The Bell states in noncommutative algebraic geometry

Charlie R Beil

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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, ϕ2 such 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.
Original languageEnglish
Article number1450033
Number of pages18
JournalInternational Journal of Quantum Information
Issue number5
Early online date31 Oct 2014
Publication statusPublished - 2014


  • Entanglement; Bell state; nonlocality; emergence; non-noetherian ring; matrix factorization; noncommutative blowup; quantum foundations; quantum information; noncommutative algebraic geometry

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