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Abstract. We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, specifically to the problem of identification and construction of codes for binary unitary noise models. (2000): 15A60, 15A90, 47A12, 81P68. Key words and phrases: Higher-rank numerical range, unitary matrix, quantum error correction. Mathematics subject classification

The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.

Abstract. The totality of normalised density matrices of order N forms a convex set Q N in R N 2 −1 . Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N . For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W (A), induced by the unitarily invariant FubiniStudy measure on the complex projective manifold CP N −1 . We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.

The effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise.