In this survey article, we give an expository account of the recent developments on the Poncelet property for numerical ranges of the compressions of the shift S(Á). It can be considered as an updated and more advanced edition of the recent expository article published in the American Mathematical Monthly by the second author on this topic. The new information includes:(1) a simplified approach to the main results (generalizations of Poncelet, Brianchon-Ceva and Lucas-Siebeck theorems) in this area, (2) the recent discovery of Mirman refuting a previous conjecture on the coincidence of Poncelet curves and boundaries of the numerical ranges of finite-dimensional S(Á), and (3) some partial generalizations by the present authors of the above-mentioned results from the unitary-dilation context to the normal-dilation one and also from the finite-dimensional S(Á) to the infinite-dimensional.

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The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a 1 , . . . , an, then its higher rank numerical range Λ k (A) is the intersection of convex polygons with vertices a j 1 , . . . , a j n−k+1 , where 1 ≤ j 1 < · · · < j n−k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m, 4} closed half planes. In addition, given a convex polygon P a construction is given for a normal matrix A ∈ Mn with minimum n such that Λ k (A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ∈ Mn with n ≤ max {p + k − 1, 2k + 2} such that Λ k (A) = P.AMS subject classifications. 15A60, 15A90, 47N50, 81P68 Key words. Quantum error correction, higher rank numerical range, normal matrices, convex polygon Λ k (A) = {λ ∈ C : P AP = λP for some rank-k orthogonal projection P },

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