Malnormal matrices
Garrett Mulcahy, Thomas Sinclair
TL;DR
This paper constructs an infinite sequence of complex matrices with bounded operator norm whose commutator maps are uniformly bounded below, using quantum expanders, and explores their applications and related conjectures.
Contribution
It introduces a novel construction of matrices with specific commutator properties based on quantum expanders, with potential applications in quantum information theory.
Findings
Constructed an infinite sequence of matrices with bounded operator norm.
Demonstrated the commutator map is uniformly bounded below.
Provided numerical evidence and posed related conjectures.
Abstract
We exhibit an operator norm bounded, infinite sequence of complex matrices for which the commutator map is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert--Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and problems related to such matrices and provide numerical evidence.
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Taxonomy
TopicsAdvanced Topics in Algebra · Matrix Theory and Algorithms · Algebraic structures and combinatorial models