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!