Two-Unitary Complex Hadamard Matrices of Order $36$

• URL: http://arxiv.org/abs/2401.01671v3
• Date: Wed, 8 May 2024 22:15:18 GMT
• Title: Two-Unitary Complex Hadamard Matrices of Order $36$
• Authors: Wojciech Bruzda, Karol Życzkowski,
• Abstract summary: A family of two-unitary complex Hadamard matrices (CHM) stemming from a particular matrix, of size $36$ is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all $36$ officers.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: A family of two-unitary complex Hadamard matrices (CHM) stemming from a particular matrix, of size $36$ is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all $36$ officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of $6$ elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

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