  The maximum achievable rate or mutual informa- tion of multidimensional
rotationally invariant distributions in the presence of additive white Gaussian
noise is analyzed. A simple expression for the special case of multisphere
distributions is derived. Such distributions consist of points in a
multidimensional Euclidean space that are uniformly distributed over several
multidimensional concentric hyperspheres. For the 2-dimensional case, such
distributions have been previously considered in order to reduce the
computational complexity of finding a bound on the channel capacity of
fiber-optic channels. These distributions take advantage of the statistical
rotational invariance of the noise and nonlinear distortions in fiber-optic
channels. Using the derived mutual information expression, 2- and 4-dimensional
multisphere distributions are compared for fiber-optic dual-polarization
channels dominated by linear noise. At high signal-to-noise ratios,
4-dimensional multisphere distributions offer higher achievable rates than
2-dimensional ones transmitted on each of the two polarizations of the optical
carrier. Such 4-dimensional multisphere distributions are also shown to be
statistically invariant under 4-dimensional nonlinear transmission in fibers.
