Conditions for the existence of quantum error correction codes
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In this dissertation, we consider quantum errors, represented by quantum channels defined on the space of n x 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 natural numbers k and m, we find a lower bound on n above which there is always an n x k matrix U with orthonormal columns such that U*A_1U,...,U*A_mU are diagonal k x 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+2. 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.