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Can entanglement occur in one to many relationship, or many to many? If so, are there entangled particles occurring across the universe that have persisted since the big bang, or does entanglement decay and become a local phenomenon as entropy increases?

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What is usually called entanglement is by definition a 1:1 relationship. And it is not a relationship between particles, because there are no particles. In quantum theory there are mathematical objects like vectors and linear operators. First of all there is a space where the math. objects live in, the Hilbert space. Depending on the world you want to model (and how accurate you want to do this) the Hilbert space has more or less dimensions. A quantum model of classical particles usually will have an infinite number of dimensions because you want to model continuous particle positions, i.e. any vector in the space is in general not a sum but an integral of continuous (="Dirac") base vectors. Simpler spaces have a finite number of dimensions. The most simple space has 2 dimensions and is also called "qubit".

Entanglement is a property of

  1. a vector (a state vector with length 1) in combination with
  2. a split of the Hilbert space where the vector lives.

A Hilbert space can be split into subspaces if its dimension is infinite or finite and not a prime number. This is because if the subspaces have dimensions $n_1$ and $n_2$ the combined space is a product space with dimension $n_1 \cdot n_2$. A qubit has a dimension of 2 (a prime number) and therefore it is the smallest Hilbert space and cannot be split any more, and no 1-qubit-vector can appear entangled in any view.

If a Hilbert space can be split in 2 parts, there are always an infinite number of ways how you can do this. When combining 2 smaller spaces you build a larger product space, and the original 2 parts are lost in some sense: You can split the product space into the original parts, but you can split it into an infinite number of equivalent pairs, too.

Depending on how you split, a given vector will show more or less entanglement. A natural measure of the entanglement is the von Neumann entropy.

The answer to the 2nd question is: probably yes. There can be planes that split the Hilbert space and across these planes there has always been entanglement since the big bang though the world state vector changed all the time. There may be other split planes that had no entanglement at the beginning and now have significant entanglement, and also the other way around.

Last question: this is now my personal opinion... I see no room for classical entropy in a quantum world. I.e. every classical entropy must be explained as entanglement entropy somehow. So if entropy grows (at the split plane you defined), entanglement will grow, too.

Whether entropy of the universe grows is an open question and related to the arrow of time which is currently not understood. For a discussion of this difficult topic see The Physical Basis of The Direction of Time Authors: Zeh, H. Dieter or Quanta Magazine.

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    $\begingroup$ "What is usually called entanglement is by definition a 1:1 relationship" - that's just wrong. Multipartite entanglement is a perfectly standard concept, best exemplified by the tripartite entanglement of GHZ states, which have zero pairwise entanglement but maximal tripartite entanglement. $\endgroup$ – Emilio Pisanty Aug 15 '18 at 19:54
  • $\begingroup$ A GHZ state is not a special state at all and therefore not worth an own expression. What you name "GHZ state" is a ordinary state vector like all others together with the subjective notion that you split the Hilbert space into 3 parts. I.e. you define 2 split planes (that do not intersect) and therefore get 3 parts. Of course you may define mathematical functions (entanglement measures) on more than 1 split positions if you like. The question is what you learn from them that you can't already learn from 1:1 entanglement measures. $\endgroup$ – Harald Rieder Aug 15 '18 at 20:03
  • $\begingroup$ And the usual meaning of entanglement is 1:1 entanglement. If you want to talk about a GHZ split, you explicitly must say that. Sorry, I don't understand at all why you voted down. $\endgroup$ – Harald Rieder Aug 15 '18 at 20:09
  • $\begingroup$ I didn't vote down at all. I voted you up, actually. Sorry for the delay, I was locked out of my account. I have to take some time to consider your answer. $\endgroup$ – Joe Nov 29 '18 at 20:58
  • $\begingroup$ Yes, I think Emilio Pisanty voted down and my further comments were addressed to him. The belief of the mainstream is, that enganglement is objective, but see here. $\endgroup$ – Harald Rieder Nov 30 '18 at 7:00

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