[Quantum Information] Entanglement processing

Entanglement processing

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Wegewijs, Maarten

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 local operations (LO) coordinated by classical communication (CC)

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

⭐️⭐️⭐️ “More entangled” states have “less ordered” marginals and can be converted to “less entangled” states with “more ordered” marginals.

LOCC cannot increase entanglement rank

Entanglement Dilution

• $|e\rangle$$=\sqrt{\frac{1}{2}}|00\rangle+\sqrt{\frac{1}{2}}|11\rangle$ $\xrightarrow[]{Dilution}$ $|\varphi\rangle$$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$

  • POVM measurement
  • can be done with successful probability $p_1+p_2=1$

• $|e\rangle$$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$ $\xrightarrow[]{Concentration}$ $|\varphi\rangle$$=\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($$\rho$$)=$$(\frac{2}{3},\frac{1}{3})$ $\prec$ $p_1(\frac{1}{2},\frac{1}{2})$ $+$ $p_2(1,0)$ $=$ $p_1$ $\lambda($$\rho_1$$)+$$p_2$ $\lambda($$\rho_2$$)$
  $\therefore$ $p_1\leq 2/3$, $p_2\geq 1/3$ are indeed optimal.

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

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

• $|e\rangle$$=\sqrt{\frac{2}{3}}|00\rangle+\sqrt{\frac{1}{3}}|11\rangle$ $\xrightarrow[]{Concentration}$ $|\varphi\rangle$$=\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  $$n=1\rightarrow n=\infty$$
• but not beyond the fundamental limit set by entanglement entropy
  $$S(\rho)=-\sum_i\lambda_i\log_2 (\lambda_i)$$
  where $\lambda_i=1/3,2/3$

Readings

Nielsen and Chuang (basic)

    • 12.5.1 Transforming bi-partite pure state entanglement

    • 12.5.2 Entanglement distillation and dilution

Cover and Thomas, Elements of information theory (basic)

    • 5.2 Kraft inequality

    • 5.3 Optimal codes

Wilde (basic)

    • 2.1 Data compression

Wilde (advanced)

    • 19 Entanglement Manipulation

Horodecki’s

    • XIII. Manipulations of entanglement and irreversibility

        A. LOCC manipulations on pure entangled states —exact case