Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, who then receives the output $y_j$. The conditional probability to obtain $\mathbf x$ given $\mathbf y$ is $$p(\mathbf y|\mathbf x)=\prod_{j=1}^np_j(y_j|x_j), $$ and for each probability $p_j(y_j|x_j)$ we may define the mutual information $I_j(x_j:y_j)$. I should prove that $$I(\mathbf x:\mathbf y)\le \sum_{j=1}^n I_j(x_j:y_j), $$ that is, the mutual information is subadditive. What is the simplest way to go about this?

  • $\begingroup$ Are you certain that what you are trying to prove is actually provable or are you trying out a new proof? $\endgroup$ Jul 25, 2021 at 1:43
  • $\begingroup$ @QuestionEverything In all honesty, this was given as an exercise in my course, so I hope my professor wasn't trolling us :-) $\endgroup$ Jul 25, 2021 at 1:48
  • $\begingroup$ I see, :) I haven't seen this proof before. $\endgroup$ Jul 25, 2021 at 2:14
  • $\begingroup$ What is quantum about this? It seems to be a question about classical information theory. $\endgroup$
    – DaftWullie
    Jul 26, 2021 at 6:50
  • $\begingroup$ @DaftWullie Yes, it is. I hope it's not off-topic for the site... I'm posting here by default as all this was done in a course in quantum information theory. After all, the tag information-theory is also dedicated to 'information theory in the classical sense'. $\endgroup$ Jul 26, 2021 at 12:59


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