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In this dissertation, we consider quantum errors, represented by quantum channels defined on the space of n × n complex matrices, M n. We begin with a quantum channel T defined on M n. The Kraus operators of this channel can be used to generate an operator system S, which is guaranteed to have a basis of Hermitian matrices A 1 , ..., A m in M n. Given k, m ≥ 1, we find a lower bound on n above which there is always an n × k matrix U with orthonormal columns such that U * A 1 U, ..., U * A m U are diagonal k × k matrices. We then extend this to a lower bound on n above which there is always a k-dimensional quantum error correction code for any quantum channel T on M n with an associated operator system S with dim(S) ≤ m + 1. Such a bound is equivalent to a lower bound on n above which the joint rank-k numerical range of any m Hermitian matrices in M n is non-empty. So, we further extend our result to a lower bound on n above which the rank-k numerical range of any m Hermitian matrices in M n is star-shaped.