Entanglement, space-time and the Mayer-Vietoris theorem

Abstract

Entanglement appears to be a fundamental building block of quantum gravity leading to new principles underlying the nature of quantum space-time. One such principle is the ER-EPR duality. While supported by our present intuition, a proof is far from obvious. In this article I present a first step towards such a proof, originating in what is known to algebraic topologists as the Mayer-Vietoris theorem. The main result of this work is the re-interpretation of the various morphisms arising when the Mayer-Vietoris theorem is used to assemble a torus-like topology from more basic subspaces on the torus in terms of quantum information theory resulting in a quantum entangler gate (Hadamard and c-NOT).

Multipartite entanglement via the Mayer-Vietoris theorem

“…Therefore any quantum state defined with respect to a space-time having such a topology and being spread over the two ends of the ER-bridge must be entangled. That quantum states can be represented as cycles over topological spaces and can be classified by (co)homology has been shown in [5] and I will not insist on the proof here. I only wish to add that in general, quantum states may also be represented by means of higher-dimensional cycles over topological spaces, while keeping the same topological properties and algebraic operations that are valid for 1-dimensional cycles.…”

“…It is worth mentioning that entanglement arises from the maps that include subspaces into the larger topological spaces with non-trivial global structure. As shown in [5], a qubit can be represented by means of a worldline. Any qubit can be written as a superposition of states by means of the Hadamard matrix.…”

“…[5] that for a twisted torus the map (i * , j * ) connecting the homology of the intersection of the two sheets A and B to the sum of the two distinct homology groups of the sheets A and B has the precise form…”

“…I showed in ref. [5] that a bipartite entangled state |ψ = 1 2 (|ψ 1 ⊗ |ψ 2 + |ψ 1 ⊗ |ψ 2 ) is induced by the inclusion maps arising in the construction of the Mayer-Vietoris theorem for the (co)homology of a 2-dimensional genus-one torus. This construction represented a preliminary mathematical proof for the ER-EPR duality [6].…”

A Field-Theoretical Approach to the P vs. NP Problem via the Phase Sign of Quantum Monte Carlo

“…This structure is capable of globally avoiding the sign problem and of reducing the complexity not by simply adding new fields, but by adding new fields such that a topologically non-trivial structure emerges that has the role of a topological anti-anomaly, namely an additional region of the manifold over which the paths will cancel precisely the terms that generate the sign problem on the original, unextended field manifold. The non-detectability of auxiliary topological structures has been discussed in [33,34].…”