ea_classical_into_quantum domain root, ea_mixed_alphabet_reed_solomon

Author: valbert4

codes/classical_into_quantum/ea_classical_into_quantum.yml

@@ -0,0 +1,50 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: ea_classical_into_quantum
+
+name: 'Entanglement-assisted (EA) c-q code'
+short_name: 'EA c-q'
+
+alternative_names:
+  - 'Entanglement-assisted classical communication (EACC) code'
+  - 'Entanglement-assisted classical code'
+
+description: |
+  Classical-quantum code whose encoding and decoding utilize pre-shared entanglement between sender and receiver.
+  The sender encodes classical information into quantum systems sent through a quantum channel, while the receiver decodes using the channel outputs together with retained halves of pre-shared entangled states.
+
+protection: |
+  A finite-block EACC code is often denoted by \([n,k,d;c]_q\), where \(n\) is the number of \(q\)-dimensional channel uses, \(q^k\) is the number of classical messages, \(d\) is the minimum distance, and \(c\) is the number of pre-shared maximally entangled qudit pairs.
+  Such a code corrects \(d-1\) erasures or \(\left\lfloor(d-1)/2\right\rfloor\) errors in the setting of Ref. \cite{arxiv:2310.19774}.
+
+features:
+  rate: |
+    The entanglement-assisted classical capacity \(C^{\rm ea}(T)\) is the highest asymptotic rate for reliable classical communication through a quantum channel \(T\) when arbitrary pre-shared entanglement is available \cite{doi:10.1103/PhysRevLett.83.3081,doi:10.1109/TIT.2002.802612}.
+    For lossy bosonic channels with high thermal noise and low transmitted photon number, pre-shared entanglement can yield a capacity ratio scaling as \(\log(1/N_S)\) relative to the unassisted Holevo capacity \cite{arxiv:2001.03934,arxiv:2208.07979}.
+    If the encoding and decoding circuits themselves are noisy, the fault-tolerant EA capacity approaches the usual EA capacity as the gate error tends to zero \cite{arxiv:2210.02939}.
+
+  encoders:
+    - 'Super-dense coding maps two \(q\)-ary classical symbols to one transmitted qudit when one maximally entangled qudit pair is available \cite{doi:10.1103/PhysRevLett.69.2881}.'
+
+relations:
+  parents:
+    - code_id: eaoaecc
+      detail: 'An EAOA QECC that has no gauge structure (e.g., gauge qubits), that has a block structure that corresponds to a classical code, that stores no quantum information, and that utilizes pre-shared entanglement is an EA c-q code.'
+  cousins:
+    - code_id: bosonic_classical_into_quantum
+      detail: 'Bosonic EA c-q schemes use pre-shared continuous-variable entanglement to assist bosonic c-q communication, including structured transceivers for lossy thermal-noise channels \cite{arxiv:2001.03934,arxiv:2208.07979}.'
+    - code_id: eacq
+      detail: 'EA c-q codes transmit only classical information with entanglement assistance, while EA hybrid QECCs transmit both classical and quantum information with entanglement assistance.'
+    - code_id: eaqecc
+      detail: 'EA c-q codes transmit classical information with entanglement assistance, while EAQECCs transmit quantum information with entanglement assistance.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/classical_into_quantum/ea_mixed_alphabet_reed_solomon.yml

@@ -0,0 +1,55 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: ea_mixed_alphabet_reed_solomon
+physical: galois
+logical: galois
+
+name: 'EA mixed-alphabet Reed-Solomon c-q code'
+short_name: 'EA mixed-alphabet RS c-q'
+introduced: '\cite{arxiv:2310.19774}'
+
+alternative_names:
+  - 'Mixed-alphabet Reed-Solomon EACC code'
+  - 'Mixed-alphabet RS entanglement-assisted classical code'
+
+description: |
+  Entanglement-assisted c-q code obtained from a mixed-alphabet Reed-Solomon construction over \(\mathbb{F}_q\) and \(\mathbb{F}_{q^2}\).
+  A codeword of an \([n,k,d;c]_q\) code consists of \(n-c\) symbols transmitted directly over \(q\)-dimensional quantum systems and \(c\) symbols transmitted through super-dense coding using \(c\) pre-shared maximally entangled qudit pairs.
+
+  More explicitly, the code evaluates all polynomials \(f\in\mathbb{F}_q[x]\) of degree at most \(k-1\) at \(n-c\) distinct points \(\alpha_i\in\mathbb{F}_q\) and \(c\) representatives \(\gamma_j\in\mathbb{F}_{q^2}\setminus\mathbb{F}_q\), choosing at most one element from each conjugate pair \(\{\gamma,\gamma^q\}\).
+  This yields codewords in \(\mathbb{F}_q^{n-c}\times\mathbb{F}_{q^2}^{c}\), where each \(\mathbb{F}_{q^2}\) symbol is identified with two \(q\)-ary symbols for dense coding.
+
+protection: |
+  Let \(n_1=n-c\) and \(n_2=c\).
+  If \(n_1\geq k-1\), then the minimum distance is \(d=n-k+1\), saturating the classical Singleton bound.
+  If \(n_1<k-1\), then
+  \begin{align}
+    d=\left\lceil\frac{n-k+1+n_2}{2}\right\rceil~,
+  \end{align}
+  which can exceed the classical Singleton bound because a known erasure of an \(\mathbb{F}_{q^2}\) position removes a two-symbol block \cite{arxiv:2310.19774}.
+
+features:
+  rate: |
+    The construction can have \(k>n\) when \(c>0\), since each dense-coded position carries two \(q\)-ary symbols.
+    Its length is bounded by \(n\leq q+(q^2-q)/2=(q^2+q)/2\), with a possible one-symbol extension using the point at infinity \cite{arxiv:2310.19774}.
+    In the range \(n\leq q+(q^2-q)/2\) and \(n-q\leq c\), the distance formula above meets the block-erasure bound of Ref. \cite{arxiv:2310.19774}.
+
+relations:
+  parents:
+    - code_id: ea_classical_into_quantum
+  cousins:
+    - code_id: reed_solomon
+      detail: 'EA mixed-alphabet RS c-q codes use Reed-Solomon polynomial evaluation, but evaluate over both \(\mathbb{F}_q\) and selected representatives from \(\mathbb{F}_{q^2}\setminus\mathbb{F}_q\) to support direct and dense-coded channel uses \cite{arxiv:2310.19774}.'
+    - code_id: mds
+      detail: 'EA mixed-alphabet RS c-q codes can saturate a block-erasure bound and, in some parameter ranges, exceed the classical Singleton bound for ordinary \(q\)-ary codes \cite{arxiv:2310.19774}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/quantum/quantum_into_quantum.yml

@@ -30,10 +30,12 @@ description: |
           \\
       \(G\) & group-valued qudit
           \\
+      \(G/H\) & coset-valued qudit
+          \\
       \(\mathcal{C}\) & category-valued qudit
       }
       \end{cells}
-      \caption{Table listing the most common alphabets (a.k.a. configuration spaces) used in quantum codes. Here, \(\mathbb{F}_q\) is a \hyperref[topic:finite-fields]{finite field}, \(G\) is a group, and \(\mathcal{C}\) is a category.}
+      \caption{Table listing the most common alphabets (a.k.a. configuration spaces) used in quantum codes. Here, \(\mathbb{F}_q\) is a \hyperref[topic:finite-fields]{finite field}, \(G\) is a group, \(H\) is a subgroup of \(G\), and \(\mathcal{C}\) is a category.}
       \label{table:quantum-alphabets}
     \end{table}
 

codetree/domains.yml

@@ -40,4 +40,4 @@
     Codes for classical communication over quantum channels
   root_codes:
     - code_id: classical_into_quantum
-
+    - code_id: ea_classical_into_quantum