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Among the tags there is also a topic of interpretation, so I hope that the question will not be closed. In the many-worlds interpretation, the wave function acquires an onotological meaning, that is, it becomes physically real. And since the wave function of the Universe is also considered in the cosmological context, does this mean that in this interpretation the Universe is a single quantum object (the same as, for example, an electron)?

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Yes. Much like an entangled pair of photons should be treated as a single system whose wavefunction is a Bell state, the same ultimately applies to the universe. It is a system where everything is entangled and tracing out all degrees of freedom except maybe a handful of particles you're interested in is going to be an approximation. Whether it's a good or bad approximation depends on how much decoherence has taken place.

A related concept is called the church of the larger Hilbert space which is essentially the statement that all states are pure once you look at enough degrees of freedom. Mixed states with $\rho \neq \left | \psi \right > \left < \psi \right |$ are an effect of looking at only part of an entangled system. The simplest example of this is again a Bell state which acts as a purifier for a single photon which appears to be in a classical admixture of horizontal and vertical polarizations.

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