This is a purely speculative question:

Has there been any work that describes non-locality/entanglement in QM by using exotic topologies in configuration space?

The 'conceptual' picture that I have in mind is of two particles that are distant in the usual topology, but when entangled, being somehow part of the same 'irreducible' open set - so they can't be 'disentangled'. For this to have any chance of working, topologies would have to change with time.

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    $\begingroup$ Entanglement by itself concerns no spatial separation. People can talk about entanglement between two qubits without referring to where the two qubits are located. As long as you can a tensor product of two Hilbert spaces, you can define entanglement for pure states (at least). But it may be worth pointing out that people have thought about relating entanglement to the geometry or topology of the Hilbert space of few qubits. $\endgroup$ – Isidore Seville Feb 6 '14 at 2:10
  • $\begingroup$ Point taken. I was thinking more along the lines of the usual EPR paradox where widely spatially separated but entangled systems have non-local correlations. $\endgroup$ – Mozibur Ullah Feb 6 '14 at 2:22
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    $\begingroup$ The word "non-local correlations" should be avoided. Locality has to do with information or energy signal "sent" at some space-time point and "received" at an other space-time point, and this cannot be done instantaneously. Correlations, on the other way, are what they are. In any probabilistic system (classical or quantum), it is always possible that 2 subsystems of a systems locally get correlations,then, after, it is possible that these 2 subsystems may evolve into spatially distant locations, anyway, the correlations remain the same, because they concern internal state of the subsystems. $\endgroup$ – Trimok Feb 6 '14 at 11:35
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    $\begingroup$ Fun question. Two comments: 1) Actually quantum theory is local in configuration space. It is only non-local in real space, so I guess you should phrase your question in that space. 2) Holographic duality is a recent development that suggests deep links between entanglement and geometry/topology. For example there is the idea by Maldacena and Susskind that if you have two entangled particles, that they are in fact connected by a wormhole, related to your intuition (if you want to google more: ``ER = EPR''). [cont] $\endgroup$ – Ruben Verresen Feb 14 '16 at 2:46
  • $\begingroup$ [cont] More generally dualities like AdS/CFT suggest that geometry (i.e. the metric) is in fact an emergent description due to underlying entanglement structures. $\endgroup$ – Ruben Verresen Feb 14 '16 at 2:46

I think this is approach is the correct one. Here is from a paper I have written:

The primary insight of this is that if we modify our concept of ‘local’, much of the strangeness may disappear. Specifically, consider that 4-dimensional space-time is a construct that is projected onto an underlying topology that I will call ‘true space’. The fundamental idea is that photons are points, not lines, in true space. Measurements of entangled photons represent observations about the same point in true space. Only when we stretch true space to fit our notions on space-time does it appear that the measurements are separated (and therefore, require some kind of non-local action or correlation).

A simplistic version of true space consists of two elements: points and causal (i.e., directional) connections between the points (which may be thought of as corresponding to matter and photons). An object, or an observer, consists of a world-line of points connected by connectors. A topology on the space can be constructed by taking two points (a and b)and looking at the forward and backward causal cones emanating from those points. An open set consists of the points that follow a and precede b.

A notion of distance can be constructed if we assume that some world-lines have clocks of some sort. For example, suppose point a is located on the world-line of observer A, and point b is located on observer B, but that a and b are also directly connected. Now imagine that b is also directly connected to point c, which is on the world line of observer A. The connections a-b and b-c are two individual elements in true space. But observer A can measure the time between point a and point c. Observer A posits that the distance between A and C is proportional to the time between point a and point c.

Thus, the distance between one world-line and another is defined (somewhat fuzzily) by the round trip time that an observer measures when interacting with another observer (i.e., by the triangle a-b-c). However, the key insight is that in the underlying topology of true space, point a and point b are not distant points. They are directly connected by connection a-b. There is no exact distance between points in space time, only fuzzy distance between world-lines. Also, there are no empty points in true space, but when true space is projected onto space-time, not all points in space time need be occupied and the connections (e.g., connection a-b) occupy either occupy no space at all (within a world-line) or they occupy whole lines themselves that ‘travel’ at the speed of light between world-lines. The resulting space-time is consistent with special relativity.

The EPR paradox can be stated in terms of three observational locations. That is, suppose observer A is connected to both observer B and observer X via connections a-b and a-x. In the underlying topology of true space, points b and x are very close indeed, even though observers B and X may be separated by an arbitrary distance in space-time. Thus, there is no non-local interaction or even non-local correlation because entangled photons are all local in true space.


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