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Non-Commutative Graphs in the Fock Space over One-Particle Hilbert Space

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич

Abstract In the present paper we continue our study of non-commutative operator graphs in infinite-dimensional spaces. We consider examples of the non-commutative operator graphs generated by resolutions of identity corresponding to the Heisenberg–Weyl group of operators acting on the Fock space over one-particle state space. The problem of quantum error correction for such graphs is discussed.

arXiv:1912.12099Lobachevskii Journal of MathematicsApr 1, 20204 citations

On non-commutative operator graphs generated by covariant resolutions of identity

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич

arXiv:1809.08586Quantum Information ProcessingOct 19, 201814 citations

On the uniqueness of unitary representations of the non-commutative Heisenberg–Weyl algebra

Laure Gouba, Frederik G. Scholtz

In this paper, we discuss the uniqueness of the unitary representations of the noncommutative Heisenberg–Weyl algebra. We show that, apart from a critical line for the noncommutative position and momentum parameters, the Stone–von Neumann theorem still holds, which implies uniqueness of the unitary representation of the Heisenberg–Weyl algebra.

arXiv:0903.3816Canadian Journal of PhysicsSep 1, 20093 citations

On inner geometry of noncommutative operator graphs

Grigori Gennadievich Amosov

The European Physical Journal PlusOct 1, 20202 citations

Stochastic mechanics of graph rewriting

Nicolas Behr, Vincent Danos A, Ilias Garnier

We propose an algebraic approach to stochastic graph-rewriting which extends the classical construction of the Heisenberg-Weyl algebra and its canonical representation on the Fock space. Rules are seen as particular elements of an algebra of "diagrams": the diagram algebra D. Diagrams can be thought of as formal computational traces represented in partial time. They can be evaluated to normal diagrams (each corresponding to a rule) and generate an associative unital non-commutative algebra of rules: the rule algebra R. Evaluation becomes a morphism of unital associative algebras which maps general diagrams in D to normal ones in R. In this algebraic reformulation, usual distinctions between graph observables (real-valued maps on the set of graphs defined by counting subgraphs) and rules disappear. Instead, natural algebraic substructures of R arise: formal observables are seen as rules with equal left and right hand sides and form a commutative subalgebra, the ones counting subgraphs forming a sub-subalgebra of identity rules. Actual graph-rewriting is recovered as a canonical representation of the rule algebra as linear operators over the vector space generated by (isomorphism classes of) finite graphs. The construction of the representation is in close analogy with and subsumes the classical (multi-type bosonic) Fock space representation of the Heisenberg-Weyl algebra.We propose an algebraic approach to stochastic graph-rewriting which extends the classical construction of the Heisenberg-Weyl algebra and its canonical representation on the Fock space. Rules are seen as particular elements of an algebra of "diagrams": the diagram algebra D. Diagrams can be thought of as formal computational traces represented in partial time. They can be evaluated to normal diagrams (each corresponding to a rule) and generate an associative unital non-commutative algebra of rules: the rule algebra R. Evaluation becomes a morphism of unital associative algebras which maps general diagrams in D to normal ones in R. In this algebraic reformulation, usual distinctions between graph observables (real-valued maps on the set of graphs defined by counting subgraphs) and rules disappear. Instead, natural algebraic substructures of R arise: formal observables are seen as rules with equal left and right hand sides and form a commutative subalgebra, the ones counting subgraphs forming a sub-subalgebra of identity rules. Actual graph-rewriting is recovered as a canonical representation of the rule algebra as linear operators over the vector space generated by (isomorphism classes of) finite graphs. The construction of the representation is in close analogy with and subsumes the classical (multi-type bosonic) Fock space representation of the Heisenberg-Weyl algebra.

ACM/IEEE Symposium on Logic in Computer Science (LICS)Jul 5, 201630 citations

On Construction of Anticliques for Noncommutative Operator Graphs

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич

arXiv:1709.08062Journal of Mathematical SciencesAug 24, 20189 citations

On Non-Commutative Operator Graphs Generated by Reducible Unitary Representation of the Heisenberg-Weyl Group

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