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Is that true that the big bang caused the quantum entanglement of all the particles of the universe so every particle is entangled to each other particle of the universe?

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No. In the proper Big Bang cosmology, the beginning of the world is a spacelike singularity – a horizontal "wiggly" line in the Penrose causal diagram – so quite the opposite to your claim holds: the Big Bang doesn't allow any correlations between particles in different regions to start with. A correlation or entanglement must always be a result of the subsystems' contact in the past – of their common ancestry, if you wish – and there are no points in the Big Bang spacetime that would belong to the intersection of the past light cones of two points at the beginning which makes any correlations and entanglement impossible.

Indeed, the cosmic microwave background suggests that different regions were correlated – they seem to have the same temperature, to say the least – and cosmic inflation is the major solution to this puzzle. Inflation extends the Penrose diagram in the temporal direction and allows the particles to communicate in the past and get entangled and correlated. But cosmic inflation is "beyond the Big Bang theory".

I also want to stress that independently of these "causal technicalities" in cosmology, there is a deeply misleading suggestion in your question. You seem to say that particles are "objectively entangled" and they remain "objectively entangled" forever. But that's not the case. Entanglement is just a correlation in the predicted properties that will be measured, expressed in the most general quantum way. Once we actually perform the measurement of at least one subsystem, the entanglement disappears.

For example, if the initial state is a (maximally entangled) singlet state of two spins $$\frac{1}{\sqrt{2}} \left( |\rm up\rangle |\rm down\rangle - |\rm down\rangle |\rm up\rangle \right), $$ the measurement of the first spin yields either "up" or "down" which means that our knowledge of the spins is changed to either "up down" or "down up" (that's the "collapse" of the wave function). The states "up down" and "down up" describe just factual values of two spins that are perfectly known and there is no longer any reason to talk about correlations or entanglements when we know the actual values.

It only makes sense to talk about correlations or entanglements of particular particles before they are actually measured. Because the properties of particles in the Universe have already been "measured" gazillion of times, the hypothetical entanglement created by the Big Bang – more precisely, by inflation, as discussed at the beginning – has been washed away many times and has almost no detectable traces anymore.

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