For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...

Let A be an n × n complex matrix and 0 ≤ q ≤ 1. The boundary of the q-numerical range of A is the orthogonal projection of a hypersurface defined by the dual surface of the homogeneous polynomial F (t, x, y, z) = det(t In + x(A + A )/2 + y(A−A)/(2i) + z AA). We construct different types of cubic surfaces SF corresponding to the homogeneous polynomial F (t, x, y, z) induced by some 3 × 3 matrice...

Notions of numerical ranges and joint numerical ranges of octonion matrices are introduced. Various properties of hermitian octonion matrices related to eigenvalues and convex cones, such as the convex cone of positive semidefinite matrices, are described. As an application, convexity of joint numerical ranges of 2×2 hermitian matrices is characterized. Another application involves existence of...