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If the Universe did start from a single point, then wouldn't all particles be fundamentally entangled? How then could there be a truly "pure" state?

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Your question can be rephrased to: "How can there be truly isolated systems?" So the answer is the same :-) – Siva Apr 20 '14 at 14:39
up vote 3 down vote accepted

Pure states are a convenient abstraction for studying tiny, specially-prepared quantum systems. States of complicated systems are never pure.

One makes (very small) systems pure by careful preparation. For example, a simple spin by means of a Stern-Gerlach magnet and a screen where only the up particles can pass. More complex systems need more complex preparation to make them (at least approximately) pure.

Note also that pure states can be entangled; indeed, entanglement is usually defined only for pure states. Thus there is no conflict between entanglement and pureness.

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