# For a little more complementarity

It’s new paper season here at Complementary Slackness! Today’s offering is all about two conditions on when a maximally entangled state can (approximately) be recovered from a given bipartite state, using only local operations at one end. Why is this interesting, you ask? Because this operation is essentially the final step in an entanglement distillation or quantum communication procedure, so knowing when it can be done is the first step in designing such protocols. It’s best to begin with the end in mind, as they say.

The two conditions are based on the idea of breaking quantum information down into two classical pieces, such that these pieces successfully reassembled into the quantum whole. Recall the original approach to quantum error correction in which quantum errors are digitized, in particular into amplitude and phase errors. Each of these errors is effectively classical, and correcting both restores the original quantum information. It turns out, though, that this is a suboptimal way to handle quantum errors. So here we shift the focus away from making the quantum errors classical to dealing with classical information inherent in the bipartite quantum state itself.

The states we’re after, maximally entangled states, have the property that, given one system, we can predict either of two complementary measurements made on the other. This is clear just by writing down the maximally entangled state, for two qubits A and B say: $|\Phi\rangle=\tfrac{1}{\sqrt{2}}(|0,0\rangle+|1,1\rangle)=\tfrac{1}{\sqrt{2}}(|+,+\rangle+|-,-\rangle)$. A and B are perfectly correlated in both the $\{|0\rangle,|1\rangle\}$ and $\{|+\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle),|-\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)\}$ bases, so measurement in either of these bases always produces correlated outcomes. The bases are complementary because given a state prepared in one basis, measuring it in the other produces a completely random outcome. Thus, the entangled state has the advertised property: to predict the outcome of measuring A in a given basis, just perform the same measurement on B, and this will work for either complementary basis. Furthermore, this property actually defines the state $|\Phi\rangle$.

The first condition just generalizes this, in several ways. First, we don’t demand that the prediction be perfect, and in exchange we’re willing to accept a good approximation to the state $|\Phi\rangle$. This is important for designing quantum communication protocols, as the amount of approximate entanglement a channel can produce is generally higher than the amount of perfect entanglement it can produce. Second, we don’t require the prediction measurements on B to have the same form as the measurement on A; any measurements which can predict the outcomes for either basis will do. This means the state spaces of A and B can be quite different. Lastly, and most importantly, we only require that one of the measurements, say the $\{|0\rangle,|1\rangle\}$ basis be predictable from B; the other measurement $\{|+\rangle,|-\rangle\}$ need only be predictable using B plus a sort of copy of A in the $\{|0\rangle,|1\rangle\}$ basis. This might sound like cheating, but it stems to the fact that the entanglement recovery operation just performs the two prediction measurements in sequence. Since the first one is good at predicting the $\{|0\rangle,|1\rangle\}$ measurement, this information is available to predict the outcome of measuring in the other basis. And it’s useful if there are correlations between the two.

This condition was implicitly used here to construct protocols for entanglement distillation which operate at the optimal rate (though you wouldn’t have guessed this from the title), so this approach successfully avoids the pitfalls encountered by error digitization. And it can be used to perform quantum communication over noisy channels, at least if there’s an extra classical channel available: Just use the channel to distribute bipartite states, distill entanglement, and then teleport the quantum information you actually want to send using the classical channel and the entanglement. The nice thing about this approach is that the necessary recovery operation (or decoder) is actually constructed, rather than just shown to exist, which might be an advantage in exploring more efficient schemes. Ok, you might be thinking that this isn’t really constructive, since you have to supply (construct) the two prediction measurements. But these are involved in a classical task and it is presumably easier to divide the full quantum problem into two classical pieces.

The second condition is sort of the inverse of the first — the two outcomes of the two measurements on A should be completely unpredictable to someone with access to the purification of A and B, i.e. a system R such that ABR is a pure state. Actually, a caveat similar to the last point above applies here, so that one of the measurments should be unpredictable even if given R and knowledge of the other measurement outcome. This is a sort of decomposition of the often-used decoupling approach into two classical pieces. I plan to say a bit more about that in a future post, but the point is that entanglement is recoverable from A and B if A is uncorrelated with R, i.e. the joint quantum state just factors into independent states for each system. Here we’re saying essentially the same thing using classical information instead. Ultimately this condition comes from the first by using the duality explored in a previous paper, discussed here. This approach isn’t constructive, but it’s aesthetically pleasing to see that it holds as well.