Polar codes for classical-quantum channels

Mark M. Wilde, Saikat Guha

Research output: Contribution to journalArticlepeer-review

72 Scopus citations


Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of this paper is to fill this gap by constructing near-explicit 'polar' codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by 'channel combining' and 'channel splitting,' in which a fraction of the synthesized channels are perfect for data transmission, while the other channels are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and 'frozen' bits through the useless ones. The main technical contributions of this paper are threefold. First, we leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity is O(N\log N), where N is the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, in the sense that the word error rate decays exponentially with the blocklength of the code. For this last result, we exploit Sen's recent 'noncommutative union bound' that holds for a sequence of projectors applied to a quantum state.

Original languageEnglish (US)
Article number6302198
Pages (from-to)1175-1187
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number2
StatePublished - 2013
Externally publishedYes


  • Channel combining
  • channel splitting
  • classical-quantum polar code
  • non-commutative union bound
  • quantum successive cancellation decoder

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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