# Entanglement processing

- *Probability outcomes measuring distinguishable states $\leftrightarrow$ Information*

- Average information gain on state $\rho_A$ for measurement in its *eigenbasis*
$$S(\rho_A)=-\sum_k\lambda_k\log_2(\lambda_k)$$

### Majorization
- $$\text{least ordered } (\frac{1}{d},...,\frac{1}{d})\prec (1,0,...0) \text{ most ordered }$$
- definition
- $\lambda:$ less ordered
- $\mu:$ more ordered
1. $$\frac{1}{d}n\leq\sum_i^n\lambda_i\leq\sum_i^n\mu_i\leq 1$$
2. $$\lambda=D\cdot \mu$$
$D$ is a doubly-stochastic matrix.

3. $$\lambda = (\sum_\pi q_\pi\cdot P_\pi)\cdot \mu$$
$P_\pi$ is a random permutation.

- 🧐 *Can one define entanglement operationally?*
- ⭕️
> ⭐️⭐️⭐️ *Entanglement state* =
> State that cannot be created using only <font color=blue>local operations (LO)</font> coordinated by <font color=green>classical communication (CC)</font>

> ⭐️⭐️⭐️ *Entanglement proccessing* =
> modifying existing entanglement using only LOCC

> ⭐️⭐️⭐️ <font color=green>"More entangled" states have "less ordered" marginals</font> and can be converted to <font color=red>"less entangled" states with "more ordered" marginals</font>.

> <font color=purple>LOCC cannot increase entanglement rank</font>

### Entanglement Dilution
- <font color=green>$|e\rangle$</font>$=\sqrt{\frac{1}{2}}|00\rangle+\sqrt{\frac{1}{2}}|11\rangle$ <font color=blue>$\xrightarrow[]{Dilution}$</font> <font color=red>$|\varphi\rangle$</font>$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$
- POVM measurement
- can be done with successful probability $p_1+p_2=1$
- <font color=green>$|e\rangle$</font>$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$ <font color=blue>$\xrightarrow[]{Concentration}$</font> <font color=red>$|\varphi\rangle$</font>$=\sqrt{\frac{1}{2}}|00\rangle+\sqrt{\frac{1}{2}}|11\rangle$
- POVM measurement
- successful probability $p_1=\frac{2}{3}<1$

- majorization condition
$\lambda($<font color=green>$\rho$</font>$)=$<font color=green>$(\frac{2}{3},\frac{1}{3})$</font> <font color=blue>$\prec$</font> <font color=red>$p_1(\frac{1}{2},\frac{1}{2})$</font> $+$ <font color=purple>$p_2(1,0)$</font> $=$ <font color=red>$p_1$</font> $\lambda($<font color=red>$\rho_1$</font>$)+$<font color=purple>$p_2$</font> $\lambda($<font color=purple>$\rho_2$</font>$)$
$\therefore$ <font color=red>$p_1\leq 2/3$</font>, <font color=purple>$p_2\geq 1/3$</font> are indeed **optimal**.

> ⭐️⭐️⭐️ Single-copy conversion of *pure entangled states* is in general possible <font color=purple>only</font> for ensembles with some <font color=purple>probabilities $p_k$</font>
> $$|\psi\rangle\xrightarrow[]{\mathcal{M}_k}\{p_k,|\psi_k\rangle\}\Leftrightarrow\lambda(\rho_A)\prec\sum_k p_k\lambda (\rho_{Ak})$$

> ⭐️⭐️⭐️ <font color=red>LOCC quantum instrument = POVM + coarse graining</font>
> $$\begin{aligned}&p_k|\psi_k\rangle\langle\psi_k|=(\mathcal{M}_k\otimes\mathcal{I})|\psi\rangle\langle\psi| &\text{ POVM}\\ &\mathcal{M}_k=\sum_{i\in k}M_{Aik}\bullet M_{Aik}^\dagger &\text{ coarse-graining}\end{aligned}$$

- <font color=green>$|e\rangle$</font>$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$ <font color=blue>$\xrightarrow[]{Concentration}$</font> <font color=red>$|\varphi\rangle$</font>$=\sqrt{\frac{1}{2}}|00\rangle+\sqrt{\frac{1}{2}}|11\rangle$
*How can we do better?*
- by optimizing for $n>1$ copies we can increase conversion rate $$n=1\rightarrow n=\infin$$
- but not beyond the fundamental limit set by entanglement entropy
$$S(\rho)=-\sum_i\lambda_i\log_2 (\lambda_i)$$
where $\lambda_i=1/3,2/3$