Jochen,
Let's go through all details to eliminate any implied assumptions on either side. So, let us have a qubit in initial state,
[math]|\psi> = \frac{1}{\sqrt{2}} (|0>|1>)[/math]
This state is such that when lab measures qubit for 0 or 1 (basis defined by apparatus setup), Lab with qubit get either in 0 or 1 state with a choice probability 50%. Any measurement is an interaction, and, like any interaction is governed by postulate 3, producing
[math]|\psi>|L_{init}> = \frac{1}{\sqrt{2}} (|0>|1>)|L_{init}> \longrightarrow \bigcup_{i=0,1} |i>|L_i>[/math]
Notice how before interaction Lab saw qubit as a sum (quantum OR), and after it goes under OR. This classical OR for Lab is humanly subjectively experienced as getting one particular state of 0 or 1. The other state is not seen, is not there, for either Lab or qubit. (Notice that equation is symmetric.)
Let the Lab, for illustrative purpose, also be such that no interaction occurs between it and external world. Then above process can be described from world's perspective as
[math]|W>(|\psi>|L_{init}>) = |W>(\frac{1}{\sqrt{2}} (|0>|1>)|L_{init}>) \longrightarrow |W>(\frac{1}{\sqrt{2}} (|0>|L_0> |1>|L_1>))[/math]
right side of which I wrote in essay equation (6) as
[math]|W> (\bigcup_{i=0,1} |i>|L_i>) = |W> (|0>|L_0> \cap |1>|L_1>)[/math]
the |W> is outside of sign OR, which applies to systems that experienced it, which are Lab and a qubit. In essay, equations 5 and 6 differ by whether OR applies to |W>, or not. Equation 5 would allow putting |W> under OR, or, in other words, vector multiplication is distributative with respect to OR in case of equation 5, and is not distributative in case of 6, which nature actually prefers.
Additional interaction, this time with world (W) and Lab, will be
[math]|W>(\frac{1}{\sqrt{2}} (|0>|L_0> |1>|L_1>)) \longrightarrow \bigcup_{i=0,1} |W>|i>|L_i>[/math]
In interaction (info exchange), like with |L> in the first interaction, |W> jumps under the OR sign, and sees a classical OR, while if OR does not apply to it, it is seen as "quantum OR", and a plus is used used.
I specifically started to talk about qubit and lab, so that you appreciate how subjective experience of measurement arises when you are a participant of an interaction. The same thing, when seen from a non-interacting perspective, involves no loosing of probability, thus, unitary operation, initial pure states stay pure states. With this let's substitute |L> for a qubit. So, have
[math]|\psi_{A}> = \frac{1}{\sqrt{2}} (|0>|1>)[/math]
and
[math]|\psi_{B}> = |0>[/math]
push these through logic-or operation, which leaks no probability to outside, i.e. unitary
[math]|\psi_{A}>|\psi_{B}> \longrightarrow \frac{1}{\sqrt{2}} (|0>|0> |1>|1>)[/math]
And we see from non-interacting perspective pure states going into unitary operation come out as a pure state again. The density operator here shall have a trace of its square equal to 1. Any further operation will have initially this pure state. As it has been said in our previous thread, pure state turns to a mix, when it is seen from a position, to which OR operator is applied. In the case we stand on the perspective of non-interacting with a pair, and its pure state stays pure.
In 1920's, one particle was the fundamental system. Turning two particles into one system, could be seen as creating less-proper, not-fundamental system. Today, with long use of QFTs, and with existence postulates 1 and 2, we say that a system is a collection of fundamental particle events. So, each of two qubits is a collection of events. Two of these qubit taken together is still a collection of fundamental particle events. The composite system in our formulation is of the same nature as sub-systems, and a composite system cannot be seen as a lesser one, as it could be in 1920's reasoning frame.
So, a composite system of our qubits, after logic-or operation, is
[math]|\Psi_{AB}> = \frac{1}{\sqrt{2}} (|00>|11>)[/math]
then its density operator is
[math]\rho_{AB} = |\Psi_{AB}>