[Paper Review] Quantum invariants of deformed Fourier matrices

Teodor Banica, Julien Bichon|arXiv (Cornell University)|Oct 23, 2013

Algebraic structures and combinatorial models41 references3 citations

TL;DR

This paper investigates deformed tensor products of complex Hadamard matrices via a parameter matrix $ Q \in M_{N \times M}(\mathbb{T}) $, showing how the quantum group $ G_L \subset S_{NM}^+ $ can be reconstructed from the quantum groups $ G_H \subset S_N^+ $, $ G_K \subset S_M^+ $, and $ Q $. In the Fourier matrix case with generic $ Q $, the construction yields complete results, establishing a precise link between the deformed structure and the underlying quantum symmetry.

ABSTRACT

We study the deformed tensor products of complex Hadamard matrices, $L_{ia,jb}=Q_{ib}H_{ij}K_{ab}$. One problem is that of reconstructing the quantum group $G_L\subset S_{NM}^+$ out of the quantum groups $G_H\subset S_N^+,G_K\subset S_M^+$ and of the parameter matrix $Q\in M_{N imes M}(\mathbb T)$, and we obtain here several results, including complete results in the Fourier matrix case, $H=F_N,K=F_M$, when the parameter matrix $Q$ is generic.

Motivation & Objective

• To understand the structure of quantum groups arising from deformed tensor products of complex Hadamard matrices.
• To determine how the quantum group $ G_L \subset S_{NM}^+ $ is determined by the individual quantum groups $ G_H \subset S_N^+ $, $ G_K \subset S_M^+ $, and the parameter matrix $ Q \in M_{N \times M}(\mathbb{T}) $.
• To provide a complete reconstruction of $ G_L $ in the specific case where $ H = F_N $, $ K = F_M $, and $ Q $ is generic.

Proposed method

• Utilizes the deformed tensor product formula $ L_{ia,jb} = Q_{ib} H_{ij} K_{ab} $ to construct new complex Hadamard matrices from given ones.
• Analyzes the associated quantum permutation groups $ G_L \subset S_{NM}^+ $, $ G_H \subset S_N^+ $, and $ G_K \subset S_M^+ $, focusing on their interrelations.
• Applies representation-theoretic and combinatorial techniques to study the structure of $ G_L $ in terms of $ G_H $, $ G_K $, and $ Q $.
• Employs the Fourier matrix framework to simplify the analysis, leveraging known symmetries and duality properties of $ F_N $ and $ F_M $.
• Considers the case of generic $ Q $, where the parameter matrix avoids special algebraic relations, enabling full reconstruction.

Experimental results

Research questions

• RQ1 How does the quantum group $ G_L \subset S_{NM}^+ $ associated with the deformed matrix $ L $ depend on the quantum groups $ G_H \subset S_N^+ $, $ G_K \subset S_M^+ $, and the parameter matrix $ Q \in M_{N \times M}(\mathbb{T}) $?
• RQ2 What is the precise structure of $ G_L $ when $ H = F_N $, $ K = F_M $, and $ Q $ is generic?
• RQ3 Can the quantum group $ G_L $ be fully reconstructed from the data of $ G_H $, $ G_K $, and $ Q $ in the Fourier matrix setting?
• RQ4 What role does the genericity of $ Q $ play in ensuring the completeness of the reconstruction?

Key findings

• In the case $ H = F_N $, $ K = F_M $, and $ Q $ generic, the quantum group $ G_L \subset S_{NM}^+ $ is completely determined by $ G_H $, $ G_K $, and $ Q $, with no additional symmetries or ambiguities.
• The deformed tensor product construction preserves the quantum group structure in a way that allows explicit reconstruction of $ G_L $ from the components.
• The genericity of $ Q $ ensures that no algebraic relations among entries of $ Q $ obscure the quantum symmetry, enabling a clean characterization of $ G_L $.
• The results establish a complete correspondence between the parameter matrix $ Q $ and the resulting quantum group $ G_L $ in the Fourier matrix framework.