  For probability vectors x and y, the catalytic majorization relation x prec_T
y is defined to hold when there exists a probability vector z such that x
otimes z is majorized by y otimes z. In this paper, an infinite family of
functions is given such that, subject to some trivial restrictions, x prec_T y
if and only if f_r(x) < f_r(y) for all functions f_r in the family. An outline
of a proof of this result is provided. The catalytic majorization relation is
known to provide a determination of which transformations of jointly held pure
quantum states are possible using local operations and classical communication
when an additional jointly held state may be specified to facilitate the
transformation without being consumed.
