### Abstract

In this paper, we address the problem of designing a quantum encoder that maximizes the minimum output purity of a given decohering channel, where the minimum is taken over all possible pure inputs. This problem is cast as a max-min optimization problem with a rank constraint on an appropriately defined matrix variable. The problem is computationally very hard because it is nonconvex with respect to both the objective function (output purity) and the rank constraint. Despite this difficulty, we provide a tractable computational algorithm that produces the exact optimal solution for codespace of dimension 2. Moreover, this algorithm is easily extended to cover the general class of codespaces, in which case the solution is suboptimal in the sense that the suboptimized output purity serves as a lower bound of the exact optimal purity. The algorithm consists of a sequence of semidefinite programmings and can be performed easily. Two typical quantum error channels are investigated to illustrate the effectiveness of our method.

Original language | English |
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Article number | 012327 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 76 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jul 26 |

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics