Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 1

Suppose $X\to Y\to Z$ is a Markov chain, so $I(X;Z\mid Y)=0$. The chain rule for mutual information givesand alsobecause conditional mutual information is nonnegative. Therefore $I(X;Z)\le I(X;Y)$. Similarly,while $I(X,Y;Z)=I(X;Z)+I(Y;Z\mid X)\ge I(X;Z)$, so $I(X;Z)\le I(Y;Z)$. These are the two data processing inequalities. In particular, applying any deterministic function or Markov kernel to either argument cannot increase mutual information.

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Independence makesa Markov chain. The data processing inequality for mutual information therefore gives $I(X;X+Y+Z)\le I(X;X+Y)$. Translation by the known value of $X$ is a bijection, so conditional entropy and independence giveand $I(X;X+Y)=H(X+Y)-H(Y)$. Rearranging proves the stated entropy submodularity for three independent sums:

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Because the Entropic Ruzsa distance depends only on marginal distributions, take $X,Y,Z$ independent with the required marginals. Since $X-Z=(X-Y)+(Y-Z)$ is a function of $(X-Y,Y-Z)$, the data processing inequality for mutual information yieldsThe map $(X,X-Y,Y-Z)\mapsto (X,Y,Z)$ is a bijection. Using independence and the chain rule for information entropy, the left side iswhereas the right side is $H(X-Z)-H(Z)$. Hencewhere the final step is subadditivity of information entropy. Substituting this inequality into the definition of $d_R$ gives the Entropic Ruzsa triangle inequality

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Let $X_1,X_2,Y$ be independent, with $X_1,X_2$ distributed as $X$. Apply part (b) to $X_1,-Y,X_2$:Adding an independent random variable cannot decrease information entropy, soIn terms of Entropic Ruzsa distance, this is $d_R(X,-X)\le 2d_R(X,Y)$. The Entropic Ruzsa triangle inequality and invariance under simultaneous negation now givewhich is the Entropic Ruzsa sum-difference inequality.

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For independent $X,Y$, expand the Entropic Ruzsa sum-difference inequality from part (d):Collecting the information entropy terms givesReplacing $Y$ by $-Y$ interchanges sum and difference and preserves $H(Y)$, yielding the requested orientation

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