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Quantum entanglement, emergence, and noncommutative blowups

Charlie Beil

Published Oct 21, 2013• 1 citation
OpenAlex

Abstract

We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial $p$ consists of two matrices $\phi_1,\phi_2$ such that $\phi_1\phi_2 = \phi_2\phi_1 = p \operatorname{id}$. 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 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.

Related papers

The Bell states in noncommutative algebraic geometry

Charlie Beil

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.

arXiv:1310.5673•International Journal of Quantum Information•Aug 1, 2014•4 citations

Complete Factorization of Entangled Quantum States PART 1: Factorizing the Bell States Using Semi-Structured Complex Numbers

Peter Jean‐Paul, Shanaz Wahid

Abstract Tracking non-local correlations through complex quantum circuits presents a massive computational bottleneck, as standard linear algebra forces multi-qubit state simulations to scale exponentially (\(\:{2}^{n}\)). This paper introduces a novel intermediate algebraic framework using semi-structured complex numbers to overcome non-separability constraints during intermediate tracking. By strategically mapping vanishing Euclidean coefficients to non-zero auxiliary forms (\(\:-pk\)), an unphysical scaffolding layer is introduced that permits the factorization of maximally entangled multiqubit systems into independent, single-qubit tensor product states. Note: \(\:p\) is the semi-structured complex unit and \(\:k\:\)is an auxiliary variable. To validate the mathematical consistency and probability conservation of this extended state space, the bipartite Bell states are completely factorized and subjected to localized operator matrices, demonstrating strict compliance with statistical scalar product rules. Finally, an algebraic inverse transformation cleanly annihilates the auxiliary parameters. This process projects the system back to the physical domain, perfectly recovering standard Euclidean quantum states with zero loss of physical data. This framework serves as a practical analytical tool for quantum compiler optimization, allowing localized qubit tracking without altering fundamental quantum mechanics.

•Aug 27, 2026•0 citations

Entanglement beyond tensor product structure: algebraic aspects of quantum non-separability

Łukasz Derkacz, Marek Gwóźdź, Lech Jakóbczyk

An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.

arXiv:1109.0818•Journal of Physics A Mathematical and Theoretical•Dec 8, 2011•12 citations

Bell's Theorem and Non-Commutative Measurement Algebras: A Topological Resolution via S³ × I Geometry

MYKHAYLO CHORNOMORCHENKO

Bell’s theorem is a foundational result in quantum theory, demonstrating that no local hidden-variable model satisfying certain assumptions can reproduce quantum correlations. While mathematically correct within its formal framework, the theorem implicitly relies on a specific representational choice: measurement outcomes are modeled as scalar, commutative variables (±1). This assumption, though natural for flat and static configurations, is not mandated by physical necessity. In this work, we examine the consequences of relaxing this representational constraint when measurement outcomes arise as projections of states with nontrivial topology. We show that when physical correlations are naturally organized as elements of an S³-valued structure — such as those associated with spin and rotational degrees of freedom — the induced measurement algebra becomes noncommutative. Under these conditions, the key algebraic factorizations employed in Bell-type derivations no longer apply. We construct an explicit local, realistic model in which observable correlations reproduce the quantum expectation E(a, b) = − cos θ and violate the CHSH bound S ≤ 2, without invoking non-local interactions or superluminal influences. The effective S³×I structure arises not as a fundamental substrate, but as the natural geometric form taken by the actualization of physical processes within an underlying relational framework. Importantly, this structure was not introduced to address quantum entanglement specifically; it emerged independently in the study of cosmological expansion, emergent gravity, and mass generation. Its relevance to Bell-type correlations follows as a consequence of geometry, not as an ad hoc modification. These results suggest that quantum entanglement may reflect topological features of physical realization rather than intrinsic non-locality. Possible experimental implications and connections to geometric and substrate-based approaches to fundamental physics

•Feb 9, 2026•0 citations

On Entangled Markov Chains (Non-Commutative Analysis and Micro-Macro Duality)

Luigi Accardi, Takashi Matsuoka, Masanori Ohya

Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of $[1|$ , were $inti^{\vee}oduced$ by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems.In [3], using our entanglement criterion for pure states [16] which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra $||generically^{I\dagger}$ are entangled.On the other hand we showed that this entanglement condition for pure state is sufficient condition for entanglement in the case of mixtures [4].This fact was then applied to prove that EMC with unitarily im- plementable transition operator induce a mixture entangled state on any local algebra.Interestingly it was also shown that all these local states provide a new class of examples (in any dimension) of entangled states which nevertheless satisfy the PPT condition [4].In this article we report the above results.

•Jul 1, 2008•0 citations

Noncommutative Mathematics for Quantum Systems

Uwe Franz, Adam Skalski

Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.

Cambridge University Press eBooks•Dec 18, 2015•6 citations
We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial ppp consists of two matrices ϕ1,ϕ2\phi_1,\phi_2ϕ1​,ϕ2​ such that ϕ1ϕ2=ϕ2ϕ1=pid⁡\phi_1\phi_2 = \phi_2\phi_1 = p \operatorname{id}ϕ1​ϕ2​=ϕ2​ϕ1​=pid. 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.
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.
Abstract Tracking non-local correlations through complex quantum circuits presents a massive computational bottleneck, as standard linear algebra forces multi-qubit state simulations to scale exponentially ( 2n\:{2}^{n}2n). This paper introduces a novel intermediate algebraic framework using semi-structured complex numbers to overcome non-separability constraints during intermediate tracking. By strategically mapping vanishing Euclidean coefficients to non-zero auxiliary forms ( −pk\:-pk−pk), an unphysical scaffolding layer is introduced that permits the factorization of maximally entangled multiqubit systems into independent, single-qubit tensor product states. Note:  p\:pp is the semi-structured complex unit and  k \:k\:kis an auxiliary variable. To validate the mathematical consistency and probability conservation of this extended state space, the bipartite Bell states are completely factorized and subjected to localized operator matrices, demonstrating strict compliance with statistical scalar product rules. Finally, an algebraic inverse transformation cleanly annihilates the auxiliary parameters. This process projects the system back to the physical domain, perfectly recovering standard Euclidean quantum states with zero loss of physical data. This framework serves as a practical analytical tool for quantum compiler optimization, allowing localized qubit tracking without altering fundamental quantum mechanics.
An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.
Bell’s theorem is a foundational result in quantum theory, demonstrating that no local hidden-variable model satisfying certain assumptions can reproduce quantum correlations. While mathematically correct within its formal framework, the theorem implicitly relies on a specific representational choice: measurement outcomes are modeled as scalar, commutative variables (±1). This assumption, though natural for flat and static configurations, is not mandated by physical necessity. In this work, we examine the consequences of relaxing this representational constraint when measurement outcomes arise as projections of states with nontrivial topology. We show that when physical correlations are naturally organized as elements of an S³-valued structure — such as those associated with spin and rotational degrees of freedom — the induced measurement algebra becomes noncommutative. Under these conditions, the key algebraic factorizations employed in Bell-type derivations no longer apply. We construct an explicit local, realistic model in which observable correlations reproduce the quantum expectation E(a, b) = − cos θ and violate the CHSH bound S ≤ 2, without invoking non-local interactions or superluminal influences. The effective S³×I structure arises not as a fundamental substrate, but as the natural geometric form taken by the actualization of physical processes within an underlying relational framework. Importantly, this structure was not introduced to address quantum entanglement specifically; it emerged independently in the study of cosmological expansion, emergent gravity, and mass generation. Its relevance to Bell-type correlations follows as a consequence of geometry, not as an ad hoc modification. These results suggest that quantum entanglement may reflect topological features of physical realization rather than intrinsic non-locality. Possible experimental implications and connections to geometric and substrate-based approaches to fundamental physics
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of [1∣[1|[1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems.In [3], using our entanglement criterion for pure states [16] which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.On the other hand we showed that this entanglement condition for pure state is sufficient condition for entanglement in the case of mixtures [4].This fact was then applied to prove that EMC with unitarily im- plementable transition operator induce a mixture entangled state on any local algebra.Interestingly it was also shown that all these local states provide a new class of examples (in any dimension) of entangled states which nevertheless satisfy the PPT condition [4].In this article we report the above results.
Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.