7
votes
Accepted
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 ...
- 14.8k
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 ...
- 66
5
votes
Accepted
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:
$$I(...
- 2,305
3
votes
Accepted
How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?
Here's a relatively simple proof just based on the data processing inequality (DPI) for the relative entropy $D(\rho\|\sigma) = \mathrm{tr}[\rho (\log \rho - \log \sigma)]$ -- if you're willing to ...
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1
vote
How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?
To expand on @Purva Tharke's comment, the strong subadditivity inequality states: $$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\...
- 1,145
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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\...
- 1,713
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