Information-entropy trade-off for random vectors

The results here are based on formulae in the two appendices in R. Schack, C. M. Caves, and G. M. D’Ariano, “Hypersensitivity to perburbation in the quantum kicked top,” Phys. Rev. E 50, 972–987 (1994). The relevant equations in the two appendices are (A18) and (B5)–(B6). A preliminary version of the analysis here can be found at the end of Sec. V of R. Schack and C. M. Caves, “Information-theoretic characterization of quantum chaos,” Phys. Rev. E 53, 3257–3270 (1996). This analysis was further elaborated to essentially the present form in A. N. Soklakov and R. Schack, “Preparation information and optimal decompositions for mixed quantum states,” J. Mod. Opt. 47, 2265–2276 (2000). Consider N vectors randomly distributed in a D-dimensional Hilbert space, where we assume that N ≥ D. Given an entropy H ≤ log D, we want to group the vectors into groups that on average have this entropy and then ask how much information I is required to specify a group. The relation between I and H is the information-entropy trade-off. We can estimate this trade-off by considering a grouping of the vectors into spheres on projective Hilbert-space whose radius is given by a Hilbert-space angle φ. The number of spheres of radius φ is [Eq. (A18) of Schack1994]