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arXiv:2401.01671 · Special Matrices

Two-unitary complex Hadamard matrices of order 36

Wojciech Bruzda, Karol Życzkowski

Published Jan 1, 2024• 6 citations
arXivHTMLPDF

Abstract

Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix 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 six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

DOI
10.1515/spma-2024-0010

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quant-pharXiv:2608.18053•Aug 19, 2026
Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix 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 six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.
Abstract Characterizing complex Hadamard matrices (CHMs) of order six is an open problem in both algebra and quantum information theory. Analyzing the eigenvalues of CHMs provides a novel approach to this problem. We construct a class of two-parameter CHMs of order six with two sorts of eigenvalues. All matrices in this class are H 2-reducible, and the Diţă matrix, Haagerup matrices and Hermitian matrices are both included. This class turns out to be complex equivalent to the class of all CHMs in the dephased form whose eigenvalues are of the form 6 \sqrt 6 , – 6 \sqrt 6 , λ 1 , λ 1 , λ 2 , λ 2 with λ j of modulus 6 \sqrt 6 excluding Tao matrix S 6 ( 0 ) S_6^{(0)} . We further show that some CHMs including all Hermitian CHMs in the former class have Schmidt rank at most three up to complex equivalence, and they are actually controlled unitary gates implementable in experiments. Our result demonstrates the feasibility of using eigenvalues for characterizing order-six CHMs and their application in quantum circuits.
A new method of obtaining a sequence of isolated complex Hadamard matrices (CHM) for dimensions N ⩾ 7, based on block-circulant structures, is presented. We discuss several analytic examples resulting from a modification of the Sinkhorn algorithm. In particular, we present new isolated matrices of orders 9, 10, and 11, which elements are not roots of unity, and also several new multiparametric families of order 10. We note novel connections between certain eight-dimensional matrices and provide new insights toward the classification of CHM for N ⩾ 7. These contributions can find real applications in quantum information theory and constructions of new families of mutually unbiased bases or unitary error bases.
Abstract Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kinds of matrices, one is H 2 -reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in ℂ 6 consists of an MUB trio and the identity) since we find that neither of the two CHMs belong to any MUB trio. Our results may lead to the more complete classification of 6 × 6 CHMs whose elements in the first row are all 1.
A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 333^333 and six of order 646^464, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
The classical combinatorial problem of 363636 officers has no solution, as there are no Graeco-Latin squares of order six. The situation changes if one works in a quantum setup and allows for superpositions of classical objects and admits entangled states. We analyze the recently found solution to the quantum version of the Euler's problem from a geometric point of view. The notion of a non-displaceable manifold embedded in a larger space is recalled. This property implies that any two copies of such a manifold, like two great circles on a sphere, do intersect. Existence of a quantum Graeco-Latin square of size six, equivalent to a maximally entangled state of four subsystems with d=6 levels each, implies that three copies of the manifold U(36)/U(1) of maximally entangled states of the 36×3636\times 3636×36 system, embedded in the complex projective space CP36×36−1{C}P^{36\times 36 -1}CP36×36−1, do intersect simultaneously at a certain point.
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3×33 \times 33×3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.