  For a noisy quantum channel, a quantum error correcting code exists if and
only if the joint higher rank numerical ranges associated with the error
operators of the channel is non-empty. In this paper, geometric properties of
the joint higher rank numerical ranges are obtained and their implications to
quantum computing are discussed. It is shown that if the dimension of the
underlying Hilbert space of the quantum states is sufficiently large, the joint
higher rank numerical range of operators is always star-shaped and contains a
non-empty convex subset. In case the operators are infinite dimensional, the
joint infinite rank numerical range of the operators is a convex set closely
related to the joint essential numerical ranges of the operators.
