Heisenberg’s Uncertainty Principle Revisited

Robert Prevedel

Author: Robert Prevedel
Affiliation: Institute for Quantum Computing, University of Waterloo, Canada

In 1927, Heisenberg [1] showed that, according to quantum theory, one cannot know both the position and the velocity of a particle with arbitrary precision; the more precisely the position is known, the less precisely the momentum can be inferred and vice versa. In other words, the uncertainty principle sets limits on our ultimate ability to predict the outcomes of certain pairs of measurements on quantum systems. Such pairs of quantities can also be energy and time or the spins and polarizations of particles in various directions. The uncertainty principle is a central consequence of quantum theory and a pillar of modern physics. It lies at the heart of quantum theory and has profound fundamental and practical consequences, setting absolute limits on precision technologies such as metrology and lithography, but at the same time also provides the basis for new technologies such as quantum cryptography [2].

Past 2Physics article by the author:
June 08, 2007: "Entanglement and One-Way Quantum Computing"
by Robert Prevedel and Anton Zeilinger

Over the years, the uncertainty principle has been reexamined and expressed in more general terms. To link uncertainty relations to classical and quantum information theory, they have been recast with the uncertainty quantified by the entropy [3,4] rather than the standard deviation. Until recently, the favored uncertainty relation of this type was that of Maassen and Uffink [5], who showed that it is impossible to reduce the so-called Shannon entropies (a measure for the amount of information that can be extracted from a system) associated with any pair of measurable quantum quantities to zero. This implies that the more you squeeze the entropy of one variable, the more the entropy of the other increases.

This intuition stood for some decades, however very recently a new work by Berta et al. [6] showed that the limitations imposed by Heisenberg’s principle could actually be overcome through clever use of entanglement, the counterintuitive property of quantum particles that leads to strong correlations between them. More precisely, the paper predicts that an observer holding quantum information about the particle can have a dramatically lower uncertainty than one holding only classical information. In the extreme case, an observer that has access to a particle that is maximally entangled with the quantum system that he wishes to measure is able to correctly predict the outcome of whichever measurement is chosen. This dramatically illustrates the need for a new uncertainty relation that takes into account the potential entanglement between the system and another particle. A derivation of such a new uncertainty relation appeared in the work of Berta et al. [6] (also see the past 2Physics article dated August 29, 2010) The new relation proves a lower bound on the uncertainties of the measurement outcomes when one of two measurements is performed.

To illustrate the main idea how an observer holding quantum information can outperform one without, the paper outlines an imaginary “uncertainty game” which we briefly outline below. In this game, two players, Alice and Bob, begin by agreeing on two measurements, R and S, one of which will be performed on an quantum particle. Bob then prepares this particle in a quantum state of his choosing. Without telling Alice which state he has prepared, he sends the particle to Alice. Alice performs one of the two measurements, R or S (chosen at random), and tells Bob which observable she has measured, though not the outcome of the measurement. The aim of the game is for Bob to correctly guess the measurement outcome. If Bob had only a classical memory (e.g. a piece of paper), he would not be able to guess correctly all of the time — this is what Heisenberg’s uncertainty relation implies. However, if Bob is able to entangle the particle he sends with a quantum memory, then for any measurement Alice makes on the particle, there is a measurement on Bob’s memory that always gives him the same outcome. His uncertainty has thus vanished and he is capable of correctly guessing the outcome of whichever measurement Alice performs.














Fig. 1: The uncertainty game. Bob sends a particle, which is entangled with one that is stored in his quantum memory, to Alice (1), who measures it using one of two pre-agreed observables (2). She then communicates the measurement choice, but not its result, to Bob who tries to correctly guess Alice’s measurement outcome. See text for more details. Illustration adapted from [6].

In our present work [7], we experimentally realize Berta et al.'s uncertainty game in the laboratory and rigorously test the new and modified uncertainty relation in an optical experiment. We generate polarization-entangled photon pairs and send one of the photons to Alice who randomly performs one of two polarization measurements. In the meantime, we delay the other photon using an optical fiber – this acts as a simple quantum memory. Dependent on Alice’s measurement choice (but not the result), we perform the appropriate measurement on the photon that was stored in the fiber. In this way, we show that Bob can infer Alice's measurement result with less uncertainty if the particles were entangled. Varying the amount of entanglement between the particles allows us to fully investigate the new uncertainty relation. The results closely follow the Berta et al. relation. By using entangled photons in this way, we observe lower uncertainties than previously known uncertainty relations would predict. We show that this fact can be used to form a simple, yet powerful entanglement witness. This more straightforward witnessing method is of great value to other experimentalists who strive to generate this precious resource. As future quantum technologies emerge, they will be expected to operate on increasingly large systems. The entanglement witness we have demonstrated offers a practical way to quantitatively assess the quality of such technologies, for example the performance of quantum memories.




















Fig. 2: A photo of the actual experiment. In the center a down-conversion source of entangled photons can be seen. The ultraviolet pump laser is clearly visible as it propagates through a Sagnac-type interferometer with the down-conversion crystal at the center. On the lower left, a small spool of fiber is light up by a HeNe laser. This fiber spool is a miniaturization of the actual spool that serves as the quantum memory in our experiment. [Copyright: R. Prevedel]

A similar experiment was performed independently in the group of G.-C. Guo, and its results were published in the same issue of Nature Physics [8] (Also see the past 2Physics article dated October 11, 2011).

References:
[1] Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172-198 (1927).
[2] Bennett, C. H. & Brassard, G. Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 175-179 (1984).
[3] Bialynicki-Birula, I. & Mycielski, J. "Uncertainty relations for information entropy in wave mechanics". Communications in Mathematical Physics, 44, 129-132 (1975). Abstract.
[4] Deutsch, D. "Uncertainty in quantum measurements". Physical Review Letters, 50, 631-633 (1983). Abstract.
[5] Maassen, H. & Uffink, J. B. "Generalized entropic uncertainty relations". Physical Review Letters, 60, 1103-1106 (1988). Abstract.
[6] Berta, M., Christandl, M., Colbeck, R., Renes, J. M. & Renner, R. "The uncertainty principle in the presence of quantum memory". Nature Physics, 6, 659-662 (2010). Abstract. 2Physics Article.
[7] Prevedel, R., Hamel, D.R., Colbeck, R., Fisher, K., & Resch, K.J. "Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement", Nature Physics, 7, 757-761 (2011). Abstract.
[8] Li, C-F., Xu, J-S., Xu, X-Y., Li, K. & Guo, G-C. "Experimental investigation of the entanglement-assisted entropic uncertainty principle". Nature Physics, 7, 752-756 (2011). Abstract. 2Physics Article.