Parameters
Evaluates the coherent information at the weighted repetition code \(\phi_n^\lambda = \lambda \lvert 0\rangle\!\langle 0\rvert^{\otimes n} + (1-\lambda)\lvert 1\rangle\!\langle 1\rvert^{\otimes n}\) .
Custom Pauli: (1 - px - py - pz, px, py, pz). Choose px, py, pz in [0,1] with sum <= 1.
Rates and entropies
The rows \(\,p_0,\ldots,p_3\,\) show the parameters used below. For Pauli channels they are the Pauli probabilities \((I,X,Y,Z)\); for the other channels they show the channel parameters.
\(S_{\mathrm{out}}\) and \(S_{\mathrm{env}}\) are the output and complementary-output entropies.
\(\mathrm{rate}(n)\) and \(\mathrm{rate}(1)\) show \(\max(0,\text{coherent information}/n)\) for the weighted \(n\)-repetition code and \(1\)-repetition code.
\(\mathrm{Gain}=\mathrm{rate}(n)-\mathrm{rate}(1)\). A positive gain shows superadditivity of coherent information with this repetition-code family.
References
Closed-form expressions for the coherent information used here are taken from the following sources.
- Pauli channels and damping-dephasing. Sujeet Bhalerao and Felix Leditzky, “Improving quantum communication rates with permutation-invariant codes,” arXiv:2508.09978 (2025). [arXiv]
- Dephrasure channel. Felix Leditzky, Debbie Leung, and Graeme Smith, “Dephrasure Channel and Superadditivity of Coherent Information,” Physical Review Letters 121, 160501 (2018). [DOI] [arXiv]
- Generalized amplitude damping channel. Johannes Bausch and Felix Leditzky, “Quantum codes from neural networks,” New Journal of Physics 22, 023005 (2020). [DOI] [arXiv]