The classic mystery of quantum mechanics concerns the passage of a particle through two slits simultaneously. We know from our understanding of the double-slit experiment that the photon must have traveled through both slits, yet we can never actually observe the photon passing through both slits at the same time. If we find the photon in one place, its probability amplitude to be anywhere else immediately vanishes.
At least, that is the familiar story. In a new experiment described in Scientific Reports, Ryo Okamoto and Shigeki Takeuchi of Kyoto University in Japan have shown that under the right conditions, a single photon can have observable physical effects in two places at once.
The power of postselection
The heightened sort of nonlocality demonstrated in Kyoto occurs only in the presence of postselection. Consider that identically prepared photons impinging on, for instance, a double slit will land at different points on the screen some time later. The question posed by Yakir Aharonov, Peter Bergmann, and Joel Lebowitz (ABL) in 1964, and revisited in even more dramatic form by Aharonov, David Albert, and Lev Vaidman (AAV) in 1988, was the following: If we look only at the subensemble of photons that land at some particular point on the screen, does that observation give us new information about those photons? Might it allow us to draw more conclusions about what the photons were doing between the time they were prepared and the time we selected a particular region on the screen?
The suggestion that we can chart the progress of photons on their journey may sound heretical at first. The standard lore is that a wave function is a complete description of the quantum state, and that the uncertainty principle prevents one from knowing both where a particle came from and where it will go. Yet ABL and AAV showed rigorously—using only the standard rules of quantum mechanics—that if some measuring apparatus had been interacting with the photons, its final state (the “pointer position” on the meter, as measurement theorists put it) would depend on which subset of photons had been considered. (See the article by Aharonov, Sandu Popescu, and Jeff Tollaksen, Physics Today, November 2010, page 27.)
In the ABL case, the interaction with the meter can be thought of as collapsing the photons into one state or another. Depending on which measurement outcome occurs, the photons may subsequently be more likely to reach one point or another on the screen: The pointer position becomes correlated with the final position of the photons. Classical statistics is sufficient for working out what the average pointer position should be, conditioned on a particular final state. (AAV extended this work to consider postselected “weak” measurements, a scenario that yields purely quantum results one could not obtain classically. Weak measurements raise even more thorny questions about the foundations of quantum mechanics that go beyond the scope of this article.)
Many experiments have confirmed these predictions about the outcomes of conditional measurements. Experimenters have applied postselection to studying interpretational issues in quantum theory and to the practical purpose of amplifying small effects to improve measurement sensitivity. Postselection has also become an important element in the toolbox of quantum information, with applications in linear-optical approaches to quantum computing and the closely related measurement-based quantum computing.
Head-scratching consequences
Although the formulas put forward by ABL and AAV are unavoidable consequences of quantum theory, they cry out for interpretation—especially since they expose new counterintuitive effects. Suppose an experimenter prepares a photon in a symmetric superposition of three states, |ψi⟩ = (|A⟩ + |B⟩ + |C⟩) / √3, but later finds the particle in a different superposition of those three states, |ψf⟩ = (|A⟩ + |B⟩ – |C⟩) / √3. What can the experimenter say about where the particle was between preparation and postselection? In 1991 Aharonov and Vaidman showed that a measurement of the particle number in state A would yield a value of 1 whenever the postselection succeeded; yet by symmetry, so would a measurement of the particle number in state B. In other words, the postselected particle would be certain to influence a measurement apparatus looking for it at A, and equally certain to influence a measurement apparatus looking for it at B. It’s akin to saying that the conditional probability is 100% to be at A and 100% to be at B.
Obviously, that situation never arises classically. But in the quantum world, every measurement disturbs the system, and the two conditional probabilities correspond to different physical situations: one in which a measurement interaction occurred at A, and one in which it occurred at B. Suppose the experimenter looks for the particle at A but doesn’t find it. That projects the original state onto |B⟩ + |C⟩ / √2, which is orthogonal to |ψf⟩. Therefore, if the experimenter does not find the particle at A, the postselection never succeeds. It follows trivially that whenever the postselection does succeed, the particle must have been found at A. My group carried out a verification of that prediction in 2004.
If the results of that quantum shell game weren’t baffling enough, Aharonov and Vaidman proposed an even stranger thought experiment in 2003. In “How one shutter can close N slits,” they imagined a series of two or more slits (let us consider the simplest example, N=2, although the argument applies for any N). They then considered an experimenter who possessed a single shutter that could block one of the slits. Instead of placing the shutter to block one slit or the other, the experimenter prepares the shutter in a superposition of both positions, along with a third position that does not block a slit.
Aharonov and Vaidman showed that if the experimenter postselected in a different, properly chosen superposition (the |ψf⟩ of the original three-box problem), then any measurement of whether the shutter was blocking a particular one of the two slits would be guaranteed to yield an affirmative answer. If a single photon was sent toward slit A, the shutter would block its path. If the photon was sent toward slit B, the same shutter would still be guaranteed to block it. Most remarkably, the photon would also be stopped by the shutter if it was sent along any coherent superposition of the paths leading to the N slits. In other words, the experimenter can block the photon whether it goes through slit A or slit B—and also block the photon if it doesn't go through one particular slit or the other, with no need to determine which slit it hit. In this very real sense, the shutter acts as though it is in two places at once.
One might be tempted to cry foul. “The shutter is affected by the choice to look for it in one place or another,” that person might say. “It wouldn’t be able to block two photons arriving at both slits.” Yet in a weak-measurement version of the proposal, in which the disturbance due to measurements is reduced to near zero, the predictions hold: The unperturbed shutter acts as though it is fully at A and also fully at B.
Building a quantum shutter
Of course, macroscopic shutters can no more easily be placed in superpositions than can macroscopic felines. Enter Okamoto and Takeuchi. Using ideas from linear-optical quantum computing, they replaced the slits and shutters with quantum routers—logic gates in which the presence of one photon (let’s call it a shutter photon) determines whether another photon (the probe photon) is transmitted or reflected.
The researchers prepared a shutter photon, which was generated via spontaneous parametric down-conversion, in a superposition of three paths. If the shutter photon took path 1, it should block a probe photon reaching router 1; if it took path 2, it should block a probe photon reaching router 2; and if it took path 3, it wouldn't block any photons at all. The three paths were then recombined with a phase shift, so that the firing of a certain detector signaled a measurement of the shutter photon in state |ψf⟩. Looking only at cases when the detector fired, the researchers could study the behavior of a postselected photon and see which slit or slits it blocked.
Lo and behold, when the postselection succeeded, Okamoto and Takeuchi found that the probe photons had a high probability of being reflected, regardless of which router they were sent toward. Because of subtle technical issues involved in the concatenation of the linear-optical quantum gates, the probability was limited to a maximum of 67%. The researchers observed about 61% experimentally, clearly exceeding the 50% threshold that could be achieved by a shutter constrained to be in one place at a time. Furthermore, the scientists confirmed the remarkable prediction that the same shutter was capable of blocking probe photons prepared in arbitrary superpositions of the two paths. The experiment demonstrates that any “shutter” (in this case a shutter photon) that is prepared in a given initial state and then measured in the appropriate final state must have been in front of multiple slits at the same time.
So, what does this experiment teach us about nonlocality? It’s hard to find a more down-to-earth approach to that question than my mother’s insightful proposal: “I hope this means I can shop in two different places at the same time.” Well, quantum mechanics may offer us fascinating new phenomena, but they always come at a price. In the case of quantum shopping, the postselection step means that my mother may get no shopping done at all. However, she may be in luck if she happens to know that only one of her N favorite stores still has the gift she wants, but she doesn’t know which store. An application of Okamoto and Takeuchi's result would mean that in the time it takes her to visit just a single store, she would have a finite probability of being certain to find the gift.
I’ll leave the calculation of the exact value of that probability as an exercise for holiday shoppers. But from the perspective of research into the foundations of quantum mechanics, we now have experimental confirmation that postselected systems exhibit a form of nonlocality even more striking than the ones we were familiar with before.
Aephraim Steinberg is a professor of physics at the University of Toronto, where he is a founding member of the Centre for Quantum Information and Quantum Control and a fellow of the Canadian Institute for Advanced Research.
