The Bell states in noncommutative algebraic geometry

Charlie Beil

doi:10.1142/s0219749914500336

Abstract

We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2such that ϕ1ϕ2= ϕ2ϕ1= p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

Highlights

Quantum entanglement is one of the most beautiful and mysterious aspects of quantum theory, with well established experimental conrmation
We study the simplest form of entanglement: the Bell states (Ref. 12)
We introduce the notion that the spacetime nonlocality inherent in an entangled pair of particles emerges from an underlying local geometry which is noncommutative

Summary

Introduction

Quantum entanglement is one of the most beautiful and mysterious aspects of quantum theory, with well established experimental conrmation (notably Refs. 1{3). This is an Open Access article published by World Scientic Publishing Company. We introduce the following diagram to relate the matrix factorization of the Bell state É to the noncommutative blowup A of R: Here n 2 RðAÞ is the evaluation map at a point n in spacetime, cg species the summand ordering. Our second main result is the following, which shows how the representation theory of the noncommutative blowup A characterizes the collapse of the Bell states. We will use the term \local" in the physics sense (e.g. a wavefunction is nonlocal if it contains space-like separated points in its support), rather than in the algebraic sense (a ring is local if it contains a unique maximal ideal). 2 H~ 1 MnðCÞ H~ 2 separable if it can be written as a product 1⁄4 1 2 with i 2 H~i, and entangled otherwise

From Pure Entangled to Mixed Separable via Matrix Factorizations

Preliminaries

A new characterization of nonlocality

Collapse from the representation theory of a noncommutative blowup