A quote from Landau & Lifshitz (Quantum Mechanics - Non-relativistic Theory, §2):
"Let us consider a system composed of two parts, and suppose that the state of this system is given in such a way that each of its parts is completely described. Then we can say that the probabilities of the coordinates q_1 of the first part are independent of the probabilities of the coordinates q_2 of the second part, and therefore the probability distribution for the whole system should be equal to the product of the probabilities of its parts."
How does independence of the two systems follow from the fact that they are both completely described? What does it mean exactly? Does independence mean no interaction?
Update: No interaction is not assumed, as Martin pointed out in his answer. I added a screenshot added below, yellow text corresponds to the above quote. Also, what is meant by complete description is of course the following:
"Completely described states occur as a result of the simultaneous measurement of a complete set of physical quantities."
Update 2: Ján Lalinský mentioned the density matrix description. I think this another quotation might be crucial for understanding of the above problem: