Entanglement assisted quantum error correcting code from Hadamard matrices

<p indent="0mm">The classical linear codes that construct quantum error correction codes need to meet certain specific conditions. The proposal of entanglement-assisted quantum error correcting codes eliminates the restriction of these conditions, so all classical linear codes can be constructed into entanglement-assisted quantum error correcting codes. The Hadamard matrix is a special type of orthogonal matrix. Hadamard codes, constructed based on the standard form of Hadamard matrices, are a type of linear complementary dual (LCD) codes. By extending the generator and parity-check matrices of Hadamard codes, two types of classical codes whose parameters are [(<italic>l</italic>+1)(<italic>n</italic>−1), <italic>n</italic>−1] and [3<italic>n</italic>−3, <italic>n</italic>−1] maintain the LCD property. Utilizing the relationship between their LCD property and the dimensions of the Euclidean hull, this paper constructs two types of EAQECCs based on Hadamard matrices. These EAQECCs can achieve maximal entanglement, resulting in higher code rates and net rates compared to other EAQECCs. Their parameters are [[3<italic>n</italic>−3, 2<italic>n</italic>−2, <italic>d</italic>≥1; <italic>n</italic>−1]] and [[2<italic>n</italic>−2, <italic>n</italic>−1, <italic>d</italic>≥1; <italic>n</italic>−1]], respectively.