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7 votes
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Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$

TL;DR: The key observation is that Schmidt basis on a subsystem consists of eigenvectors of the reduced state of that subsystem. Consequently, if the reduced state is a product state then its Schmidt ...
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5 votes
Accepted

What exactly is the relation between the Holevo quantity and the mutual information?

Right, they are quite similar. The Holevo bound is a bound on the amount of accessible information between your quantum system and your classical system. The I(X;B) object written in the HSW theorem ...
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5 votes
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Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?

The mutual information can be written in terms of the relative entropy, please see Nielsen and Chuang (the entropy Venn diagram figure 11.2). I am writing the equation in the question's notation: $$I(...
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3 votes
Accepted

How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?

Here's a relatively simple proof just based on the data processing inequality (DPI) for the relative entropy $D(\rho\|\sigma) = \mathrm{tr}[\rho (\log \rho - \log \sigma)]$ -- if you're willing to ...
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  • 3,837
1 vote

How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?

To expand on @Purva Tharke's comment, the strong subadditivity inequality states: $$H(ABC)+H(C) \le H(AC) + H(BC)$$ $$=H(ABC)+H(C) +H(C) -H(C) \le H(AC) + H(BC)$$ $$=H(AB|C) \le H(A|C) + H(B|C)$$ $$=0\...
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  • 1,113
1 vote

What does vanishishing mutual information of the Choi imply about the channel?

Here's a guess: they might be related to entanglement-breaking channels (also known as measure-and-prepare channels, quantum-classical channels, etc.). Any channel of the form, $$ \Phi(\rho) = \sum\...
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