The intersection \(\textbf{C}\cap \textbf{C}^{\perp _H}\) of a linear code \(\textbf{C} \subset \textbf{F}_{q^2}^n\) and its Hermitian dual \(\textbf{C}^{\perp _H}\) is called the Hermitian hull of this code. A linear code \(\textbf{C} \subset \textbf{F}_{q^2}^n\) satisfying \(\textbf{C} \subset \textbf{C}^{\perp _H}\) is called Hermitian self-orthogonal. Many Hermitian self-orthogonal codes were given for the construction of MDS quantum error correction codes (QECCs). In this paper we prove that for a nonnegative integer h satisfying \(0 \le h \le k\), a linear Hermitian self-orthogonal \([n, k]_{q^2}\) code is equivalent to a linear h-dimension Hermitian hull code. Therefore a lot of new MDS entanglement-assisted quantum error correction (EAQEC) codes can be constructed from previous known Hermitian self-orthogonal codes. Actually our method shows that previous constructed quantum MDS codes from Hermitian self-orthogonal codes can be transformed to MDS entanglement-assisted quantum codes with nonzero consumption parameter c directly. We prove that MDS EAQEC \([[n, k, d; c]]_q\) codes with nonzero c parameters and \(d\le \frac{n+2}{2}\) exist for arbitrary length n satisfying \(n \le q^2+1\). Moreover any QECC constructed from k-dimensional Hermitian self-orthogonal codes can be transformed to k different EAQEC codes. We also prove that MDS entanglement-assisted quantum codes exist for all lengths \(n\le q^2+1\).

线性码 \(\textbf{C} \subset \textbf{F}_{q^2}^n\) 与其埃尔米特对偶 \(\textbf{C}^{\perp _H}\) 的交集\(\textbf{C}\cap \textbf{ C } ^{\perp _H}\)称为该码的埃尔米特外壳。满足\(\textbf{C} \subset \textbf{C}^{\perp _H}\) 的线性码\(\textbf{C} \subset \textbf{F}_{ q^2}^n\)称为 Hermitian 自正交。给出了许多 Hermitian 自正交码用于 MDS 量子纠错码（QECC）的构造。在本文中，我们证明对于满足\(0 \le h \le k\)的非负整数h，线性 Hermitian 自正交\([n, k]_{q^2}\)代码相当于线性h维 Hermitian 赫尔代码。因此，许多新的MDS纠缠辅助量子纠错（EAQEC）码可以从先前已知的埃尔米特自正交码构造出来。实际上，我们的方法表明，先前从埃尔米特自正交码构造的量子MDS码可以直接转换为具有非零消耗参数c的MDS纠缠辅助量子码。我们证明具有非零c参数和\(d\le \frac{n+2}{2}\) 的MDS EAQEC \([[n, k, d; c]]_q\)代码存在于满足\(n \le q^2+1\) 的任意长度n。此外，任何由k构造的 QECC维 Hermitian 自正交码可以变换为k 个不同的 EAQEC 码。我们还证明了所有长度\(n\le q^2+1\) 都存在 MDS 纠缠辅助量子码。