8 research outputs found
The Bell states in noncommutative algebraic geometry
We introduce new mathematical aspects of the Bell states using matrix
factorizations, nonnoetherian singularities, and noncommutative blowups. A
matrix factorization of a polynomial consists of two matrices
such that .
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 nonnoetherian 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.Comment: 18 pages. Previously titled "Quantum entanglement, emergence, and
noncommutative blowups
Direct-sum behavior of modules over one-dimensional rings
Let R be a reduced, one-dimensional Noetherian local ring whose integral closure is finitely generated over R. Since is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated-modules are easily described, and every finitely generated-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules. © 2011 Springer Science+Business Media, LLC