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 \[n=1\rightarrow n=\infin\]
  • but not beyond the fundamental limit set by entanglement entropy
    \[S(\rho)=-\sum_i\lambda_i\log_2 (\lambda_i)\]
    where \(\lambda_i=1/3,2/3\)