Quantum Error Correction and Higher Rank Numerical Ranges of Normal Matrices

The higher rank numerical range is useful for constructing quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a1, . . . , an, then its rank-k numerical range Λk(A) is the intersection of convex polygons with vertices aj1 , . . . , ajn−k+1 , where 1 ≤ j1 < · · · < jn−k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct vertices can be written as the intersection of no more than m closed half planes. In addition, given a convex polygon P a construction is given for a normal matrix A ∈ Mn with minimum n such that Λk(A) = P . In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ∈ Mn with n ≤ max {p + k − 1, 2k + 2} such that Λk(A) = P .