A lower bound on the C-numerical radius of nilpotent matrices appearing in coherent spectroscopy Chi-Kwong Li and Hugo J. Woerdeman ∗ April 13, 2005 Department of Mathematics The College of William and Mary Williamsburg, VA 23185-8795 ckli@math.wm.edu and Department of Mathematics Drexel University 3141 Chestnut Street Philadelphia, PA 19104 hugo@math.drexel.edu Keywords: C-numerical radius, NMR spectroscopy, Nilpotent matrices MSC: 15A60, 81V99 Abstract We provide a lower bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. Mathematically the problem is the determination of the C -numerical radius of A for certain nilpotent matrices C and A. The presented lower bound is conjectured to be exact as it coincides with numerical data provided in [U. Helmke et al., J. Global Opt. 23 (2002), 283-308]. 1 Introduction In the study of the efficiency of polarization or coherence transfer between quantized states under unitary transformations (see [1, 2, 3, 5, 6]), one is interested in determining or estimating the quantity b(A n ,C n ) := max UU * =I |tr(C * n UA n U * )|, (1.1) * Research of both authors are supported in part by National Science Foundation. 1