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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 ...
Adam Zalcman's user avatar
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6 votes
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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 ...
The Yomster's user avatar
6 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(...
David Bar Moshe's user avatar
3 votes
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Does quantum mutual information encompass information only about quantum correlations, or does it encompass both classical and quantum correlations?

Definition and Intuition Quantum mutual information is defined as: $$ S(A:B) = S(A) + S(B) - S(A,B) $$ Here, $ S(A) $ is the Von Neumann entropy of the quantum state described by density matrix $ A $,...
banercat's user avatar
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3 votes
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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 ...
Rammus's user avatar
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3 votes
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Comparison of Quantum Mutual Information and Coherent Information with Classical Mutual Information

You are asking about the quantum analogue of a classical quantity, but quantum information contains features that have no classical analogue and so you won't find an objective answer. Here are some ...
forky40's user avatar
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3 votes
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Question regarding the monotonicity of Mutual Information of a tripartite state under multiple non-local commuting unitaries

Define the relative entropy between two states $\rho$ and $\sigma$ as $$ D(\rho\|\sigma) := \mathrm{tr}[\rho(\log \rho - \log \sigma)]\,. $$ Then we can write a the mutual information between $X$ and $...
Rammus's user avatar
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2 votes
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Can separable states have quantum mutual information larger than one?

If $\rho_{AB}$ is separable then $$ I(A:B) \leq \min\{H(A), H(B)\}. $$ To see this first note that $$ I(A:B) = H(A) + H(B) - H(AB) = H(A) - H(A|B). $$ Now consider the conditional entropy term $H(A|B) ...
Rammus's user avatar
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1 vote
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Is quantum mutual information an entanglement measure?

Coherent information is the measure of quantum correlations. Positivity of coherent information indicates that quantum correlations are present. But this is not an iff relation. Coherent information ...
FDGod's user avatar
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1 vote
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How to evaluate the mutual information of a classical-quantum state?

Note that $S(XB) = S(X) + S(B|X)$. You know that $S(B|X=i) = S(\rho_i)$. $S(B|X)$ is then just the entropy of $\rho_i$ on average i.e. $\sum_i p_i S(\rho_i)$. That leaves you with $$I(X:B) = S(X) + S(...
rnva's user avatar
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How to evaluate the mutual information of a classical-quantum state?

Let us express the state $\rho^{XB}$ in terms of eigen decomposition of $\rho_i$: $\rho^{XB}{=}\sum_{i,\lambda^{(i)}}p_i\lambda^{(i)}|i\rangle\langle i|{\otimes}|\lambda^{(i)}\rangle\langle\lambda^{(i)...
Abir's user avatar
  • 135
1 vote

How to prove that the mutual information is subadditive?

Unlike entropy, mutual information can be either subadditive or superadditive. If $A$, $B$, $C$ are bits with equal probabilities and $A=B=C$, then $$ I(A,B:C) = 1,\\ I(A:C) + I(B:C) = 2, $$ ...
Adam's user avatar
  • 11
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\...
GaussStrife's user avatar
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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\...
keisuke.akira's user avatar

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