# Entanglement distillation

Difficulty

by Filip Rozpedek

## Entanglement, is it always pure?

You have probably already heard about entanglement. Entanglement is this fascinating phenomenon, in which two distant objects can manifest correlations, even if they are far far away from each other. You may have also heard that remote entanglement is a necessary ingredient for many quantum information processing tasks. For example, in quantum cryptography, two people who hold entangled particles can use those correlations to obtain shared secret keys, whose security is guaranteed by the laws of quantum mechanics. Today, we will not discuss how to use remote entanglement, but rather, what to do if our entanglement is too weak.

Unfortunately, fully entangled states which are perfectly correlated are a great idealization and from an experimental perspective almost impossible to create. In general, there can be many reasons for this, e.g. our experimental equipment isn’t perfect or we cannot maintain our quantum system long enough. All those things combined lead to various forms of contamination of the entanglement. That is, the correlations become weaker and completely diluted in a mixture of various other quantum states.
So what do we do with those so-called “partially entangled states”? Let us say that two parties working at QuTech, whom we call Alice and Bob, share those partially entangled states and would like to use them to generate shared secret keys. Let us also say that their experimental setup allows them to produce partially entangled states very fast, but the amount of entanglement in each of them is insufficient to generate shared secret keys. It is known from Quantum theory that it is not possible to increase the amount of entanglement in a given quantum state by only performing operations on the entangled particles locally and exchanging classical messages. It seems that there is no choice for Alice and Bob, but to go home without a key.

## Entanglement is like alcohol, it is distillable

Fortunately, Alice and Bob have two friends (well maybe they have more, but this remains unknown), called Alessandro and Bruno, who are the experts on decoherence (as can be easily verified in this article). It turns out that Alessandro and Bruno have a special skill that could rescue Alice and Bob. Namely, they can perform distillation.

In Fort William, Alessandro and Bruno distill alcohol for the production of whisky. So what do Alessandro and Bruno exactly do? Well, they start with a form of beer that is stored in the massive stills as the ones on the picture. This beer is a mixture of pure alcohol (8-10%), water and some other substances.
For the high quality whisky Alessandro and Bruno need to extract this pure ethanol from everything else. Fortunately ethanol boils at around 78.5 degrees Celsius and so by heating the stills to this temperature, the alcohol can be evaporated. Later, the alcohol condenses outside of the stills and one is able to separate it from the “contamination” that shouldn’t end up in any high quality whisky. Well, this is a big approximation, because to separate alcohol from the water one requires to perform multiple rounds of distillation.

Alice and Bob became very interested in the work of Alessandro and Bruno because they started to suspect that entanglement is a bit like alcohol. Weakly entangled states are “contaminated”, but it may be possible to use the techniques of Alessandro and Bruno, to distill the entanglement out of the mixture of all those different quantum states. Hence, Alice and Bob invited Alessandro and Bruno to QuTech and asked them to apply their precious skill to their shared partially entangled states. You might be puzzled now. How will Alessandro and Bruno be performing entanglement distillation? Will they heat up the entangled particles? It’s a bit different.

In entanglement distillation Alice and Bob will use multiple copies of the partially entangled states. The hope of Alessandro and Bruno is that although each of these states has some, small amount of entanglement, it is possible to mix those copies together, and eliminate the contamination by extracting only the entangled part, so that at the end they obtain a single highly entangled state. Guided by their experience in Fort William, Alessandro and Bruno expect that for realistic procedures, a single run of such a distillation would not be enough. Similar to alcohol distillation, where the total amount of alcohol cannot be increased, entanglement can be concentrated into a smaller number of copies, using only local operations and classical communication, without increasing the total amount of entanglement. This shows that, unfortunately, Alessandro and Bruno are not superhuman and remain bounded by the laws of nature.

## From two copies to one

Having absorbed these abstract concepts you might think: “What a story!”. To illustrate these methods we will focus on some simple examples. Alessandro and Bruno are very interested in protocols that can be implemented with current technology. Therefore they will not consider any operations for which Alice and Bob would need a large number of partially entangled states.

Let us look at the methods of entanglement distillation applicable to small systems. Firstly, let’s say that Alice and Bob can only generate two-qubit states in their lab at QuTech. Specifically, there exist simple distillation procedures in which Alice and Bob perform simple local operations on two shared copies of partially entangled states. Those operations include a so-called controlled NOT gate. This quantum gate is very analogous to its classical counterpart. In the classical scenario depending on the value of the first bit, the second bit is flipped or not. Now in the quantum case our first “controlling bit” can be in superposition of 0 and 1, in which case in superposition we flip and not flip the second bit.

In our distillation procedure Alessandro and Bruno instruct Alice and Bob to apply this controlled NOT gate to the two qubits that each of them have. The qubits are chosen such that the controlling qubit of Alice is entangled with the controlling qubit of Bob and the same is true for the target qubits. At the end of the whole procedure both Alice and Bob measure their target qubits. Finally, conditioned on the measurement result, which they report to each other over the phone, they either keep or discard the resulting shared two-qubit state. That is e.g. if they both obtained the same outcome they keep the state and discard it otherwise (the specific details depend on the chosen procedure, that should depend on the type of partially entangled states that Alice and Bob share). The result is that successful measurement output always results in a state that has more entanglement than each of the two individual input states. The cost is that this success is normally only probabilistic, since the measurement outcome must pass the described test. However, since Alice and Bob can generate their states at a very fast rate this is not a problem for them.

## Can we just filter one copy?

Unfortunately, after careful examination of their experimental setup, Alice and Bob realize that they can only maintain one such state at a time. So the ability to concentrate entanglement from multiple copies into a single copy of the state is unfortunately of no use for them at the moment. We’ve said that it is not possible to increase the total amount of entanglement using only local operations and classical communication. Fortunately probabilistic procedures come to the rescue. There is nothing that forbids increasing entanglement only probabilistically, such that on average it does not increase. In fact there exist so called filtering protocols where by just performing a smart measurement on their single qubits, Alice and Bob can from one copy of some specific partially entangled states, obtain one more entangled copy, paying the cost of probabilistically ending up with nothing. Hence we see that Alessandro and Bruno can help Alice and Bob in their great endeavor so that Alice and Bob can go home with a (shared) key.