Complex Hadamard matrices for prime number
Petre Di\c{t}\u{a}
TL;DR
This paper constructs new complex Hadamard matrices of prime order five, disproving a previous conjecture and providing new mutually unbiased bases relevant to quantum information processing.
Contribution
It introduces circulant matrices that generate novel Hadamard matrices and mutually unbiased bases, expanding the known set of such matrices for prime order five.
Findings
Disproved Haagerup's conjecture for order five matrices
Constructed new mutually unbiased bases
Generated novel complex Hadamard matrices
Abstract
In this paper we disprove the Haagerup statement that all complex Hadamard matrices of order five are equivalent with the Fourier matrix by constructing circulant matrices that lead to new Hadamard matrices. An important item is the construction of new mutually unbiased bases that are a basic concept of quantum theory and play an essential role in quantum tomography, quantum cryptografy, teleportation, construction of dense coding schemes, classical signal proccesing, etc.
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Taxonomy
Topicsgraph theory and CDMA systems · Coding theory and cryptography · Advanced Topics in Algebra