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Proceedings of the American Mathematical Society

A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids

Ferenc Szöllősi

Published Oct 20, 2009• 22 citations
Publisher

Abstract

We construct a 2 2 -parameter family of complex Hadamard matrices of order 6 6 by a natural block construction. We combine this family with an earlier result of Zauner to derive a 2 2 -parameter family of triplets of mutually unbiased bases (MUBs) in C 6 \mathbb C^6 . This invalidates some numerical evidence given by Brierley and Weigert and sheds new light on the problem of determining the maximal number of MUBs in C 6 \mathbb C^6 .

DOI
10.1090/s0002-9939-09-10102-8

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We construct a 2 2 -parameter family of complex Hadamard matrices of order 6 6 by a natural block construction. We combine this family with an earlier result of Zauner to derive a 2 2 -parameter family of triplets of mutually unbiased bases (MUBs) in C 6 \mathbb C^6 . This invalidates some numerical evidence given by Brierley and Weigert and sheds new light on the problem of determining the maximal number of MUBs in C 6 \mathbb C^6 .
We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.
We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.
We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.
Abstract We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that each MUB-triplet in Cd\mathbb {C}^dCd C d is characterized by a d×d×dd\times d\times dd×d×d d × d × d object that we call a Hadamard cube . We describe the basic properties of Hadamard cubes and show how a MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.
We show that K6(2)∩X6(2)=D6(1)∪C6(1)K_6^{(2)} \cap X_6^{(2)} = D_6^{(1)} \cup C_6^{(1)}K6(2)​∩X6(2)​=D6(1)​∪C6(1)​ up to standard Hadamard equivalence, where C6(1)C_6^{(1)}C6(1)​ is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Szöllősi is strictly larger than the known affine Diţă curve D6(1)D_6^{(1)}D6(1)​. Our new parametrisation of the Szöllősi family X6(2)X_6^{(2)}X6(2)​ in Karlsson coordinates and our constructive classification of the Karlsson-Szöllősi intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.
We study the problem of constructing mutually unbiased bases in dimension 6. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family K (2) 6 which we confirm analytically. Also, we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.