Quantum Permutation Matrices
TL;DR: In this paper , the authors define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces, and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.
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Abstract: Abstract Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.
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Citations
Quantum isomorphism of graphs from association schemes
Ada Chan,William J. Martin +1 more
- 10 Sep 2022
TL;DR: In this article , it was shown that any two Hadamard graphs on the same number of vertices are quantum isomorphic, and the main result is built from three tools.
Quantum isomorphism of graphs from association schemes
Ada Chan,William J. Martin +1 more
TL;DR: Quantum isomorphism of graphs from association schemes is shown for Hadamard graphs and other graphs arising from association schemes.
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Q A ] 1 8 A pr 2 02 3 A CONCRETE MODEL FOR THE QUANTUM PERMUTATION GROUP ON 4 POINTS NICOLAS FAROSS
TL;DR: In 2019, Jung-Weber gave an example of a concrete magic unitary M , which defines a C∗-algebraic model of the quantum permutation group S 4 , and showed that there exist no polynomials up to degree 50 separating the entries of M from the generators of C(S 4 ).
Advances in quantum permutation groups
Amaury Freslon
TL;DR: Quantum permutation groups and subgroups are surveyed with a focus on quantum information, dynamics, and probability theory.
References
Compact matrix pseudogroups
TL;DR: The compact matrix pseudogroup as mentioned in this paper is a non-commutative compact space endowed with a group structure, and the existence and uniqueness of the Haar measure is proved and orthonormality relations for matrix elements of irreducible representations are derived.
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John Watrous
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TL;DR: In this article, the authors present a self-contained book on the theory of quantum information focusing on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject.
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Alexandru Nica,Roland Speicher +1 more
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TL;DR: In this article, the authors present a case study of non-normal distribution and non-commutative joint distributions and define a set of basic combinatorics, such as non-crossing partitions, sum-of-free random variables, and products of free random variables.
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José M. Gracia-Bondía,Joseph C. Várilly,Héctor Figueroa +2 more
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TL;DR: In this article, a wide range of topics including sources of non-commutative geometry, fundamentals of Non-Commutative topology, K-theory and Morita equivalance, non-commodity integrodifferential calculus, noncommutativity Riemannian spin manifolds, commutative geometrics, tori, second quantization, quantum field theory, and pseudodifferential operators are discussed.
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