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Entanglement Theory Quantum Correlations
Condition for the higher rank numerical range to be non-empty
arXiv
Authors: Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze
Year
2007
Paper ID
50009
Status
Preprint
Abstract Read
~2 min
Abstract Words
110
Citations
N/A
Abstract
It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if n ge 3k - 2. The proof is based on a recent characterization of the rank-k numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that Λ2(A) is non-empty if n ge 4. This confirms a conjecture of Choi et al. If 3k-2>n>0, an n-by-n complex matrix is given for which the rank-k numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.
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- This paper contributes to the Entanglement Theory & Quantum Correlations research area in the Quantum Articles archive.
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- It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if n ge 3k - 2.
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