Coherent information of repetition codes under noise

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This interactive tool computes the coherent information \(I_{\mathrm{coh}}=S_{\mathrm{out}}-S_{\mathrm{env}}\) of weighted repetition code inputs to various quantum channels, where \(S_{\mathrm{out}}\) and \(S_{\mathrm{env}}\) are the channel output and complementary channel output entropies. Informally, it measures how much quantum information can be reliably communicated through \(n\) channel copies with the chosen input state, and is a lower bound on the quantum capacity of a quantum channel.

Parameters

Evaluates the coherent information at the weighted repetition code \(\phi_n^\lambda = \lambda \lvert 0\rangle\!\langle 0\rvert^{\otimes n} + (1-\lambda)\lvert 1\rangle\!\langle 1\rvert^{\otimes n}\) .

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.

\(p_0\) \(p_1\) \(p_2\) \(p_3\)
\(S_{\mathrm{out}}(n)/n\) \(S_{\mathrm{env}}(n)/n\) \(\mathrm{rate}(n)\)
\(S_{\mathrm{out}}(1)\) \(S_{\mathrm{env}}(1)\) \(\mathrm{rate}(1)\)
\(\mathrm{Gain}\)

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]