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Question

I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices $A,B$, does the following inequality holds? $$H\left(\langle i|A|i\rangle\parallel\langle i|B|i\rangle\right)\leq S(A\parallel B)$$ where $H(\cdot\parallel\cdot)$ and $S(\cdot\parallel\cdot)$ are general relative entropy and general quantum relative entropy respectively defined as following.

Denote the general negative relative entropy as $$H(p\parallel q)=\sum_i\left(p_i\log\frac{p_i}{q_i}-p_i+q_i\right),$$ and the general quantum relative entropy (von Neumann divergence) as $$S(A\parallel B)=\mathrm{tr}\left(A(\log A-\log B)-A+B\right),$$ where $A,B$ are both positive semi-definite matrices which are not necessarily to be density matrices.

I have repeated the experiments about this inequality with more than 100,000 random generalized SPD matrices using Matlab, and it always holds. However, does it really hold in theory?

Motivation

What motivates this conjecture is the similar inequality of squared Euclidean distance and squared Frobenius norm. Specifically, given a unit vector $|i\rangle$ and two Hermitian matrices $A,B$, the following inequality holds with simple matrix calculations. $$\left\|\langle i|A|i\rangle-\langle i|B|i\rangle\right\|^2\leq\|A-B\|_\mathrm{F}^2.$$ In context of Bregman divergence, squared Euclidean distance and squared Frobenius norm are Bregman divergence and Bregman matrix divergence with the same seed function $\varphi(\mathbf{x})=\sum_ix_i^2$, while general relative entropy and general quantum relative entropy with the same seed function $\varphi(\mathbf{x})=\sum_i\left(x_i\log x_i-x_i\right)$.

Could anyone please give me some help for this question or recommend some relevant papers? Any suggestion will be appreciated.

Thank you very much!

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I found a proof of this problem and the inequality does holds. A related problem has been studied several years ago. I have presented the outline of the proof in another question as the following url: mathoverflow.net/questions/93149/… If there is any mistake in the proof, please let me know. Thank you! –  ppyang Apr 17 '12 at 17:39
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