# <font color=blue>Evolution</font>
- probability should stay probability $\Rightarrow\mathcal{E}$ is a stochastic map.
- Positivity Preservation (PP): $\mathcal{E}(\rho)\geq 0$
- Trace Preservation (TP): $\text{tr }\mathcal{E}(\rho)=1$

- 🧐 *These are general restrictions for quantum evolutions?*
- ❌ Difficult in principle: because of <font color=red>entanglement</font>
- ❌ Difficult in practice: PP is more complicated than <font color=red>complete positivity</font>

> ⭐️⭐️⭐️ <font color=blue>Evolutions (superoperators) </font><font color=red>correspond to </font><font color=green>states (operators)</font>
> $\rho'=$<font color=blue>$\sum_m M_m$</font><font color=green>$\rho$</font><font color=blue>$M_m^\dagger$</font> <font color=red>$\longleftrightarrow$</font> <font color=green>$\sum_m|M_m\rangle\langle M_m|$</font>
> We take <font color=red>maximally entangled</font> pure state <font color=red>$$\frac{1}{\sqrt{d}}|\mathbb{1}_{AB}\rangle=\frac{1}{\sqrt{d}}\sum_i|i_A\rangle\otimes|i_B\rangle$$</font>
> <font color=blue>$\sigma_{AB}$</font>$=($<font color=blue>$\mathcal{E}_A$</font>$\otimes\mathcal{I}_B)$<font color=red>$\frac{1}{d}|\mathbb{1}_{AB}\rangle\langle\mathbb{1}_{AB}|$</font>
> <font color=blue>$$\sigma_{AB}\xleftarrow{one-}\xrightarrow{to-one} \mathcal{E}_A$$</font>
> example:
> <font color=green>$\mathcal{E}_A$</font>$=\lambda_0\mathbb{1}_A\bullet\mathbb{1}_A+\lambda_1 Z_A\bullet Z_A$
> <font color=red>$\sigma_{AB}$</font>$=$<font color=pink>$\frac{1}{2}$</font>$\lambda_0|\mathbb{1}_{AB}\rangle\langle \mathbb{1}_{AB}|+\lambda_1 |Z_{AB}\rangle\langle Z_{AB}|$

- Finite-time evolution
$$
U(t) = e^{-iHt} \\
H: \text{Hamiltonian},\ H^\dagger = H \\
H \text{ and } H+\varphi \text{ describes the same evolution for any } \varphi \in \mathbb{R}
$$

### <font color=blue>Evolution in presence of entanglement</font>: <font color=red>Quantum Channels</font>
1. Linear map
2. Hermicity-preserving map (HP): $\rho_A=\rho_A^\dagger\Rightarrow\mathcal{E}_A(\rho_A)=(\mathcal{E}_A(\rho_A))^\dagger$
3. **Completely-positive map(CP)**: $\rho_{AB}\geq 0\Rightarrow \mathcal{E}_A\otimes\mathcal{I}_B(\rho_{AB})\geq 0$
> $\Rightarrow$ PP $\Rightarrow$ HP
4. Trace-preserving map(TP): $\text{tr }\mathcal{E}(\rho)=\text{tr }\rho=1$

### $$PP\neq CP$$
- PP-non-CP: <font color=red>negative probabilities</font>
- example: transposition superoperator $\mathcal{E}_A(\rho_A)=\rho_A^{T_A}$

> ⭐️⭐️⭐️ The difference between PP and CP constitudes what is *quantum* about *evolution:* <font color=red>Entanglement</font>

### Measurement model for evolution
- $$\mathcal{E}_A(\rho_A)=\sum_b p_b\rho_{Ab}=\sum_bM_{Ab}\rho_AM_{Ab}^\dagger$$

- 🧐 *Are evolutions derived from measurement models indeed CP-TP maps?*
- ⭕️

- 🧐 *Does every operator-sum evolution $\mathcal{E}_A$ correspond to a unique model $\{U_{AB},|0_B\rangle\}$?*
- ❌
- Nonuniqueness comes from
- <font color=green>$\langle b_B|$</font> $U_{AB}|c_B\rangle$
- $M_{Ab}=\sum_{b'}$<font color=green>$\langle b_B|b_B'\rangle$</font>$M_{Ab}'$

- 🧐 *Given any CP-TP map $\mathcal{E}$ how to compute a set of measurement operators $\{M_k\}$?*
1. compute a bipartite state corresponding to $\mathcal{E}$ $$\sigma_{AB}=(\mathcal{E}_A\otimes\mathcal{I}_B)\frac{1}{d}|\mathbb{1}_{AB}\rangle\langle\mathbb{1}_{AB}|$$
2. Diagonalize it $$\frac{1}{d}\sum_k|(M_k)_{AB}\rangle\langle (M_k)_{AB}|$$
3. write down operator sum with operators $M_k$ $$\mathcal{E}=\sum_kM_k\bullet M_k^\dagger$$

> These are called *canonical* measurement operator.

> *Purification of evolution =* tomography+purification
> $$\frac{1}{d}\mathbb{1}_A=\text{tr}_C\frac{1}{d}|\mathbb{1}_{AC}\rangle\langle\mathbb{1}_{AC}|$$

- $$\rho_A'=\mathcal{E}_A(\rho_A)=\text{tr}_B\{{U_{AB}(\rho_A\otimes |0_B\rangle\langle 0_B|)U_{AB}^\dagger}\}$$