Q:

Are all $[[n, k, d]]$ quantum codes equivalent to additive self-orthogonal $GF(4)^n$ classical codes?

Theorem 2 of [1] states:

Suppose $C$ is an additive self-orthogonal sub-code of $\textrm{GF}(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C^\perp/C$. Then any eigenspace of $\phi^{-1}(C)$ is an additive quantum-error-correcting code with parameters $[[n, k, d]]$.

where here $\phi: \mathbb{Z}_2^{2n} \rightarrow \textrm{GF}(4)^n$ is the map between the binary representation of $n$-fold Pauli operators and their associated codeword, and $C$ is self-orthogonal if $C \subseteq C^\perp$ where $C^\perp$ is the dual of $C$.
This tells us that each additive self-orthogonal $\textrm{GF}(4)^n$ classical code represents a $[[n, k, d]]$ quantum code.
My question is whether the reverse is also true, that is: is every $[[n, k, d]]$ quantum code represented by an additive self-orthogonal $\textrm{GF}(4)^n$ classical code?
Or equivalently: Are there any $[[n, k, d]]$ quantum codes that are not represented by an additive self-orthogonal $\textrm{GF}(4)^n$ classical code?
[1]: Calderbank, A. Robert, et al. "Quantum error correction via codes over GF (4)." IEEE Transactions on Information Theory 44.4 (1998): 1369-1387.

A:

The additive self-orthogonal constraint on the classical codes in order to create stabilizer quantum codes is needed due to the fact that the stabilizer generators must commute between them in order to create a valid code space. When creating quantum codes from classical codes, the commutation relationship for the stabilizers is equivalent to having a self-orthogonal classical code.
However, quantum codes can be constructed from non-self-orthogonal classical codes over $GF(4)^n$ by means of entanglement-assistance. In this constructions, an arbitrary classical code is selected, and by adding some Bell pairs in the qubit system, commutation between the stabilizers is obtained.
This entanglement-assisted paradigm for constructing QECCs from any classical code is presented in arXiv:1610.04013, which is based on the paper "Correcting Quantum Errors with Entanglement" published in Science by Brun, Devetak and Hsieh.

