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Theoretical and Natural Science

Order-six complex hadamard matrices constructed by Schmidt rank and partial transpose in operator algebra

Yuming Chen

Published May 10, 2024• 1 citation
Publisher

Abstract

Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 × 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 × 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.

DOI
10.54254/2753-8818/34/20241113

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Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 × 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 × 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.
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.
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.
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 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.
The partial transpose map is a linear map widely used in quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely [Formula: see text], where [Formula: see text]’s and [Formula: see text]’s are respectively the matrices of the same size, and [Formula: see text] is the Schmidt rank. We find explicit conditions that are equivalent to equality when the [Formula: see text]’s are column or row vectors, or [Formula: see text] matrices. For the case where the Schmidt rank equals the dimension of [Formula: see text], we extend the results from [Formula: see text] matrices to square matrices, and further to rectangular matrices. In detail, we show that [Formula: see text] is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for [Formula: see text], and it turns out that the key is to characterize the expression of matrices [Formula: see text]’s and [Formula: see text]’s.
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.