# Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone!

## 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