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In (Vidal 1999) they prove that, given any pair of pure bipartite states written as $$|\Psi\rangle = \sum_{i=1}^n \sqrt{\alpha_i}|i,i\rangle, \qquad |\Phi\rangle = \sum_{i=1}^n \sqrt{\beta_i}|i,i\rangle,$$ with $\alpha_1\ge \alpha_2\ge\cdots\ge \alpha_n$, $\beta_1\ge \beta_2\ge\cdots\ge \beta_n$, the maximum probability of converting $|\Psi\rangle$ into $|\Phi\rangle$ by means of any local strategy is given by $$P(\Psi\to\Phi) = \min_{\ell\in[1,n]} \frac{\sum_{i=\ell}^{n}\alpha_i}{\sum_{i=\ell}^{n}\beta_i}.$$ What are some toy examples to better understand the idea behind the strategy, and the optimality proof?

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  • $\begingroup$ "Toy examples" and "optimality proof" feel a bit contradictory ... (Note that the optimality proof for LOCC protocols involves proving first that one round (actually half round) of communication suffices ... now this can indeed also be illustrated by examples, though I don't feel this would be any more insightful than the general proof.) $\endgroup$ Mar 1 at 18:48
  • $\begingroup$ @NorbertSchuch I disagree. One can certainly find explicit examples and specialise the proof to find the optimal conversion protocols. I personally find it easier to grok the intuition behind a calculation knowing how it applies to a number of specific toy examples showing the ideas involved $\endgroup$
    – glS
    Mar 2 at 18:02
  • $\begingroup$ I think the proof that LOCC can be condensed to a single round is one of the arguments which is more annying to write down with qubits than in the general case (or you just take the general case and replace $d$ by $2$). Do you know the proof? $\endgroup$ Mar 2 at 21:58
  • $\begingroup$ ... that being said, I would argue the question is too broad. Giving an example how to do conversion and the optimality argument are two entirely different things. $\endgroup$ Mar 2 at 21:59
  • $\begingroup$ (On the other hand, if you meet me next to a blackboard, I will be happy to explain both through examples.) $\endgroup$ Mar 3 at 17:16

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I always rather enjoyed the examples in this paper:

https://arxiv.org/abs/quant-ph/9905071

They are more stated in terms of deterministic conversion, but you could just as well calculate probabilities of conversion in the ones that cannot be done deterministically.

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