### Bell inequality violation finally done right

Tuesday, September 15th, 2015A few weeks ago, Hensen et al., of the Delft University of Technology and Barcelona, Spain, put out a paper reporting the first experiment that violates the Bell inequality in a way that closes off the two main loopholes simultaneously: the locality and detection loopholes. Well, at least with ~96% confidence. This is big news, not only because of the result itself, but because of the advances in experimental technique needed to achieve it. Last Friday, two renowned experimentalists—Chris Monroe of U. of Maryland and Jungsang Kim of Duke—visited MIT, and in addition to talking about their own exciting ion-trap work, they did a huge amount to help me understand the new Bell test experiment. So OK, let me try to explain this.

While some people like to make it more complicated, the Bell inequality is the following statement. Alice and Bob are cooperating with each other to win a certain game (the “CHSH game“) with the highest possible probability. They can agree on a strategy and share information and particles in advance, but then they can’t communicate once the game starts. Alice gets a uniform random bit *x*, and Bob gets a uniform random bit *y* (independent of *x*). Their goal is to output bits, *a* and *b* respectively, such that *a* XOR *b* = *x* AND *y*: in other words, such that *a* and *b* are different if and only if *x* and *y* are both 1. The Bell inequality says that, in any universe that satisfies the property of *local realism*, no matter which strategy they use, Alice and Bob can win the game at most 75% of the time (for example, by always outputting *a*=*b*=0).

What does local realism mean? It means that, after she receives her input *x*, any experiment Alice can perform in her lab has a definite result that might depend on *x*, on the state of her lab, and on whatever information she pre-shared with Bob, but at any rate, not on *Bob’s* input *y*. If you like: *a*=*a*(*x*,*w*) is a function of *x* and of the information w available before the game started, but is *not* a function of *y*. Likewise, *b*=*b*(*y*,*w*) is a function of *y* and *w*, but not of *x*. Perhaps the best way to explain local realism is that it’s the thing you believe in, if you believe all the physicists babbling about “quantum entanglement” just missed something completely obvious. Clearly, at the moment two “entangled” particles are created, but before they separate, one of them flips a tiny coin and then says to the other, “listen, if anyone asks, I’ll be spinning up and you’ll be spinning down.” Then the naïve, doofus physicists measure one particle, find it spinning down, and wonder how the other particle instantly “knows” to be spinning up—oooh, spooky! mysterious! Anyway, if that’s how you think it has to work, then you believe in local realism, and you must predict that Alice and Bob can win the CHSH game with probability at most 3/4.

What Bell observed in 1964 is that, even though quantum mechanics doesn’t let Alice send a *signal* to Bob (or vice versa) faster than the speed of light, it still makes a prediction about the CHSH game that conflicts with local realism. (And thus, quantum mechanics exhibits what one might not have realized beforehand was even a logical possibility: it doesn’t allow communication faster than light, but simulating the predictions of quantum mechanics in a classical universe *would* require faster-than-light communication.) In particular, if Alice and Bob share entangled qubits, say $$\frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}},$$ then there’s a simple protocol that lets them violate the Bell inequality, winning the CHSH game ~85% of the time (with probability (1+1/√2)/2 > 3/4). Starting in the 1970s, people did experiments that vindicated the prediction of quantum mechanics, and falsified local realism—or so the story goes.

The violation of the Bell inequality has a schizophrenic status in physics. To many of the physicists I know, Nature’s violating the Bell inequality is so trivial and obvious that it’s barely even worth doing the experiment: if people had just *understood and believed* Bohr and Heisenberg back in 1925, there would’ve been no need for this whole tiresome discussion. To others, however, the Bell inequality violation remains so unacceptable that some way must be found around it—from casting doubt on the experiments that have been done, to overthrowing basic presuppositions of science (e.g., our own “freedom” to generate random bits *x* and *y* to send to Alice and Bob respectively).

For several decades, there was a relatively conservative way out for local realist diehards, and that was to point to “loopholes”: imperfections in the existing experiments which meant that local realism was still theoretically compatible with the results, at least if one was willing to assume a sufficiently strange conspiracy.

Fine, you interject, but surely no one *literally* believed these little experimental imperfections would be the thing that would rescue local realism? Not so fast. Right here, on this blog, I’ve had people point to the loopholes as a reason to accept local realism and reject the reality of quantum entanglement. See, for example, the numerous comments by Teresa Mendes in my Whether Or Not God Plays Dice, I Do post. Arguing with Mendes back in 2012, I predicted that the two main loopholes would both be closed in a single experiment—and not merely eventually, but in, like, a decade. I was wrong: achieving this milestone took only a few years.

Before going further, let’s understand what the two main loopholes are (or rather, were).

The *locality loophole* arises because the measuring process takes time and Alice and Bob are not infinitely far apart. Thus, suppose that, the instant Alice *starts* measuring her particle, a secret signal starts flying toward Bob’s particle at the speed of light, revealing her choice of measurement setting (i.e., the value of *x*). Likewise, the instant Bob starts measuring his particle, his doing so sends a secret signal flying toward Alice’s particle, revealing the value of *y*. By the time the measurements are finished, a few microseconds later, there’s been plenty of time for the two particles to coordinate their responses to the measurements, despite being “classical under the hood.”

Meanwhile, the *detection loophole* arises because in practice, measurements of entangled particles—especially of photons—don’t always succeed in *finding* the particles, let alone ascertaining their properties. So one needs to select those runs of the experiment where Alice and Bob both find the particles, and discard all the “bad” runs where they don’t. This by itself wouldn’t be a problem, if not for the fact that the very same measurement that reveals whether the particles are there, is also the one that “counts” (i.e., where Alice and Bob feed *x* and *y* and get out *a* and *b*)!

To someone with a conspiratorial mind, this opens up the possibility that the measurement’s success or failure is somehow *correlated* with its result, in a way that could violate the Bell inequality despite there being no real entanglement. To illustrate, suppose that at the instant they’re created, one entangled particle says to the other: “listen, if Alice measures me in the *x*=0 basis, I’ll give the *a*=1 result. If Bob measures you in the *y*=1 basis, you give the *b*=1 result. In any other case, we’ll just evade detection and count this run as a loss.” In such a case, Alice and Bob will win the game with certainty, whenever it gets played at all—but that’s only because of the particles’ freedom to choose which rounds will count. Indeed, by randomly varying their “acceptable” *x* and *y* values from one round to the next, the particles can even make it look like *x* and *y* have no effect on the probability of a round’s succeeding.

Until a month ago, the state-of-the-art was that there were experiments that closed the locality loophole, and other experiments that closed the detection loophole, but there was no single experiment that closed both of them.

To close the locality loophole, “all you need” is a fast enough measurement on photons that are far enough apart. That way, even if the vast Einsteinian conspiracy is trying to send signals between Alice’s and Bob’s particles at the speed of light, to coordinate the answers classically, the whole experiment will be done before the signals can possibly have reached their destinations. Admittedly, as Nicolas Gisin once pointed out to me, there’s a philosophical difficulty in defining what we mean by the experiment being “done.” To some purists, a Bell experiment might only be “done” once the results (i.e., the values of *a* and *b*) are registered in human experimenters’ brains! And given the slowness of human reaction times, this might imply that a *real* Bell experiment ought to be carried out with astronauts on faraway space stations, or with Alice on the moon and Bob on earth (which, OK, would be cool). If we’re being reasonable, however, we can grant that the experiment is “done” once *a* and *b* are safely recorded in classical, macroscopic computer memories—in which case, given the speed of modern computer memories, separating Alice and Bob by half a kilometer can be enough. And indeed, experiments starting in 1998 (see for example here) have done exactly that; ~~the current record, unless I’m mistaken, is 18 kilometers~~. (**Update:** I *was* mistaken; it’s 144 kilometers.) Alas, since these experiments used hard-to-measure photons, they were still open to the detection loophole.

To close the detection loophole, the simplest approach is to use entangled qubits that (unlike photons) are slow and heavy and can be measured with success probability approaching 1. That’s exactly what various groups did starting in 2001 (see for example here), with trapped ions, superconducting qubits, and other systems. Alas, given current technology, these sorts of qubits are virtually impossible to move miles apart from each other without decohering them. So the experiments used qubits that were close together, leaving the locality loophole wide open.

So the problem boils down to: how do you create long-lasting, reliably-measurable entanglement between particles that are very far apart (e.g., in separate labs)? There are three basic ideas in Hensen et al.’s solution to this problem.

The first idea is to use a hybrid system. Ultimately, Hensen et al. create entanglement between electron spins in nitrogen vacancy centers in diamond (one of the hottest—or coolest?—experimental quantum information platforms today), in two labs that are about a mile away from each other. To get these faraway electron spins to talk to each other, they make them communicate via photons. If you stimulate an electron, it’ll sometimes emit a photon with which it’s entangled. Very occasionally, the two electrons you care about will even emit photons at the same time. In those cases, by routing those photons into optical fibers and then measuring the photons, it’s possible to entangle the electrons.

Wait, what? How does measuring the photons entangle the *electrons* from whence they came? This brings us to the second idea, *entanglement swapping*. The latter is a famous procedure to create entanglement between two particles A and B that have never interacted, by “merely” entangling A with another particle A’, entangling B with another particle B’, and then performing an entangled measurement on A’ and B’ and conditioning on its result. To illustrate, consider the state

$$ \frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}} \otimes \frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}} $$

and now imagine that we project the first and third qubits onto the state $$\frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}}.$$

If the measurement succeeds, you can check that we’ll be left with the state $$\frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}}$$ in the second and fourth qubits, even though those qubits were not entangled before.

So to recap: these two electron spins, in labs a mile away from each other, both have some probability of producing a photon. The photons, if produced, are routed to a third site, where *if* they’re both there, then an entangled measurement on both of them (and a conditioning on the results of that measurement) has some nonzero probability of causing the original electron spins to become entangled.

But there’s a problem: if you’ve been paying attention, all we’ve done is cause the electron spins to become entangled with some tiny, nonzero probability (something like 6.4×10^{-9} in the actual experiment). So then, why is this any improvement over the previous experiments, which just directly measured faraway entangled photons, and also had some small but nonzero probability of detecting them?

This leads to the third idea. The new setup is an improvement because, whenever the photon measurement succeeds, we know that the electron spins are there and that they’re entangled, *without having to measure the electron spins to tell us that*. In other words, we’ve decoupled the measurement that tells us whether we succeeded in creating an entangled pair, from the measurement that *uses* the entangled pair to violate the Bell inequality. And because of that decoupling, we can now just condition on the runs of the experiment where the entangled pair was there, without worrying that that will open up the detection loophole, biasing the results via some bizarre correlated conspiracy. It’s as if the whole experiment were simply switched off, except for those rare lucky occasions when an entangled spin pair gets created (with its creation heralded by the photons). On those rare occasions, Alice and Bob swing into action, measuring their respective spins within the brief window of time—about 4 microseconds—allowed by the locality loophole, seeking an additional morsel of evidence that entanglement is real. (Well, actually, Alice and Bob swing into action regardless; they only find out later whether this was one of the runs that “counted.”)

So, those are the main ideas (as well as I understand them); then there’s lots of engineering. In their setup, Hensen et al. were able to create just a few heralded entangled pairs per hour. This allowed them to produce 245 CHSH games for Alice and Bob to play, and to reject the hypothesis of local realism at ~96% confidence. Jungsang Kim explained to me that existing technologies could have produced many more events per hour, and hence, in a similar amount of time, “particle physics” (5σ or more) rather than “psychology” (2σ) levels of confidence that local realism is false. But in this type of experiment, everything is a tradeoff. Building not one but two labs for manipulating NV centers in diamond is extremely onerous, and Hensen et al. did what they had to do to get a significant result.

The basic idea here, of using photons to entangle longer-lasting qubits, is useful for more than pulverizing local realism. In particular, the idea is a major part of current proposals for how to build a scalable ion-trap quantum computer. Because of cross-talk, you can’t feasibly put more than 10 or so ions in the same trap while keeping all of them coherent and controllable. So the current ideas for scaling up involve having lots of separate traps—but in that case, one will sometimes need to perform a Controlled-NOT, or some other 2-qubit gate, between a qubit in one trap and a qubit in another. This can be achieved using the Gottesman-Chuang technique of gate teleportation, *provided* you have reliable entanglement between the traps. But how do you create such entanglement? Aha: the current idea is to entangle the ions by using photons as intermediaries, very similar in spirit to what Hensen et al. do.

At a more fundamental level, will this experiment finally convince everyone that local realism is dead, and that quantum mechanics might indeed be the operating system of reality? Alas, I predict that those who confidently predicted that a loophole-free Bell test could never be done, will simply find some new way to wiggle out, without admitting the slightest problem for their previous view. This prediction, you might say, is based on a different kind of realism.