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So, I am learning about quantum entanglement in my quantum course and had a thought. If the Big Bang is the creation of everything - time included, and this came from a single "source". Would it not follow that all particles in the universe are entangled to each other? And perhaps, we cannot observe this entanglement because we are "inside the system" sort to speak. Are we not all connected?

I know the last part is cheesy, but the point is serious.

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By far the most particles in the visible universe aren't quantum entangled. That's obvious from observation, since if all electron spins e.g. would be entangled, all electrons would flip their spin at the same time, and we couldn't observe varying statistics of electron spins in a sample, resulting in varying degrees of magnetism, e.g. by applying gradually changing exterior magnetic fields.

A second example: All (bosonic) atoms of spin 0 would form a Bose-Einstein condensate not just near absolute zero, since they would be in the same quantum state, hence in the lowest.

The "single source" isn't a single quantum state. In the early phase of the big bang temperatures were very high, allowing for a huge amount of possible quantum states within a small volume. Without warranty you may think of the initial state of the universe as a superposition of all possible quantum states, the wave functions of which collapse relative to an observer, regarding the observer as a particular quantum state.

That way we get to the many world interpretation of quantum theory and a universal wave function. See also a proposed Hartle-Hawking state as a pre-Planck epoch of the universe.

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If you try to consider some hypothetical state of all particles, that state is almost certainly entangled. However, that does not mean that you can't (say) treat two particles at random as very probably independent of each other--effectively unentangled.

The standard definition of an entangled state is a state that is not fully separable, i.e. one that is not probability mixture of product states. This is just the quantum version of the notion of "independent random variables" in more ordinary probability theory. Thus, for any compound system, almost all states are entangled, as the non-entangled ones are vanishly small (measure zero) subset of all possible states.

For example, any time you measure a particle with apparatus, after measurement the apparatus indicates something about the measured system. Thus, the joint state of the compound "apparatus + particle" system is not separable because its parts are not independent of one another. Measurement produces a particular case of entanglement between the measuring and the measured.

As a macroscopic system interacts with its environment, information about it diffuses into the environment, producing entanglement between parts of the system and parts of its environment. Overall, this leakage of information is responsible for quantum decoherence and the second law of thermodynamics. But its chaotic nature means that for all practical purposes, that information is lost, and we can think of entropy increase as 'missing information'.

However, if you just pick particles at random, chances overwhelming are that you can treat them as essentially independent--unentangled. Intuitively, one can think information about something becoming so mixed up in its environement in the large that any particular microscopic parts of it and its environment are going to tell you next to nothing about each other. In other words, they'd be almost entirely independent from each other--unentangled.

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