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Journal of Operator Theory

Volume 66, Issue 2, Fall 2011  pp. 335-351.

Generalized numerical ranges and quantum error correction

Authors:  Chi-Kwong Li (1) and Yiu-Tung Poon (2)
Author institution: (1) Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23185, U.S.A.
(2) Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A.

Summary:  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 $\widehat k$-numerical range of the operators for sufficiently large $\widehat k$. In case the operators are infinite dimensional, the joint rank $\infty$-numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.

Keywords:  quantum error correction, joint higher rank numerical range, joint essential numerical range, self-adjoint operator, Hilbert space

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