## Abstract

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ε Mn has eigenvalues a_{1}, ⋯ , a _{n}, then its higher rank numerical range Γ_{κ}(A) is the intersection of convex polygons with vertices a_{j1} , ⋯ , a_{jn-k+1}, where 1 ≤ j_{1} < ⋯ ≤ j _{n-k+1} ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m, 4} closed half planes. In addition, given a convex polygon P, a construction is given for a normal matrix A ε M_{n} with minimum n such that Δ_{κ}(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ε M_{n} with n ≤ max {p + k -1, 2k + 2} such that Γ_{κ}(A) = P.

Original language | English |
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Pages (from-to) | 23-43 |

Number of pages | 21 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

## Keywords

- Convex polygon
- Higher rank numerical range
- Normal matrices
- Quantum error correction