I don't know, how
@A. Neumaier can misunderstand what I wrote in my notes on statistical physics. As he rightfully says, it's in accordance with the standard interpretation, and that's my intention: I don't see any problems with the standard interpretation (which for me is the minimal statistical interpretation).
A system's state is as completely determined as possible according to QT if it is prepared in a pure state. If there is incomplete knowledge about the system one has to describe it with a mixed state, and the problem is, how to choose this mixed state, according to the knowledge about the system at hand, and one objective way is to argue with information theory and the maximum-entropy principle.
I don't see, where there is a contradiction to what I wrote in one of my today's earlier postings. There I explained the well-known standard procedure, if you want to describe a part of a larger system. The answer in all textbooks I know is that you take the partial trace.
Nothing at all contradicts the statements in my manuscript: If you have a big quantum system, this big quantum system can well be completely prepared, i.e., prepared in a pure state and then the part of the system you describe by tracing out the other part(s) of the system according to this rule, is usually in a mixed state. Of course, tracing out the non-wanted part of the big system and describing only one part means to ignore the rest of the system. This means of course that you lose information, and thus the partial system is not in a pure state. Why should it be? The reduced density matrix is the correct choice based on the knowledge we have in this case, which is that the big system is prepared in some pure state but that I choose to ignore parts of the system and only look at one part, of which we have only partial information and thus describe it by a mixed state.
Take the Bohm's spin-1/2 example, the preparation of a spin-1/2 pair in the singlet state (total spin ##S=0##). Then the pair is in the pure state
$$\hat{\rho}=|\Psi \rangle \langle \Psi| \quad \text{with} \quad |\Psi \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle - |-1/2,1/2 \rangle.$$
Tracing out particle 2, i.e., only looking at particle 1, leads to the state for particle 1,
$$\hat{\rho}_1= \mathrm{Tr}_2 \hat{\rho} = \frac{1}{2} (|1/2 \rangle \langle 1/2| + |-1/2 \rangle \langle -1/2|)=\frac{1}{2} \hat{1},$$
i.e., to the state of maximum entropy.
The same of course holds for the reduced stat. op. of particle 2, which is described by
$$\hat{\rho}_2=\mathrm{Tr}_1 \hat{\rho}=\frac{1}{2} \hat{1}.$$
This is in full accordance with my statistics script and (as you claim) with Landau and Lifshitz (I guess, you refer to vol. III, which I consider as one of the better QT books, with somewhat too much overemphasis of wave mechanics, but that's a matter of taste; physicswise it's amazingly up to date given its date of publication; one has just to ignore the usual collapse-hypothesis argument of older QM textbooks ;-))):
You have to distinguish precisely who describes which system and how to associate the statistical operators with the various systems. For the above example you have the following:
(1) An observer Alice, who only measures the spin of particle 1 (you disginguish particle 1 and particle 2 simply by where they are measured; I don't want to make the example to complicated and ignore the spatial part, which however is important when it comes to identical particles in this example). What she shall measure are simply completely unpolarized particles and thus her stat. op. for the spin state is that of maximal entropy, which is ##\hat{\rho}_1## with the maximal possible entropy for a spin 1/2-spin component, ##S_1=\ln 2##.
(2) An observer Bob, who only measures the spin of particle 2. What he shall measure
are simply completely unpolarized particles and thus her stat. op. for the spin state is that of maximal entropy, which is ##\hat{\rho}_2## with the maximal possible entropy for a spin 1/2-spin component, ##S_2=\ln 2##.
(3) Observer Cecil, who knows that the particle pair was produced through the decay of a scalar particle at rest and thus its total spin is ##s=0##. He describes the state of the complete system (consisting of two spins here) by the pure state ##\hat{\rho}##, and thus his knowledge is complete and accoringly the entropy is ##S=0##.
He is the one who knows, without even knowing the measurement results of A and B, that there's a 100% correlation of the two measured spins, namely if A finds ##+1/2##, B must necessarily find ##-1/2## and vice versa. That's independent of the temporal order A and B measure there respective spin and thus there's no causal "action at a distance" of eithers spin measurement on the others particle.
All three description of the situation are thus (a) consistent, (b) there's no non-local action at a distance caused by the local measurement processes of A's and B's spin, (c) there's no contradiction to the statement that A's and B's knowledge prior to their measurement is less complete compared to C's. In this case it's even taken to the extreme that C's knowledge is even complete, i.e., he associates the entropy 0 ("no missing information") to his knowledge, while A and B have the least possible information, and that's what they also will figure out when doing their spin measurements.
This example shows that there are no contradictions within minimally interpreted QT nor between Einstein causality and QT.
The fact that a part of a bigger system prepared completely is not prepared completely, by the way, was Einstein's true quibble with QT, not what's written in this (in)famous EPR paper, which Einstein himself didn't like much, being quite unhappy with Podolsky's formulations when writing it up. He called this feature of quantum theory "inseparability", and that's what's the real profound physical value of this debate: It triggered Bell to develop his famous inequality valid for all local deterministic hidden-variable models and to the empirical conclusion that all these are wrong but QT is right, and Einstein's quibble, the inseparability, is an empirically validated fact.
