Uncertainty about uncertainty

Here’s a news story I have written for Physics World. It makes me realize I still don’t understand the uncertainty principle, or at least not in the way I thought I did – so it doesn’t, then, apply to successive measurements on an individual quantum particle?!

But while on the topic of Heisenberg, I discuss my new book Serving the Reich on the latest Nature podcast, following a very nice review in the magazine from Robert Crease. I’m told there will be an extended version of the interview put up on the Nature site soon. I’ve also discussed the book and its context for the Guardian science podcast, which I guess will also appear soon.

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How well did Werner Heisenberg understand the uncertainty principle for which he is best known? When he proposed this central notion of quantum theory in 1927 [1], he offered a physical picture to help it make intuitive sense, based on the idea that it’s hard to measure a quantum particle without disturbing it. Over the past ten years an argument has been unfolding about whether Heisenberg’s original analogy was right or wrong. Some researchers have argued that Heisenberg’s ‘thought experiment’ isn’t in fact restricted by the uncertainty relation – and several groups recently claimed to have proved that experimentally.

But now another team of theorists has defended Heisenberg’s original intuition. And the argument shows no sign of abating, with each side sticking to their guns. The discrepancy might boil down to the irresolvable issue of what Heisenberg actually meant.

Heisenberg’s principle states that we can’t measure certain pairs of variables for a quantum object – position and momentum, say – both with arbitrary accuracy. The better we know one, the fuzzier the other becomes. The uncertainty principle says that the product of the uncertainties in position and momentum can be no smaller than a simple fraction of Planck’s constant h.

Heisenberg explained this by imagining a microscope that tries to image a particle like an electron [1]. If photons bounce off it, we can “see” and locate it, but at the expense of imparting energy and changing its momentum. The more gently it is probed, the less the momentum is perturbed but then the less clearly it can be “seen.” He presented this idea in terms of a tradeoff between the ‘error’ of a position measurement (Δx), owing to instrumental limitations, and the resulting ‘disturbance’ in the momentum (Δp).

Subsequent work by others showed that the uncertainty principle does not rely on this disturbance argument – it applies to a whole ensemble of identically prepared particles, even if every particle is measured only once to obtain either its position or its momentum. As a result, Heisenberg abandoned the argument based on his thought experiment. But this didn’t mean it was wrong.

In 1988, however, Masanao Ozawa, now at Nagoya University in Japan, argued that Heisenberg’s original relationship between error and disturbance doesn’t represent a fundamental limit of uncertainty [2]. In 2003 he proposed an alternative relationship in which, although the two quantities remain related, their product can be arbitrarily small [3].

Last year Ozama teamed up with Yuji Hasegawa at the University of Vienna and coworkers to see if his revised formulation of the uncertainty principle held up experimentally. Looking at the position and momentum of spin-polarized neutrons, they found that, as Ozawa predicted, error and disturbance still involve a tradeoff but with a product that can be smaller than Heisenberg’s limit [4].

At much the same time, Aephraim Steinberg and coworkers at the University of Toronto conducted an optical test of Ozawa’s relationship, which also seemed to bear out his prediction [5]. Ozawa has since collaborated with researchers at Tohoku University in another optical study, with the same result [6].

Despite all this, Paul Busch at the University of York in England and coworkers now defend Heisenberg’s position, saying that Ozawa’s argument does not apply to the situation Heisenberg described [7]. “Ozawa's inequality allows arbitrarily small error products for a joint approximate measurement of position and momentum, while ours doesn’t”, says Busch. “Ours says if the error is kept small, the disturbance must be large.”

“The two approaches differ in their definition of Δx and Δp, and there is the freedom to make these different choices”, explains quantum theorist Johannes Kofler of the Max Planck Institute of Quantum Optics in Garching, Germany. “Busch et al. claim to have the proper definition, and they prove that their uncertainty relation always holds, with no chance for experimental violation.”

The disagreement, then, is all about which definition is best. Ozawa’s is based on the variance in two measurements made sequentially on a particular quantum state, whereas that of Busch and colleagues considers the fundamental performance limits of a particular measuring device, and thus is independent of the initial quantum state. “We think that must have been Heisenberg's intention”, says Busch.

But Ozawa feels Busch and colleagues are focusing on instrumental limitations that have little relevance to the way devices are actually used. “My theory suggest if you use your measuring apparatus as suggested by the maker, you can make better measurement than Heisenberg's relation”, he says. “They now prove that if you use it very badly – if, say, you use a microscope instead of telescope to see the moon – you cannot violate Heisenberg's relation. Thus, their formulation is not interesting.”

Steinberg and colleagues have already responded to Busch et al. in a preprint that tries to clarify the differences between their definition and Ozawa’s. What Busch and colleagues quantify, they say, “is not how much the state that one measures is disturbed, but rather how much ‘disturbing power’ the measuring apparatus has.”

“Heisenberg's original formula holds if you ask about "disturbing power," but the less restrictive inequalities of Ozawa hold if you ask about the disturbance to particular states”, says Steinberg. “I personally think these are two different but both interesting questions.” But he feels Ozawa’s formulation is closer to the spirit of Heisenberg’s.

In any case, all sides agree that the uncertainty principle is not, as some popular accounts imply, about the mechanical effects of measurement – the ‘kick’ to the system. “It is not the mechanical kick but the quantum nature of the interaction and of the measuring probes, such as a photon, that are responsible for the uncontrollable quantum disturbance”, says Busch.

In part the argument comes down to what Heisenberg had in mind. “I cannot exactly say how much Heisenberg understood about the uncertainty principle”, Ozawa says. “But”, he adds, “I can say we know much more than Heisenberg.”

References

1. W. Heisenberg, Z. Phys. 43, 172 (1927).
2. M. Ozawa, Phys. Rev. Lett. 60, 385 (1988).
3. M. Ozawa, Phys. Rev. A 67, 042105 (2003).
4. J. Erhart et al., Nat. Phys. 8, 185 (2013).
5. L. A. Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)
6. S.-Y. Baek, F. Kaneda, M. Ozawa & K. Edamatsu, Sci. Rep. 3, 2221 (2013).
7. P. Busch, P. Lahti & R. F. Werner, Phys. Rev. Lett. 111, 160405 (2013).
8. L. A. Rozema, D. H. Mahler, A. Hayat & A. M. Steinberg, http://www.arxiv.org/1307.3604 (2013).