Uncertainty over the Uncertainty Principle

Eighty-six years after Werner Heisenberg first described his eponymous uncertainty principle, experts are still arguing over what, exactly, the infamous inequality really means. Briefly, of course, the principle says that the product of the uncertainties of position and momentum will always be greater than a constant —though a very, very tiny constant (*see note below). The more tightly you tie one factor down, the more the other swings.

But does it mean that the act of measuring position changes the momentum, and vice-versa—the observer effect? Or does it mean that the particle simply doesn’t have precisely defined momentum and position to measure?

The uncertainty principal became a matter of urgent practical debate with the first glimmerings of quantum computing. In particular, some schemes for quantum encryption derive much of their promised security from the uncertainty principle’s assurance that any attempt at eavesdropping will disrupt the information and betray the interloper’s presence. As Ron Cowen pointed out recently in Nature, "Quantifying by how much a measuring device can disturb the properties of a quantum system is crucial to the burgeoning field of quantum cryptography and computing."

But is quantum encryption really secure? In 2003 and 2004,(in Physics Letters and Annals of Physics) Masanao Ozawa, now at the University of Nagoya, recalculated the "measurement-disturbance relation" (MDR) to add two additional terms to Heisenberg’s inequality and suggest a scheme of successive observations of surrogate variables. Ozawa's scheme would, on paper anyway, perturb the system far less than Heisenberg and his successors had calculated. (ArXiv versions are available here and here.) In theory, a spy could read quantum-encrypted information without scrambling the message and leaving footprints.

At the end of 2012, following Ozawa’s lead, a group of physicists at the University of Toronto reported that they had successfully performed just such a measurement-disturbance experiment. By chaining successive "weak" and "strong" measurements, they simultaneously measured the polarization of entangled photon pairs in two different planes. The experimental results, said Leo Rozema, Aephraim Steinberg, and their colleagues, were more consistent with Ozawa’s calculations than Heisenberg’s.

"In conclusion," they wrote in Physical Review Letters (and posted to ArXiv), "using weak measurements to experimentally characterize a system before and after it interacts with a measurement apparatus, we have directly measured its precision and the disturbance...a violation of Heisenberg’s hypothesized MDR."

"Our work," they continued, "conclusively shows that, although correct for uncertainties in states, the form of Heisenberg’s precision limit is incorrect if naively applied to measurement. Our work highlights an important fundamental difference between uncertainties in states and the limitations of measurement in quantum mechanics." (Other groups have also claimed to circumvent uncertainty.)

So, they say, Heisenberg’s uncertainty principle is both true and not true...as undead as Schrödinger’s cat. Even without the provocative "naively," those are fighting words.

It wasn't long before Paul Busch (University of York in England), Pekka Lahti (University of Turku in Finland), and Reinhard Werner (Leibniz University in Germany) responded on ArXiv with a "Proof of Heisenberg’s error-disturbance relation."

For the past several years, Busch, Lahti, and a variety of collaborators have been trying to formalize the more traditional Heisenberg formulation, as a counterbalance to the revisionism of Ozawa and his school.

In five pages of dense derivations (though not as dense as the 30-odd pages of theorems and proofs Ozawa published in Annals of Physics in 2004), Busch, Lahti, and Werner provide what they call "a generally quantitative quantum version of Heisenberg’s original semiclassical uncertainty discussion...a remarkable vindication of Heisenberg’s intuition, far beyond the usual view...."

As for the contrary results, they say Ozawa "defines a relation and claims it to be a rigorous version of Heisenberg’s ideas, only to show that it fails to hold in general. It is merely this failure which has recently been verified experimentally" by Rozema and others.

It’s fairly certain that the dispute will continue.

* Heisenberg’s principle is most familiarly expressed in a later formulation, as Δq·Δp ≥ ħ/2, where Δq is the uncertainty of position, Δp is the uncertainty of momentum, and ħ (=1.05457 ×ばつ 10⁻³⁴ Joule-seconds) is the reduced Planck constant.

Photo: Heisenberg at work—Philippe Le Tellier/Paris Match/Getty Images