi. In particular, one could have used those
in the decomposition ofρinto a basis of its eigenvectors. For physics, no kind of
measurement would distinguish the two mixtures.
Another delicate point is that, dealing with mixed states, definitions (2.64) and
(2.66) for, respectively the expectation value〈A〉ψ and the standard deviation
ΔAψof an observableAreferred to the pure state〈ψ, 〉ψwith||ψ||=1areno
longer valid. We just say that extended natural definitions can be stated referring
to the probability measure associated to both the mixed stateρ∈B 1 (H)(with
ρ≥0andtrρ= 1) and the observable,
μ(ρA):B(R)E→tr(ρPE(A)).We refer the reader to[5; 6]for a technical discussion on these topics.