Non-entangling environmental noise in QM/QFT

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Does non-entangling environmental noise actually exist? The existence of such a thing seems to be fundamentally inconsistent with the properties of quantum entanglement.

It seems to me that if non-entangling environmental noise does exist, then standard QM - or QFT for that matter - automatically cannot be seen as a fundamental theory anymore.
 

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Environmental noise, e.g. some stochastic process within some stochastic differential equation, which only interacts with quantum systems in such a manner such that it cannot ever become entangled with the system that it is interacting with. In other words, the source of such non-entangling noise would be an intrinsically non-quantum effect occurring in nature.

The existence of such a noise source would automatically imply that quantum theory - in any of its currently accepted incarnations or formulations - is necessarily incapable of actually being a fundamental theory of physics.

NB: I'm not an experimentalist, which is why I'm asking, but this is as much an empirical question as it is a mathematical question.