Bell demonstrated that a hidden-variables model that satisfied locality—in which the properties of A remained unaffected by what measurements were conducted on B—could easily reproduce the perfect correlation when A's and B's spins were measured along the same direction. At root, this meant imagining that each particle carried with it a definite value of spin along any given direction, even if most of those values remained hidden from view. The spin values were considered to be properties of the particles themselves; they existed independent of and prior to any effort to measure them, just as Einstein would have wished.
Next Bell considered other possible arrangements. One could choose to measure a particle's spin along any direction: the z-axis, the y-axis, or any angle in between. All one had to do was rotate the magnets between which the particle passed. What if one measured A's spin along the z-axis and B's spin along some other direction? (Fig. 2.2.) Bell homed in on the expected correlations of spin measurements when shooting pairs of particles through the device, while the detectors on either side were oriented at various angles. He considered detectors that had two settings, or directions along which spin could be measured.
Using only a few lines of algebra, Bell proved that no local hidden variables theory could ever reproduce the same degree of correlations as one varied the angles between detectors. The result has come to be known as "Bell's theorem." Simply assuming that each particle carried a full set of definite values on its own, prior to measurement—even if most of those values remained hidden from view—necessarily clashed with quantum theory. Nonlocality was indeed endemic to quantum mechanics, Bell had shown: somehow, the outcome of the measurement on particle B depended on the measured outcome on particle A, even if the two particles were separated by huge distances at the time those measurements were made. Any effort to treat the particles (or measurements made upon them) as independent, subject only to local influences, necessarily led to different predictions than those of quantum mechanics. Here was what Bell had been groping for, on and off since his student days: some quantitative means of distinguishing Bohr's interpretation of quantum mechanics from other coherent, self-consistent possibilities. The problem—entanglement versus locality—was amenable to experimental test. In his bones he hoped locality would win.
In the years since Bell formulated his theorem, many physicists (Bell included) have tried to articulate what the violation of his inequality would mean, at a deep level, about the structure of the microworld. Most prosaically, entanglement suggests that on the smallest scales of matter, the whole is more than the sum of its parts. Put another way: one could know everything there is to know about a quantum system (particles A + B), and yet know nothing definite about either piece separately. As one expert in the field has written, entangled quantum systems are not even "divisible by thought": our natural inclination to analyze systems into subsystems, and to build up knowledge of the whole from careful study of its parts, grinds to a halt in the quantum domain.
Physicists have gone to heroic lengths to translate quantum nonlocality into everyday terms. The literature is now full of stories about boxes that flash with red and green lights; disheveled physicists who stroll down the street with mismatched socks; clever Sherlock Holmes-inspired scenarios involving quantum robbers; even an elaborate tale of a baker, two long conveyor belts, and pairs of soufflés that may or may not rise. My favorite comes from a "quantum-mechanical engineer" at MIT, Seth Lloyd. Imagine twins, Lloyd instructs us, separated a great distance apart. One steps into a bar in Cambridge, Massachusetts just as her brother steps into a bar in Cambridge, England. Imagine further (and this may be the most difficult part) that neither twin has a cell phone or any other device with which to communicate back and forth. No matter what each bartender asks them, they will give opposite answers. "Beer or whiskey?" The Massachusetts twin might respond either way, with equal likelihood; but no matter which choice she makes, her twin brother an ocean away will respond with the opposite choice. (It's not that either twin has a decided preference; after many trips to their respective bars, they each wind up ordering beer and whiskey equally often.) The bartenders could equally well have asked, "Bottled beer or draft?" or "Red wine or white?" Ask any question—even a question that no one had decided to ask until long after the twins had traveled far, far away from each other—and you will always receive polar opposite responses. Somehow one twin always "knows" how to answer, even though no information could have traveled between them, in just such a way as to ensure the long-distance correlation.