For properly normalized extensive macroscopic properties (and this includes the center of mass operator), there is such a proof in many treatises of statistical mechanics. It is the quantum analogue of the system size expansion for classical stochastic processes. For example, see Theorem 9.3.3 and the subsequent discussion in my online book. But you can find similar statements in all books on stochastic physics where correlations are discussed in a thermodynamic context if you care to look, though usually for different, thermodynamically relevant variables. [more on this here] Indeed. But without simplifying assumptions one can never do anything in physics. Successful science and hence successful physics lives from concentrating on the typical, not on the too exceptional. No physicist ever considers (except in thought experiments) a system where a single electron is in a superposition of being here and 1000 miles away. It is completely uninteresting from the point of view of applications. Everywhere in physics one makes approximations which (in view of the inherent nonlinearity and chaoticity of the dynamics of physical systems when expressed in observable terms) exclude situations that are too exceptional. This is the reason why randomness is introduced in classical physics, and it is the reason why randomness appears in quantum physics. It is a consequence of approximations necessary to be able to get useful and predictable results. It is the same with statistical mechanics. Predictiions in statistical mechanics exclude all phenomena that require special efforts to prepare. For example, irreversibility is the typical situation, and this is heavily exploited everywhere in physics. But by taking special care one can devise experiments such as spin echos where one can see that the irreversibity assumption can be unwarranted. Similarly, it takes a lot of effort to prepare experiments where nonlocal effects are convincingly demonstrated - the typical situation is that nonlocal correlations die out extremely fast and can be ignored. As everywhere in physics if you want to observe the untypical you need to make special efforts. These may be valuable but they don't take anything away from the fact that under usual circumstancs these effects do not occur. If you want to have statements that are valid without exceptions you need to do mathematics, not physics.Mathematical arguments do not allow exceptions (or make statements of their very low probability).