Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum symmetries of Hadamard matrices

Daniel Gromada

Abstract

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size $N\ge 4$ has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras.

Quantum symmetries of Hadamard matrices

Abstract

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras.
Paper Structure (24 sections, 40 theorems, 69 equations)

This paper contains 24 sections, 40 theorems, 69 equations.

Table of Contents

  1. Diagrammatic categories
  2. Diagrammatic categories
  3. Spiders and partitions
  4. Fibre functors on partitions
  5. Complementary spiders
  6. Two-coloured graphs with even degrees
  7. Hadamard matrices and generalizations
  8. Hadamard matrices
  9. Hadamard morphisms
  10. Quantum Hadamard matrices
  11. Structure of $\mathsf{NCBipartEven}$
  12. Quantum symmetries
  13. Quantum groups
  14. Tannaka--Krein reconstruction and quantum automorphisms
  15. Hopf-bi-Galois objects and quantum isomorphisms
  16. ...and 9 more sections

Key Result

Proposition 1.3

Let $\mathscr{C}$ be a pure category. Then for every non-trivial fibre functor $F\colon \mathscr{C}\to\mathsf{Mat}$, we have $\ker F=\mathscr{N}$, where is the tensor ideal of negligible morphisms.

Theorems & Definitions (101)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9: Ban02
  • ...and 91 more