How Certain Can You Be?

The Heisenberg Uncertainty Principle is one of the central features of quantum mechanics, but it has been misunderstood for a long time. Only now – almost a century later – has a first complete quantitative description of his uncertainty principle been found.

By Martin Ringbauer

A team of physicists has challenged the limits of Heisenberg’s famous uncertainty principle by measuring quantum particles with unprecedented accuracy.

At the heart of all the natural sciences is measurement – assigning a number to a physical property. Just how accurately we can measure, however, has long been an open question. Can we just keep building better and better devices to make more and more accurate measurements?

What appears to be true for standard technology breaks down as things get smaller. In the quantum world there are fundamental limits to what you can know about a system. This crucial feature of quantum mechanics was discovered by German physicist Werner Heisenberg in 1927 and named the Heisenberg Uncertainty Principle after him.

Only now, almost a century later, we understand the full scope of its implications and have experimental evidence for the ultimate limits of uncertainty.

What Are We Uncertain About?

Heisenberg’s principle may have shattered the dream of perfect measurements, but it doesn’t mean that we cannot know anything for sure. In particular, there are pairs of properties of quantum systems that are incompatible, which means that they cannot be measured with arbitrary accuracy at the same time on the same system.

One such pair is position and velocity. Take an atom, for example. To measure its position one would take a picture of it. To measure its velocity one would compare two successive pictures. Increasing the resolution of the first picture improves the accuracy of the position measurement, but this disturbs the atom and changes its velocity.

Along the lines of this simple illustration, Heisenberg introduced three fundamental limitations in nature:

a quantum system cannot be prepared such that two incompatible properties have arbitrarily accurate values;
two such properties cannot be measured jointly with arbitrary accuracy; and
if measured subsequently, the measurement of the first disturbs the measurement of the second.

The distinction between these three cases is crucial, and has caused misunderstandings and misinterpretations of the Heisenberg Uncertainty Principle for a long time. Let’s see what it means for the previous example.

Case 1 refers to the atom itself, and states that it cannot have an absolutely accurate position and velocity at the same time.

Crucially, the other two cases are independent of the actual properties of the measured physical system. When measuring position and velocity, Case 2 means that we can never get arbitrarily accurate outcomes for both from a single measurement.

Similarly, Case 3 means that if we measure position before velocity, then a more accurate position measurement would result in a less accurate velocity measurement – our original example.

Having established this viewpoint, the next step is quantifying the limits imposed by the theory, which is done by so-called uncertainty relations.

For all three cases above, Heisenberg conjectured that the product of the uncertainties in position and momentum (mass × velocity) should be larger than the Planck constant divided by two. Hence, as we make one more accurate, the other one has to get more uncertain. The Planck constant is a very small number, which is the reason why these effects are not usually observed in everyday life.

Heisenberg’s guess was rigorously proven by Kennard and Robertson in 1929. Their proof, however, is only valid for Case 1. Even in textbooks, this fact was largely ignored until about 10 years ago, when the Japanese theorist Masanao Ozawa pointed out the issue and proposed an alternative relation to quantify the measurement scenarios. While Ozawa’s relation correctly describes restrictions on the accuracy of quantum measurements, it was soon found to be too conservative – it doesn’t capture everything that’s forbidden.

A solution to this problem was suggested by the Queensland theorist Cyril Branciard in 2013, who derived optimal bounds on measurement uncertainty. This new theory is optimal in the sense that is describes not only what is forbidden but also what is allowed in quantum mechanics.

Laboratory Measurements

New theory must be tested experimentally. Measuring the accuracy of a measurement, however, turns out to be quite difficult and was for some time even believed to be impossible. In particular we are asking “how accurately can I measure x?” rather than “how accurate is the measured value of x?”

The crucial point is that a measurement always has to be performed on some object – a quantum system. From Case 1 we know that this system cannot have arbitrarily accurate values of the properties to be measured. Hence, at least one of the measurements will suffer from this uncertainty in addition to the accuracy of the measurement itself.

To get an estimate of the measurement uncertainty we would need to filter out this intrinsic uncertainty of the test system. This, in turn, requires a perfectly accurate measurement to characterise this system – a vicious circle.

To get around this problem, two methods have been developed to extract information about the accuracy of the measurement independently of the state of the quantum system used in the experiment.

For the first technique, the measurement of interest is performed on three closely related variations of the input system. The three measurement results, together with the knowledge of their relation, allows us to separate measurement inaccuracy from the intrinsic uncertainty of the measured system.

The second technique involves a pre-measurement of the system before the actual measurement. The measurement uncertainty can then be estimated from the combined outcomes of the two measurements.

Using these techniques, experiments in 2013 using neutrons and photons demonstrated that measurements can be more accurate than predicted by Heisenberg’s original guess while obeying Ozawa’s predictions. While these initial results still suffered from idealised assumptions in the above methods, they were conclusively confirmed earlier this year by an experiment at the University of Queensland using generalised versions of these methods.

Our experiment also tested Branciard’s theory for the first time, and demonstrated that it is optimal, which was only possible with an experiment of unprecedented quality.

With a topic of such fundamental importance, of course not everyone is convinced by Branciard’s new theory and the experimental results. There is still a vicious debate over the correct definitions of measurement uncertainty. For example, Paul Busch of the University of York showed how different definitions allow the recovery of Heisenberg’s guess for measurement scenarios.

While Branciard’s definitions seem sensible from an experimental point of view, others have crucially different physical interpretations that may correspond to other relevant situations.

Settling the Debate

Our latest experiments, published recently in the journal Physical Review Letters, have obtained measurements on single light particles with unprecedented accuracy. This experimental work settles a decade-long debate, revealing that “Heisenberg-like” relations do not hold for joint measurements. Now that we have a complete theory, as well as experimental evidence, it is probably time to update the textbooks.

This marks a milestone in our understanding of the limits of quantum mechanics, which is essential for engineering next-generation quantum technology and for designing the most accurate quantum measurements.

Martin Ringbauer is a PhD student at the Quantum Technology Lab at The University of Queensland.

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