  The intersection ${\bf C}\bigcap {\bf C}^{\perp_H}$ of a linear code ${\bf C}
\subset {\bf F}_{q^2}$ and its Hermitian dual ${\bf C}^{\perp_H}$ is called the
Hermitian hull of this code. A linear code ${\bf C} \subset {\bf F}_{q^2}$
satisfying ${\bf C} \subset {\bf C}^{\perp_H}$ 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 $0 \leq h \leq k$, 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 $d\leq \frac{n+2}{2}$ exist for
arbitrary length $n \leq q^2+1$. Moreover any QECC constructed from
$k$-dimensional Hermitian self-orthogonal codes can be transformed to $k$
different EAQEC codes.
