Paying Heisenberg's Quantum Toll: The Cost of Gaining Information

Heisenberg's Uncertainty Principle is one of the most revolutionary notions in modern science, so much so, that it has even permeated into recent popular TV shows such asand.
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Heisenberg's Uncertainty Principle is one of the most revolutionary notions in modern science, so much so, that it has even permeated into recent popular TV shows such as Futurama and Breaking Bad. Heisenberg's Uncertainty Principle states, loosely speaking, that it is impossible to learn the position of a quantum particle (such as an electron) without causing a disturbance to its momentum. The principle is counterintuitive: in our ordinary experiences, there does not seem to be any limitation to how well we can measure properties of physical systems, the only limitation seeming to come from how accurate we can build our measurement devices to be. Heisenberg showed us that this "classical" intuition is wrong -- there is a fundamental trade-off between how much you learn about the position of a quantum particle with how much disturbance you inevitably cause to its momentum.

In more recent years, a few theoretical physicists have realized that Heisenberg's original formulation of the uncertainty principle has some unsatisfying aspects. Perhaps most strikingly, in 1988 Masanao Ozawa of Nagoya University in Japan demonstrated that it is possible to violate a formulation of Heisenberg's uncertainty principle (he proposed a particular setup for doing so), and in 2003 he proposed a revision to the uncertainty principle that cannot be violated in any conceivable physical setup. In 2012, Ozawa teamed up with some experimentalists from the University of Vienna in Austria and they demonstrated with spin-polarized neutrons that Ozawa's revision of the uncertainty principle held up to all of their tests, while the original Heisenberg formulation did not.

Now, the Heisenberg uncertainty principle is not only a cornerstone of modern theoretical physics, but it also underpins new quantum technologies, such as communication schemes that are guaranteed to be secure based on physical principles. The intuition here is that if a malicious eavesdropper attempts to learn something about secret information that two honest parties are exchanging, then the eavesdropper will inevitably cause a disturbance to this information that is detectable by the honest parties. However, Ozawa's reformulation of the uncertainty principle is not directly applicable in this setting or related ones, and it has been a pressing open question since Ozawa's work to develop an uncertainty relation similar to his that is directly applicable in this setting.

In a recent paper that has appeared in Physical Review Letters, Francesco Buscemi of Nagoya University, Michael Hall of Griffith University (Australia), Ozawa and myself teamed up to establish such a reformulation of the Heisenberg uncertainty principle. The result is a new kind of noise-disturbance uncertainty principle in which both the measurement noise and the measurement disturbance are quantified in terms of information quantities. Our approach has the advantages of being operational, applicable to any pair of physical observables, independent of how the measurement outcomes are labeled and processed, and also independent of the quantum state of the system being measured.

For example, we show that any perfect measurement of the linear polarization of light is possible only at the cost of destroying all information about the circular polarization. A similar result holds for position and momentum observables. We now expect our results to have applications not only in quantum cryptography but also in quantum metrology, where both information gain and irreversible disturbance are central concepts.

Full paper, "Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach," is available here

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