What is the Procrustean method for entanglement distillation?

[meteor]

Please explain what's entanglement distillation and what's entanglement concentration. I'm mistaken to think that these two techniques are related somehow?

 

[slyboy]

The difference is mainly historical in nature. The original paper that proved that multiple copies of any pure bipartite state can be reversibly converted into Bell states called the technique "entanglement concentration". The case of converting mixed bipartite states into Bell states is often called "entanglement purification". "Entanglement distillation" refers to either of these two tasks, and it is now common to just call everything distillation, without referring to concentration or purification.

 

[meteor]

Thanks. I've looked a few papers, and this seems interesting
http://pi1.physi.uni-heidelberg.de/physi/atph/teaching/seminar/Sem_SS_2004/literatur/2306/repeaterPRL.pdf

it says that there're mainly 3 ways to achieve entanglement distillation:
1)the Procrustean method
2)entanglement concentration (aka Schmidt decomposition scheme)
3)the purification method

the 2) and the 3) are the methods that you say, so I will look what's the Procrustean method

 

[slyboy]

The entanglement concentration scheme (based on Schmidt projection) becomes optimal for pure states in the limit where the number of copies of the state tends to infinity. However, even for moderately large numbers, the measurements require a great deal of interation between the copies of the state and so are hard to implement in practice.

It is much easier to use a scheme where only measurements on a single copy of the state are required. Because entanglement cannot increase on average under LOCC, such a scheme must necessarily be probabalistic, i.e. it has some probability of failing, resulting in an unentangled state.

The method is relatively easy to describe. Suppose Alice and Bob share a state
$$\sqrt{p} |00 \rangle + \sqrt{1-p} |11 \rangle$$
where 1/2 < p < 1.

Alice performs a measurement, with outcomes corresponding to the operators
$$M_1 = \sqrt{\frac{1-p}{p}} | 0 \rangle \langle 0 | + |1 \rangle \langle 1 |$$
$$M_2 = \sqrt{\frac{2p-1}{p}} |0 \rangle \langle 0 |$$

It is relatively easy to check that these are valid measurement operators, i.e. $$M_1^\dagger M_1 + M_2^\dagger M_2 = I$$, where $$I$$ is the identity operator. Hence, it can be implemented by bringing in an ancillary qubit, performing a unitary interaction between Alice's qubit and the ancilla, and then finally measuring the ancilla.

It can also be checked that on obtaining outcome 1, the state is left in a maximally entangled state and on obtaining outcome 2, the state is left in a product state. The scheme is successful with probability 2(1-p) and this is actually the optimal success probability. The best procustean measurement for any finite dimensional, bipartite pure state has been found by Vidal (quant-ph/9902033).

By the way, the method is named after Procrustes, a mythical Greek giant who stretched or shortened his captives in order to make them fit into his beds. Suppose only maximally entangled or product states fit into his beds. Then, he can make any state fit into a bed using the method, but he might have to chop off all the entanglement to do it. The name is due to Charlie Bennett, who seems to be responsible for most of the more creative jargon in the field, especially when it comes to Greek mythology.