Telescopic Relative Entropy--II Triangle inequalities

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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(ρ||σ)=ρ(ρ-σ), by replacing the second argument σ by a convex combination of the first and the second argument, τ=aρ+(1-a)σ and dividing the result by - 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.