Just when you thought you'd heard every quantum mystery that was possible, out pops another one. Jeff Tollaksen mentioned it in passing during his talk at the recent Foundation Questions Institute conference. Probably Tollaksen assumed we'd all heard it before. After all, his graduate advisor, Yakir Aharonov—who has made an illustrious career of poking the Schrdinger equation to see what wild beasts come scurrying out—first discovered it in the 1990s and discussed it in chapter 17 of his 2005 book, Quantum Paradoxes. But it was new to me.
The situation is an elaboration of Schrdinger's thought experiment. You have a cat. It is either purring or meowing. It is curled up in one of two boxes. As in Schrdinger's scenario, you couple the cat to some quantum system, like a radioactive atom, to make its condition ambiguous—a superposition of all possibilities—until you examine one of the boxes. If you reach into box 2, you feel the cat. If you listen to the boxes, you hear purring. But when you listen more closely, you notice that the purring is coming from box 1. The cat is in one box, the purring in the other. Like a Cheshire Cat, the animal has become separated from the properties that constitute a cat. What a cat does and what a cat is no longer coincide.
In practice, you'd pull this stunt on an electron rather than a cat. You'd find the electron in one box, its spin in the other. Even by the standards of quantum mechanics, this is surprising. It requires what quantum physicists call "weak measurement," whereby you interact with a system so gently that you avoid collapsing it from a quantum state to a classical one. On the face of it, such an interaction scarcely qualifies as a measurement; any results get lost in the noise of Heisenberg's Uncertainty Principle. What Aharonov realized is that, if you sift through the results, you can find patterns buried within them.
In practice, this means repeating the experiment on a large number of electrons (or cats) and then applying a filter or “postselection.” Only a few particles will pass through this filter, and among them, the result of the softly softly measurement will stand out.
Because you avoid collapsing the quantum state, quintessentially quantum phenomena such as wave interference still occur. So, for a Cheshire Cat, you apply the following filter: you change the sign of one term in the superposition, causing the location and spin of the electron to interfere constructively in one box and destructively in the other, zeroing out the probability of finding the electron in box 1 and zeroing out the net spin of the electron in box 2. Voil, the electron is in box 2 and its spin in box 1.
If this leaves your head spinning, it should. The word “weak” describes not only the measurement but also my intuitive grasp for what's really going on. The best I can do is recommend the article on weak measurement by Aharonov, Tollaksen, and Sandu Popescu in last November's Physics Today, but be prepared to read it several times before you have the slightest idea of what they're saying. I've commissioned an article about Aharonov's work for an upcoming issue of Scientific American to collapse some of the uncertainty. In the meantime, try sitting in a different room from where your confusion is.