In 1964, the Northern Irish physicist John Bell found a way to put this paradoxical notion to the test. He showed that if particles have definite states even when no one is looking (a concept known as “realism”) and if indeed no signal travels faster than light (“locality”), then there is an upper limit to the amount of correlation that can be observed between the measured states of two particles. But experiments have shown time and again that entangled particles are more correlated than Bell’s upper limit, favoring the radical quantum worldview over local realism.

Only there’s a hitch: In addition to locality and realism, Bell made another, subtle assumption to derive his formula—one that went largely ignored for decades. “The three assumptions that go into Bell’s theorem that are relevant are locality, realism, and freedom,” said Andrew Friedman of the Massachusetts Institute of Technology, a co-author of the new paper. “Recently it’s been discovered that you can keep locality and realism by giving up just a little bit of freedom.” This is known as the “freedom-of-choice” loophole.

In a Bell test, entangled photons A and B are separated and sent to far-apart optical modulators—devices that either block photons or let them through to detectors, depending on whether the modulators are aligned with or against the photons’ polarization directions. Bell’s inequality puts an upper limit on how often, in a local-realistic universe, photons A and B will both pass through their modulators and be detected. (Researchers find that entangled photons are correlated more often than this, violating the limit.) Crucially, Bell’s formula assumes that the two modulators’ settings are independent of the states of the particles being tested. In experiments, researchers typically use random-number generators to set the devices’ angles of orientation. However, if the modulators are not actually independent—if nature somehow restricts the possible settings that can be chosen, correlating these settings with the states of the particles in the moments before an experiment occurs—this reduced freedom could explain the outcomes that are normally attributed to quantum entanglement.

The universe might be like a restaurant with 10 menu items, Friedman said. “You think you can order any of the 10, but then they tell you, ‘We’re out of chicken,’ and it turns out only five of the things are really on the menu. You still have the freedom to choose from the remaining five, but you were overcounting your degrees of freedom.” Similarly, he said, “there might be unknowns, constraints, boundary conditions, conservation laws that could end up limiting your choices in a very subtle way” when setting up an experiment, leading to seeming violations of local realism.

This possible loophole gained traction in 2010, when Michael Hall, now of Griffith University in Australia, developed a quantitative way of reducing freedom of choice. In Bell tests, measuring devices have two possible settings (corresponding to one bit of information: either 1 or 0), and so it takes two bits of information to specify their settings when they are truly independent. But Hall showed that if the settings are not quite independent—if only one bit specifies them once in every 22 runs—this halves the number of possible measurement settings available in those 22 runs. This reduced freedom of choice correlates measurement outcomes enough to exceed Bell’s limit, creating the illusion of quantum entanglement.