# Questions tagged [mutual-information]

For questions about the quantum mutual information, a measure of correlations between subsystems of a quantum state. Can also be used for questions about classical mutual information, as long as the general question is of relevance to quantum information science.

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### What does a quantum mutual information larger than its classical upper bound represent?

Let $\rho$ be a bipartite state. Its quantum mutual information is defined as $$\newcommand{\tr}{\operatorname{tr}}I(\rho) = S(\tr_B(\rho)) + S(\tr_A(\rho)) - S(\rho),$$ where $S(\sigma)$ is the von ...
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### What does vanishishing mutual information of the Choi imply about the channel?

Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'. For the ...
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### How is $I(\rho^{QC})=I_{CC}(\rho^{QC})$

On page 3 of this paper, for the proof of theorem 1, it states that, using Lemma 2 from the previous page, that if $$I(\Lambda_{A}\otimes\Gamma_{B})[\rho]=I(\rho))$$ then there exists $\Lambda_{A}^{*}$...
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### How to prove that the mutual information is subadditive?

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, ...
### How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?
I was reading Wilde's 'Quantum Information Theory' and saw the following theorem at chapter 11 $(11.7.2)$: $$I(A; B | C) \ge 0,$$ where, $$I(A;B|C) := H(A|C) + H(B | C) - H(AB|C).$$ I know that ...