Bell’s theorem famously proclaims that there exists no local hidden variable (LHV) model for quantum theory. This takes place in a scenario where Alice and Bob share an entangled pair of particles and privately choose how to measure their shares, but in space-like separated events, to ensure that no communication could take place between them. In the simplest case, the measurements are projections onto a basis and the state consists of a pair of qubits, which can (up to local rotations) be written as
\begin{equation}
|\psi\rangle = \sqrt{p}\, | 00\rangle+\sqrt{1-p}\,| 11\rangle
\end{equation}
for some $p\in[\frac{1}{2},1]$.

Knowing that for any choice of $p\neq 1$, the LHVs cannot in general reproduce the correlations between Alice and Bob, it becomes interesting to ask what more such a model would need in order to do the job. There are many ways to approach that question, but a particularly elegant one is to ask for the smallest number of superluminal bits of communication. That is, what if the LHVs could, somehow, break the no-signaling imposed from space-like separation, and get some bits through between Alice and Bob? And if they can pull that trick off, how many bits would they need to simulate the quantum correlations?

At first sight, one might think infinitely many bits are needed. After all, Hilbert space hosts infinitely many measurements. But indeed, at the turn of the century, it was realized that the cost is finite: we do not need to simulate Hilbert spaces, only the statistics of the Born rule. A milestone result came in 2003, when Toner and Bacon [1] showed that for the maximally entangled state ($p=1/2$) only one superluminal bit is needed. In other words, no matter which basis measurements Alice and Bob apply to that state, there exists an LHV model that can simulate their statistics, provided it can cheat the no-signaling principle by just a single bit.

But what about non-maximally entangled states, i.e. $p\neq \frac{1}{2}$. It has not been an uncommon belief that non-maximally entangled states should be more costly to simulate than the maximally entangled state. Naively, this might appear strange, but the pitfall lies here: maximally entangled states are only ‘maximal’ with respect to local operations and classical communication, but in Bell scenarios there is no classical communication. In fact, there are many Bell inequalities that are maximally violated by non-maximally entangled states. The distinct feature of non-maximally entangled states is that their reductions are not maximally mixed, and hence the local statistics must not be uniformly random. Perhaps complicated marginals are more pricy for LHVs? Actually, Toner and Bacon proved that if these non-maximally entangled states indeed are more costly, they cannot be that much more expensive: two bits of superluminal communication is always sufficient [1]. But are two bits also necessary? The interest here lies not in having an extra bit here or there, but in the fundamental question of whether non-maximally entangled states, by this perspective, can be considered ‘more nonlocal’ than the maximally entangled state. This question has remained open for the past 20 years.

Now, thanks to the works of M. J. Renner and M. T. Quintino [2], and P. Sidajaya et al, [3], we are much closer to having a complete answer. Renner and Quintino first show that Toner and Bacon’s two bits are not necessary. They do this by constructing an LHV protocol supplemented with a three-valued message, which simulates all quantum predictions for any $p$. A clear improvement, but it does not answer the basic conceptual question: are non-maximally entangled states more expensive to simulate? To tackle this, the authors put forward an elegant simulation protocol. By identifying cleverly biased LHV distributions over the Bloch sphere, they achieve the simulation with just one bit. Their protocol works for any state in the range $p\in[\approx 0.835,1]$. Recall that this is the ‘weaker’ regime of entangled states, far away from the maximally entangled state, where the simulation cost is now identical.

However, there is little reason to believe that $p\approx 0.835$ is a fundamental limit, rather than a feature of the model discovered in [2]. This is where the results of [3] come into play, and they go a long way towards showing that one bit is sufficient for every non-maximally entangled state. Sidajaya et al., use an artificial neural network which takes as input the Bloch vectors of Alice and Bob and outputs a best approximation with a distribution obtained from an LHV model with one bit of communication. This way, they are able to systematically find, with convincing precision, simulations for the statistics of non-maximally entangled states. What’s particularly interesting is that the authors even manage to draw inspiration from these numerics, to propose analytical simulation protocols that approximate the quantum distributions. Now, the simulations are no longer limited by computational bottlenecks, such has how many measurements are performed per party. True, these analytical models are not exact simulations, but they have a good statistical power, and therefore appear to render pretty much untenable the hypothesis of non-maximally entangled states having a higher classical cost.

However, there is still a way for two-qubit nonlocality to cost more than a bit: these results are restricted to projective measurements. In principle, the use of a non-projective qubit measurement, with up to four outcomes, could reveal nonlocality with a higher price tag for the LHVs. It does not appear likely that this would be the case, but the question is certainly interesting. It was recently shown that when it comes to simulating single-qubit correlations, no such increase is associated with non-projective measurements [4].

Now, given the advances of [2,3], it might be time to start looking at the next frontier, namely the simulation of quantum nonlocality from higher-dimensional entanglement. Nonlocality from higher dimensions can break the one-bit barrier [5], but it may come as a surprise that simple examples are not so easy to find. The arguably most elegant example was recently found in [6]. It builds on the famous Magic Square game but requires local dimension 16. One might wonder if there exists an example that is both simple and noise-tolerant enough to be experimentally viable. Actually, from a fundamental point of view, it is not even certain that the size of the message alphabet needed to simulate a pair of entangled qutrits must be finite. An infinite simulation cost would be intriguing and raise many more questions. The author thinks that it is unlikely, but is used to being wrong.

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