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I am reading the following paper: on quantum information theory

Does anybody understand the estimate at the bottom of page $7$

$$\left\lVert \rho-\rho_n \right\rVert \le 2 \left\lVert (id_A-P_n)\otimes id_R \rho (P_n \otimes id_R)\right\rVert_1+ 2 r_n\ ?$$

I have troubles understanding what happened in this step.

Can anybody explain this one to me?

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1 Answer 1

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Here is one estimate; it doesn't go exactly as in the paper, but it achieves the same. Since $r_n\to0$, for $n$ big enough one may assume that $1-r_n\geq1/2$. One has \begin{align} \|\rho-\rho_n\|_1 &\leq\|\rho-\rho(P_n\otimes I_R)\|_1+\|\rho(P_n\otimes I_R)-(P_n\otimes I_r)\rho(P_n\otimes I_R)\|_1+\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \|(P_n\otimes I_R)\rho(P_n\otimes I_R)-(1-r_n)^{-1}(P_n\otimes I_r)\rho(P_n\otimes I_R)\|_1\\ \ \\ &=\|\rho[(I_A-P_n)\otimes I_R)]\|_1+\|((I_A-P_n)\otimes I_R)\rho(P_n\otimes I_R)\|_1\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\|(P_n\otimes I_R)\rho(P_n\otimes I_R)\|_1\,(1-(1-r_n)^{-1})\\ \ \\ &\leq 2\|\rho[(I_A-P_n)\otimes I_R)]\|_1+2r_n. \end{align} The estimates used at the end come from the inequality $\|AB\|_1\leq\|A\|\,\|B\|_1$, and the fact that $\|P_n\otimes I_R\|=1$, $\|\rho\|_1=1$. The condition $1-r_n\geq1/2$ gives $(1-(1-r_n)^{-1})\leq 2r_n$.

Finally, if $Q$ is a projection, write $Q\rho=V|Q\rho|$ the polar decomposition, and then (using Cauchy-Schwarz, that $VV^*\leq I$ and that $\rho\geq0$) $$ \|Q\rho\|_1=\operatorname{Tr}(V^*Q\rho)=\operatorname{Tr}(V^*Q\rho^{1/2}\rho^{1/2}) \leq \operatorname{Tr}(\rho^{1/2}QVV^*Q\rho^{1/2})^{1/2}\,\operatorname{Tr}(\rho)^{1/2} \leq\operatorname{Tr}(Q\rho)^{1/2}. $$ So $$ \|\rho[(I_A-P_n)\otimes I_R)]\|_1\leq\sqrt{\operatorname{Tr}(\rho[(I_A-P_n)\otimes I_R)])}=\sqrt{r_n}. $$

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