The uncertainty principle in the presence of quantum memory

Journal name:
Nature Physics
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Published online

The uncertainty principle, originally formulated by Heisenberg1, clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future2, it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.


  1. Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172198 (1927).
  2. Julsgaard, B., Sherson, J., Cirac, J. I., Fiurášek, J. & Polzik, E. S. Experimental demonstration of quantum memory for light. Nature 432, 482486 (2004).
  3. Robertson, H. P. The uncertainty principle. Phys. Rev. 34, 163164 (1929).
  4. Bial strokeynicki-Birula, I. & Mycielski, J. Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129132 (1975).
  5. Deutsch, D. Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631633 (1983).
  6. Kraus, K. Complementary observables and uncertainty relations. Phys. Rev. D 35, 30703075 (1987).
  7. Maassen, H. & Uffink, J. B. Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 11031106 (1988).
  8. Devetak, I. & Winter, A. Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207235 (2005).
  9. Horodecki, M., Horodecki, P. & Horodecki, R. Separability of mixed states: Necessary and sufficient conditions. Phys. Lett. A 223, 18 (1996).
  10. Terhal, B. M. Bell inequalities and the separability criterion. Phys. Lett. A 271, 319326 (2000).
  11. Lewenstein, M., Kraus, B., Cirac, J. I. & Horodecki, P. Optimization of entanglement witnesses. Phys. Rev. A 62, 116 (2000).
  12. Gühne, O. & Tóth, G. Entanglement detection. Phys. Rep. 747, 175 (2009).
  13. Wiesner, S. Conjugate coding. Sigact News 15, 7888 (1983).
  14. Bennett, C. H. & Brassard, G. in Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing, Bangalore, India 175179 (IEEE, 1984).
  15. Deutsch, D. et al. Quantum privacy amplification and the security of quantum cryptography over noisy channels. Phys. Rev. Lett. 77, 28182821 (1996).
  16. Lo, H-K. & Chau, H. F. Unconditional security of quantum key distribution over arbitrarily long distances. Science 283, 20502056 (1999).
  17. Shor, P. W. & Preskill, J. Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441444 (2000).
  18. Christandl, M., Renner, R. & Ekert, A. A generic security proof for quantum key distribution. Preprint at (2004).
  19. Renner, R. & König, R. Universally composable privacy amplification against quantum adversaries. Theory of Cryptography Conference, TCC 2005 407425 (Lecture Notes in Computer Science, Vol. 3378, Springer, 2005).
  20. Renner, R. Security of quantum key distribution. Int. J. Quantum Inform. 6, 1127 (2008).
  21. Koashi, M. Unconditional security of quantum key distribution and the uncertainty principle. J. Phys. Conf. Ser. 36, 98102 (2006).
  22. Ekert, A. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661663 (1991).
  23. Renes, J. M. & Boileau, J-C. Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009).
  24. Christandl, M., König, R. & Renner, R. Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102, 020504 (2009).
  25. Renner, R. & Scarani, V. Quantum cryptography with finite resources: Unconditional security bound for discrete-variable protocols with one-way postprocessing. Phys. Rev. Lett. 100, 200501 (2008).
  26. Chandran, N., Fehr, S., Gelles, R., Goyal, V. & Ostrovsky, R. Position-based quantum cryptography. Preprint at (2010).
  27. DiVincenzo, D. P., Horodecki, M., Leung, D. W., Smolin, J. A. & Terhal, B. M. Locking classical correlations in quantum states. Phys. Rev. Lett. 92, 067902 (2004).
  28. Christandl, M. & Winter, A. Uncertainty, monogamy and locking of quantum correlations. IEEE Trans. Inf. Theory 51, 31593165 (2005).
  29. König, R., Renner, R. & Schaffner, C. The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55, 43374347 (2009).
  30. Tomamichel, M., Colbeck, R. & Renner, R. A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55, 58405847 (2009).

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  1. Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

    • Mario Berta,
    • Matthias Christandl,
    • Roger Colbeck &
    • Renato Renner
  2. Faculty of Physics, Ludwig-Maximilians-Universität München, 80333 Munich, Germany

    • Mario Berta &
    • Matthias Christandl
  3. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

    • Roger Colbeck
  4. Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland

    • Roger Colbeck
  5. Institute for Applied Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany

    • Joseph M. Renes


All authors contributed equally to this work.

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