pure distance

codes/classical/q-ary_digits/universally_optimal/univ_opt_q-ary.yml

@@ -11,7 +11,7 @@ name: 'Universally optimal \(q\)-ary code'
 introduced: '\cite{manual:{V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.},doi:10.1109/18.412678,doi:10.1007/BF00053379,preset:HPLevBounds,doi:10.1007/s10623-016-0286-4,doi:10.1109/18.915662,arxiv:1212.1913}'
 
 description: |
-  A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on binary space \cite{arxiv:1212.1913}.
+  A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on Hamming space \cite{arxiv:1212.1913}.
 
   All codes that attain the linear programming (LP) bound by Delsarte \cite{manual:{P. Delsarte, “Bounds for unrestricted codes, by linear programming,” Philips Research Reports, vol. 27, pp. 272–289, 1972}} are universally optimal \cite{arxiv:1212.1913}.
   Such codes are called \textit{LP universally optimal} or \textit{extremal}.

codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml

@@ -28,7 +28,7 @@ description: |
 
 protection: |
   Code distance \(\mathcal{Q} = ( \mathcal{G},\mathcal{C}) \) is upper bounded by the distance of the classical code \(\mathcal{C} \).
-  The \hyperref[code:qubits_into_qubits]{diagonal distance} is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\) \cite{arxiv:2107.11286}.
+  The \hyperref[code:qubits_into_qubits]{pure distance} is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\) \cite{arxiv:0712.1979,arxiv:2107.11286}.
   Some bounds on the distance are provided in Ref. \cite{arxiv:1108.5490}.
 
 features:

codes/quantum/qubits/qubits_into_qubits.yml

@@ -25,11 +25,11 @@ protection: |
   As a result, qubit codes cannot tolerate adversarial errors on more than \((1-R)/4\) registers, where \(R = \log_2 K/n\) is the code rate.
 
   \subsection{Pauli-string error basis}
+  \label{topic:pauli}
 
   A convenient and often considered error set is the \textit{Pauli error} or \textit{Pauli string} basis.
 
   \begin{defterm}{Pauli strings}
-  \label{topic:pauli}
   For a single qubit, this set consists of products of powers of the Pauli matrices
   \begin{align}
     X=\begin{pmatrix}0 & 1\\
@@ -46,21 +46,17 @@ protection: |
   The Pauli error set is a unitary and Hermitian basis for linear operators on the multi-qubit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a prototypical \hyperref[topic:nice-error-basis]{nice error basis}.
   The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.
 
-  The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the \textit{diagonal distance} \cite{arxiv:0712.1979,arxiv:2107.11286} (see also pure distance \cite{arxiv:2107.14252}).
-  Codes whose distance is greater than the diagonal distance are \hyperref[topic:degeneracy]{degenerate}.
-  \hyperref[topic:degeneracy]{Degenerate} codes admit undetectable Pauli errors (i.e., errors whose projection into the codespace is nonzero) of weight less than the code distance (i.e., the projection satisfies the \term{Knill-Laflamme conditions}).
-
   \subsection{Noise channels}
 
   A quantum channel that admits a set of Pauli strings as its Kraus operators is called a \textit{Pauli channel}, and such channels are typically more tractable than the more general, non-Pauli channels.
   Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel \cite{arxiv:quant-ph/0203003}).
   Relevant non-Pauli channels are \hyperref[topic:ad]{AD} noise, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
   Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon \cite{arxiv:quant-ph/0505153,arxiv:2012.01894}.
 
-  \subsection{Quantum weight enumerators}
+  \subsection{Quantum weight enumerators and pure distance}
+  \label{topic:quantum-weight-enumerator}
 
   \begin{defterm}{Quantum weight enumerator}
-  \label{topic:quantum-weight-enumerator}
   Determining protection and bounds on code parameters can also be done using the code's Shor-Laflamme \textit{quantum weight enumerator} \cite{arxiv:quant-ph/9610040} (cf. \hyperref[topic:weight-enumerator]{weight enumerators})
     \begin{align}
     \begin{split}
@@ -82,14 +78,20 @@ protection: |
   It gives rise to quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049}; see the book \cite{preset:GottesmanBook}.
 
   The distance \(d\) of a code is the smallest \(j=d\) at which \(A_j \neq B_j\) \cite{arxiv:quant-ph/9906126}.
-  Such a code is called \textit{pure} if \(A_j = B_j = 0\) for all \(j < d\); otherwise, the code is called \textit{impure}.
-  \hyperref[topic:degeneracy]{Degeneracy} is sufficient but not necessary for impurity \cite{preset:GottesmanBook}.
-
-  Other types of quantum weight enumerators are the Rains unitary enumerators \cite{ arXiv:quant-ph/9612015} and the \textit{Rains shadow enumerators} \cite{arxiv:quant-ph/9611001} (see also \cite{arxiv:quant-ph/0406063}), with the latter related to Bell sampling \cite{arxiv:2408.16914}.
+  A code is called \textit{pure} if \(A_j = 0\) for all \(1 < j < d\); otherwise, the code is called \textit{impure}.
+  The \textit{pure distance} \cite{arxiv:2107.14252} (a.k.a. diagonal distance \cite{arxiv:0712.1979}) \(d_{\smallsetminus}\) is the smallest \(1 < j=d_{\smallsetminus}\) at which \(A_j > 0\).
+  Codes for which \(d_{\smallsetminus} < d\) are impure, otherwise they are pure.
+  For impure codes, there exists a Pauli error of weight less than the code distance that has a non-zero expectation value with respect to a code state.
+
+  Degenerate qubit codes are impure, but impure codes may not be degenerate \cite{preset:GottesmanBook}.
+  There are subtleties with defining \hyperref[topic:degeneracy]{degeneracy} for non-stabilizer qubit codes with even distance \cite{preset:GottesmanBook}.
+  
+  Other types of quantum weight enumerators are the Rains unitary enumerators \cite{arXiv:quant-ph/9612015} and the \textit{Rains shadow enumerators} \cite{arxiv:quant-ph/9611001} (see also \cite{arxiv:quant-ph/0406063}), with the latter related to Bell sampling \cite{arxiv:2408.16914}.
   These notions can be generalized to qudit codes and other error bases \cite{doi:10.1016/j.aam.2020.102085,arxiv:2211.02756,arxiv:2308.05152}.
   There are techniques to compute them for general codes \cite{arxiv:2308.05152}.
   Semidefinite programming (SDP) hierarchies and a quantum Delsarte bound have been developed \cite{arxiv:2408.10323}.
 
+
 features:
   rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}. There are many bounds on the quantum capacity of the depolarizing channel (e.g., \cite{arxiv:quant-ph/0607039}); see review \cite{arxiv:1801.02019}.'
   transversal_gates:

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -107,6 +107,12 @@ protection: |
   Define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all Pauli operators that commute with all \(S\in\mathsf{S}\).
   A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\).
 
+  There are subtleties with defining \hyperref[topic:degeneracy]{degeneracy} for non-stabilizer qubit codes with even distance \cite{preset:GottesmanBook}, but they are resolved for stabilizer codes.
+  A stabilizer code is a \hyperref[topic:degeneracy]{degenerate} with respect to \(\mathcal{E}\) if and only if \(E^\dagger F \in \mathsf{N(S)}\) for some Pauli strings \(E,F \in \mathcal{E}\).
+  As a distance-\(d\) code, a stabilizer code is degenerate if it admits a non-identity stabilizer whose weight is lower than the distance \cite{preset:GottesmanBook}.
+  Since that stabilizer is in the normalizer, a stabilizer code is degenerate if and only if it is \hyperref[topic:quantum-weight-enumerator]{impure}.
+  The \hyperref[topic:quantum-weight-enumerator]{pure distance} of a stabilizer code is the minimum weight of a non-identity stabilizer. 
+
   \begin{defterm}{Cleaning lemma}
   \label{topic:cleaning-lemma}
   If all logical operators act trivially on some subset of qubits in a stabilizer code, then any logical Pauli operator can be represented on the complementary qubit subset via a stabilizer.