Number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function f_d(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.