Quantum entanglement is a well observed but not well
understood phenomena.  The frontier in
this area has been to entangle systems at greater and greater distances.  Theoretically however it is poorly
understood.  Susskind and Maldacena proposed
the ER=EPR conjecture, which to oversimplify, states that entangled particles
are connected by tiny wormholes(Maldacena and Susskind)  In this brief blog post I present a simple
proof that the “non-locality” that experimentalist write of, and Susskind
conjectured about solving via wormholes, can be explained with standard quantum
mechanics and standard relativity.   What
is new here is how we look at the spaces involved.

The precise nature of the spaces involved is the key to understanding
the phenomena of entanglement.   In short
Alice and Bob start out with separate states that they entangle. Creating one
state in a common Hilbert space which is the tensor product of the original
states.   When Bob travels to his lab he
takes his state with him.  Mathematically
and physically this can be thought of as a series of general transformations and Lorentz transformations as he accelerates, and travels at a
steady velocity, then comes to rest in his labs frame of reference. 

Here is where the misconception comes in.  To an experimentalist eye Alice and Bob now
occupy distinctly separate spaces several light seconds or light minutes apart.
The correlation between Alice’s observations of her state and Bobs observations
of his state appear to violate special relativity.  However, if we sit and think about the math
and do the math that’s not the case at all. 
Even without invoking wormhole solutions to Einstein’s equations this
can be shown.     Consider the figure.

Alice and Bob prepare quantum states that they entangle.  Alice’s state is at spacetime point X and bobs
is at space time Point Y and they are separated by an invariant interval S.  After a series of transformations Bob arrives
at his lab via path S’ and observes “non-locality” because of the misconception
that quantum information would only have the single classical path available
for him and Alice to observe.  Like, neighborhoods
on opposite sides of a city connected by a 16 lane expressway  can be more “local” than ones a mile or two
apart with congested one way streets between them.

Bob takes his part of the entangled state and travels to his
office.  Mathematically and physically this
can be thought of as a series of transformations which increase the observed
invariant interval from S between Alice’s state located at X and Bob’s state
now located at Y’ an apparent interval S’ away.

My full handwritten notes that lead me to this are at this
link. 

The key misunderstanding comes from thinking that S’ the
apparent invariant interval between X and Y’, the path Bob himself took as he transported
his state is the same path that information must necessarily take from X to Y’.  

Quantum information also has available to it
the shortcut S.  S is an invariant
interval as such it would not change under transformation.  The X Y Z and t components may change but S
itself is by construction and definition invariant.    So, to the quantum particles and to quantum
information S is just as available as S’. 

In fact, since S is a shorter path more information would likely pass by
S than by S’.    Yet it does not violate
special relativity at all and does not need a wormhole to accomplish this
feat.   The answer lies in combining Quantum
Mechanics with Special Relativity. 
Putting this in full algebraic quantum field theory terms would make it neater
but not that much more clear.  To see
what that looks like my earlier papers on relativization address that (Farmer).

So why don’t Alice and Bob observe the same shorter interval
S as their quantum states do?   

Why don't we see these "wormholes" or shorter intervals ourselves? 

That question is answered in detail by the second
reference below (Sperling and Walmsley).  For this blog, I will shorten it and anyone
who wants more detail really should read  (Sperling and Walmsley).  The entanglement we’d see would be a
statistical phenomenon manifesting in the quantum states of their constituent particles.  Furthermore, in a macroscopic system like a
person the constituent particles are constantly interacting with each other
forming new stronger entanglements.   

Without veering into metaphysics and paranormal ideas where
people who are friends, family, or lovers have a “sixth sense” about the state
of a physically distant person, statistics about large ensembles of particles
are all we can discuss. 

References

 BIBLIOGRAPHY Farmer, Hontas. n.d. "Fundamentals of
Relativization." The Winnower. doi:
10.15200/winn.141487.76774.

Maldacena, J., and L. Susskind. n.d. "Cool
horizons for entangled black holes." Fortschritte der Physik.
doi:10.1002/prop.201300020♠.

Sperling, J., and I. A. Walmsley. n.d.
"Entanglement in macroscopic systems." Physical Review A.
doi: 10.1103/PhysRevA.95.062116.