  In previous work (see arxiv:1102.3040), we have defined the telescopic
relative entropy (TRE), which is a regularisation of the quantum relative
entropy $S(\rho||\sigma)=\trace\rho(\log\rho-\log\sigma)$, by replacing the
second argument $\sigma$ by a convex combination of the first and the second
argument, $\tau=a\rho+(1-a)\sigma$ and dividing the result by $-\log a$. We
also explored some basic properties of the TRE. In this follow-up paper we
state and prove two upper bounds on the variation of the TRE when either the
first or the second argument changes. These bounds are close in spirit to a
triangle inequality. For the ordinary relative entropy no such bounds are
possible due to the fact that the variation could be infinite.
