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I am confused about a point in the proof of Theorem 27.3.6 which claims that $\text{dim}_{\mathbb{C}}Q(\chi) = 2^{n-(n-k)}$ in a "Concise Encyclopedia of Coding Theory" (page 663/664). The details are as follows:

Theorem 27.3.6: Let $C$ be an $(n-k)-$dimensional self-orthogonal subspace of $\mathbb{F}_{2}^{n-k}$ under the symplectic inner product. Let $d := w_{Q} (C^{\perp_{s}} $\ $C) = \min${$wt_{Q}(v) \in > C^{\perp_{s}}$\ $C$}. Then there is an $[[n,k,d]]$-qubit stabilizer code $Q$.


Useful definitions:

  • A quantum operator $E$ on $\mathbb{C}^{2^{n}}$ is of the form $i^{\lambda}X(a)Z(b)$
  • $\epsilon_{n}$ is the set of error operators $E$
  • $\overline{\epsilon_{n}}=\epsilon_{n} / Z(\epsilon_{n})$
  • $C:=\overline{G} \in \overline{\epsilon}_{n}$ Please find the relevant section of the proof of this theorem attached below. Please note that $Q(\chi) = L_{\chi}(C^{2^{n}})$ is the subspace $Q(\chi) =${$|v\rangle \in \mathbb{C}^{2^{n}}|g|v\rangle = \chi(g) |v\rangle \text{ } \forall g \in G$}.

Proof enter image description here

Here is proposition 27.3.5

enter image description here


I don't understand how they came to the conclusion that $\text{dim}_{\mathbb{C}}Q(\chi) = 2^{n-(n-k)}$. I understand that the dimension of $C$, which we lift to $\overline{G}$ is $n-k$. And I understand that $\text{dim}_{\mathbb{C}}Q(\chi)$ is the same for all $\chi \in \hat{G}$. But I can't figure out why the dimension of $Q(\chi)$ is $2^{2-(n-k)}$.

Any help or hints would be greatly appreciated! Perhaps I am missing something simple.

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I will sketch another proof of this fact, that I think is slightly more intuitive, I hope you can fill in any remaining details.

The codespace is the simultaneous $+1$-eigenspace of the independent commuting Pauli generators $G_1,\ldots, G_{(n-k)}$ which act on a vector space of dimension $2^n$ (i.e. bit strings). The dimension of the $+1$-eigenspace of a single $G_i$ is $2^n/2=2^{n-1}$. This can be seen via Lagrange's theorem, as each generator induces a surjective group homomorphism $\mathbb{Z}_2^n\to \mathbb{Z}_2$, that sends each codeword in $\{0,1\}^n$ to $\pm1$, (these are the characters $f_\alpha(z)=(-1)^{\langle \alpha,z\rangle}$ of the abelian subgroup generated by $G_1,\ldots, G_{(n-k)}$). Since all the generators commute and are independent (no generator is a product of the other generators) it follows that the dimension of the simultaneous $+1$-eigenspace is $2^n/2^{n-k}=2^{n-(n-k)}=2^k$, that is the dimension of the space is halved by each of the $n-k$ generators (see Why do stabilizer cut the Hilbert space into two halves?).

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