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From topology, via quantum entanglement, to arithmetic.
by
Bob Coecke
Oxford University
Coauthors: Aleks Kissinger and Alex Merry
(0) There is a canonical correspondence between certain symmetric states (i.e. vectors in C^n (x) C^n (x) C^n, or mor abstractly, morphisms of type I -> A (x) A (x) A in soem monoidal category) and arbitrary commutative Frobenius algebras CFA) on C^n (or A).
(1) In particular, for n=2 there are only two kinds of CFAs and they precisely correspond with the two kinds of non-trivial tripartite entanglement that exist in nature, so-called GHZ and W entanglement. The distinction between the CFAs is that the loop is either connected or disconnected.
(2) When passing from the Bloch sphere representation of a qubit to the stereographic projection, the CFAs are in fact nothing but multiplication and addition. In other words, up to `local transformations' all CFAs on C^2 are either addition or multiplication.
(3) Putting (1) and (2) together we obtain a topological interpretation of addition and multiplication, merely in terms of connectedness. We expect that this fact must have been observed in other contexts and are eager to learn about this.
Paper reference: In part based on arXiv:1002.2540 and arXiv:1103.2812
Date received: April 17, 2011
Copyright © 2011 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbcc-13.