Let $\rho, \sigma \in \mathfrak{D}(A)$ with $\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$ and spectral decomposition $$ \rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\psi_{x}\right| \quad \text { and } \quad \sigma=\sum_{y} q_{y}\left|\phi_{y}\right\rangle\left\langle\phi_{y}\right| $$ For any integer $n$ denote by $y^{n}:=\left(y_{1}, \ldots, y_{n}\right), q_{y^{n}}:=q_{y_{1}} \cdots q_{y_{n}},\left|\phi_{y^{n}}\right\rangle:=\left|\phi_{y_{1}}\right\rangle \otimes \cdots \otimes\left|\phi_{y_{n}}\right\rangle,$ and for any $\epsilon>0$ the subspace $\mathfrak{T}_{n, \epsilon} \subset A^{n}$ is defined as $$ T_{n, \epsilon}:=\operatorname{span}\left\{\left|\phi_{y^{n}}\right\rangle \in A^{n}:\left|\operatorname{Tr}[\rho \log \sigma]-\frac{1}{n} \log \left(q_{y^{n}}\right)\right| \leqslant \epsilon\right\} $$ Finally, denote by $\Pi_{n, \epsilon}$ the projection onto the subspace $\mathfrak{T}_{n, \epsilon}$ in $A^{n}$

Now, I want to show two things. First, I want to show that $$ 2^{n(\operatorname{Tr}[\rho \log \sigma]-\epsilon)} \Pi_{n, \epsilon} \leqslant \sigma^{\otimes n} \leqslant 2^{n(\operatorname{Tr}[\rho \log \sigma]+\epsilon)} \Pi_{n, \epsilon} $$ And I also want to show that for any $\epsilon>0$ $$ \lim _{n \rightarrow \infty} \operatorname{Tr}\left[\rho^{\otimes n} \Pi_{n, \epsilon}\right]=1 $$

$\textbf{ATTEMPT}$: I think I should start as follows. I denoted $r_{y}:=\left\langle\phi_{y}|\rho| \phi_{y}\right\rangle,$ and by $\tilde{Y}$ the random variable whose alphabet is the same as that of $Y,$ but his corresponding distribution is $\left\{r_{y}\right\} .$ Then, I want to show that $$ \operatorname{Tr}[\rho \log \sigma]=\mathbb{E}\left(\log q_{\tilde{Y}}\right) $$

I'm really confused and struggling with this problem. Could you please help me?


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