GENERALIZED NUMERICAL RANGES AND QUANTUM ERROR CORRECTION

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 underlying Hilbert space of the quantum states is sufficiently large, then the joint rank k-numerical range of operators is always star-shaped and contains the convex hull of the rank (k) over cap -numerical range of the operators for sufficiently large (k) over cap. In case the operators are infinite dimensional, the joint rank cc-numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.