Uncertain about uncertainty

This is the English version of the cover article (in French) of the latest issue of La Recherche (October). It’s accompanied by an interview that I conducted with Robert Crease about the cultural impact of the uncertainty principle, which I’ll post next.

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If there’s one thing most people know about quantum physics, it’s that it is uncertain. There’s a fuzziness about the quantum world that prevents us from knowing everything about it with absolute detail and clarity. Almost 90 years ago, the German physicist Werner Heisenberg pointed this out in his famous Uncertainty Principle. Yet over the few years there has been heated debate among physicists about just what Heisenberg meant, and whether he was correct. The latest experiments seem to indicate that one version of the Uncertainty Principle presented by Heisenberg might be quite wrong, and that we can get a sharper picture of quantum reality than he thought.

In 1927 Heisenberg argued that we can’t measure all the attributes of a quantum particle at the same time and as accurately as we like [1]. In particular, the more we try to pin down a particle’s exact location, the less accurately we can measure its speed, and vice versa. There’s a precise limit to this certainty, Heisenberg said. If the uncertainty is position is denoted Δx, and the uncertainty in momentum (mass times velocity) is Δp, then their product ΔxΔp can be no smaller than ½h, where h [read this as h bar] is the fundamental constant called Planck’s constant, which sets the scale of the ‘granularity’ of the quantum world – the size of the ‘chunks’ into which energy is divided.

Where does this uncertainty come from? Heisenberg’s reasoning was mathematical, but he felt he needed to give some intuitive explanation too. For something as small and delicate as a quantum particle, he suggested, it is virtually impossible to make a measurement without disturbing and altering what we’re trying to measure. It we “look” at an electron by bouncing a photon of light off it in a microscope, that collision will change the path of the electron. The more we try to reduce the intrinsic inaccuracy or “error” of the measurement, say by using a brighter beam of photons, the more we create a disturbance. According to Heisenberg, error (Δe) and disturbance (Δd) are also related by an uncertainty principle in which ΔeΔd can’t be smaller than ½h.

The American physicist Earle Hesse Kennard showed very soon after Heisenberg’s original publication that in fact his thought experiment is superfluous to the issue of uncertainty in quantum theory. The restriction on precise knowledge of both speed and position is an intrinsic property of quantum particles, not a consequence of the limitations of experiments. All the same, might Heisenberg’s “experimental” version of the Uncertainty Principle – his relationship between error and disturbance – still be true?

“When we explain the Uncertainty Principle, especially to non-physicists,” says physicist Aephraim Steinberg of the University of Toronto in Canada, “we tend to describe the Heisenberg microscope thought experiment.” But he says that, while everyone agrees that measurements disturb systems, many physicists no longer think that Heisenberg’s equation relating Δe and Δd describes that process adequately.

Japanese physicist Masanao Ozawa of Nagoya University was one of the first to question Heisenberg. In 2003 he argued that it should be possible to defeat the apparent limit on error and disturbance [2]. Ozawa was motivated by a debate that began in the 1980s on the accuracy of measurements of gravity waves, the ripples in spacetime predicted by Einstein’s theory of general relativity and expected to be produced by violent astrophysical events such as those involving black holes. No one has yet detected a gravity wave, but the techniques proposed to do so entail measuring the very small distortions in space that will occur when such a wave passes by. These disturbances are so tiny – fractions of the size of atoms – that at first glance the Uncertainty Principle would seem to determine if they are feasible at all. In other words, the accuracy demanded in some modern experiments like this means that this question of how measurement disturbs the system has real, practical ramifications.

In 1983 Horace Yuen of Northwestern University in Illinois suggested that, if gravity-wave measurement were done in a way that barely disturbed the detection system at all, the apparently fundamental limit on accuracy dictated by Heisenberg’s error-disturbance relation could be beaten. Others disputed that idea, but Ozawa defended it. This led him to reconsider the general question of how experimental error is related to the degree of disturbance it involves, and in his 2003 paper he proposed a new relationship between these two quantities in which two other terms were added to the equation. In other words, ΔeΔd + A + B ≥ h/2, so that ΔeΔd itself could be smaller than h/2 without violating the limit..

Last year, Cyril Branciard of the University of Queensland in Australia (now at the CNRS Institut Néel at Grenoble) tightened up Ozawa’s new uncertainty equation [3]. “I asked whether all values of Δe and Δd that satisfy his relation are allowed, or whether there could be some values that are nevertheless still forbidden by quantum theory”, Branciard explains. “I showed that there are actually more values that are forbidden. In other words, Ozawa's relation is ‘too weak’.”

But Ozawa’s relationship had by then already been shown to give an adequate account of uncertainty for most purposes, since in 2012 it was put to the test experimentally by two teams [4,5]. Steinberg and his coworkers in Toronto figured out how to measure the quantities in Ozawa’s equation for photons of infrared laser light travelling along optical fibres and being sensed by detectors. They used a way of detecting the photons that perturbed their state as little as possible, and found that indeed they could exceed the relationship between precision and disturbance proposed by Heisenberg but not that of Ozawa. Meanwhile, Ozawa himself teamed up with a team at the Vienna University of Technology led by Yuji Hasegawa, who made measurements on the quantum properties of a beam of neutrons passing through a series of detectors. They too found that the measurements could violate the Heisenberg limit but not Ozawa’s.

Very recent experiments have confirmed that conclusion with still greater accuracy, verifying Branciard’s relationships too [6,7]. Branciard himself was a collaborator on one of those studies, and he says that “experimentally we could get very close indeed to the bounds imposed by my relations.”

Doesn’t this prove that Heisenberg was wrong about how error is connected to disturbance in experimental measurements? Not necessarily. Last year, a team of European researchers claimed to have a theoretical proof that in fact this version of Heisenberg’s Uncertainty Principle is correct after all [8]. They argued that Ozawa’s theory, and the experiments testing it, were using the wrong definitions of error. So they might be correct in their own terms, but weren’t really saying anything about Heisenberg’s error-disturbance principle. As team member Paul Busch of the University of York in England puts it, “Ozawa effectively proposed a wrong relationship between his own definitions of error and disturbance, wrongly ascribed it to Heisenberg, then showed how to fix it.”

So Heisenberg was correct after all in the limits he set on the tradeoff, argues Busch: “if the error is kept small, the disturbance must be large.”

Who is right? It seems to depend on exactly how you pose the question. What, after all, does measurement error mean? If you make a single measurement, there will be some random error that reflects the limits on the accuracy of your technique. But that’s why experimentalists typically make many measurements on the same system, so that you average out some of the randomness. Yet surely, some argue, the whole spirit of Heisenberg’s original argument was about making measurements of different properties on a particular, single quantum object, not averages for a whole bunch of such objects?

It now seems that Heisenberg’s limit on how small the combined uncertainty can be for error and disturbance holds true if you think about averages of many measurements, but that Ozawa’s smaller limit applies if you think about particular quantum states. In the first case you’re effectively measuring something like the “disturbing power” of a specific instrument; in the second case you’re quantifying how much we can know about an individual state. So whether Heisenberg was right or not depends on what you think he meant (and perhaps on whether you think he even recognized the difference).

As Steinberg explains, Busch and colleagues “are really asking how much a particular measuring apparatus is capable of disturbing a system, and they show that they get an equation that looks like the familiar Heisenberg form. We think it is also interesting to ask, as Ozawa did, how much the measuring apparatus disturbs one particular system. Then the less restrictive Ozawa-Branciard relations apply.”

Branciard agrees with Steinberg that this isn’t a question of who’s right and who’s wrong, but just a matter of how you make your definitions. “The two approaches simply address different questions. They each argue that the problem they address was probably the one Heisenberg had in mind. But Heisenberg was simply not clear enough on what he had in mind, and it is always dangerous to put words in someone else's mouth. I believe both questions are interesting and worth studying.”

There’s a broader moral to be drawn, for the debate has highlighted how quantum theory is no longer perceived to reveal an intrinsic fuzziness in the microscopic world. Rather, what the theory can tell you depends on what exactly you want to know and how you intend to find out about it. It suggests that “quantum uncertainty” isn’t some kind of resolution limit, like the point at which objects in a microscope look blurry, but is to some degree chosen by the experimenter. This fits well with the emerging view of quantum theory as, at root, a theory about information and how to access it. In fact, recent theoretical work by Ozawa and his collaborators turns the error-disturbance relationship into a question about the cost of gaining information about one property of a quantum system on the other properties of that system [9]. It’s a little like saying that you begin with a box that you know is red and think weighs one kilogram – but if you want to check that weight exactly, you weaken the link to redness, so that you can’t any longer be sure that the box you’re weighing is a red one. The weight and the colour start to become independent pieces of information about the box.

If this seems hard to intuit, that’s just a reflection of how interpretations of quantum theory are starting to change. It appears to be telling us that what we can know about the world depends on how we ask. To that extent, then, we choose what kind of a world we observe.

The issue isn’t just academic, since an approach to quantum theory in which quantum states are considered to encode information is now starting to produce useful technologies, such as quantum cryptography and the first prototype quantum computers. “Deriving uncertainty relations for error-disturbance or for joint measurement scenarios using information-theoretical definitions of errors and disturbance has a great potential to be useful for proving the security of cryptographic protocols, or other information-processing applications”, says Branciard. “This is a very interesting and timely line of research.”

References
1. W. Heisenberg, Z. Phys. 43, 172 (1927).
2. M. Ozawa, Phys. Rev. A 67, 042105 (2003).
3. C. Branciard, Proc. Natl. Acad. Sci. U.S.A. 110, 6742 (2013).
4. J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa & Y. Hasegawa, Nat. Phys. 8, 185 (2012).
5. L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar & A. M. Steinberg, Phys. Rev. Lett. 109, 100404 (2012).
6. F. Kandea, S.-Y. Baek, M. Ozawa & K. Edamatsu, Phys. Rev Lett. 112, 020402 (2014).
7. M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard & A. G. White, Phys. Rev. Lett. 112, 020401 (2014).
8. P. Busch, P. Lahti & R. F. Werner, Phys. Rev. Lett. 111, 160405 (2013).
9. F. Buscemi, M. J. W. Hall, M. Ozawa & M. W. Wilde, Phys. Rev. Lett. 112, 050401 (2014).