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

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.