The uncertainty principle in the presence of quantum memory

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Abstract

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.

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Figure 1: Illustration of the uncertainty game.

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Acknowledgements

We thank R. König, J. Oppenheim and M. Tomamichel for discussions and L. del Rio for the illustration (Fig. 1). M.B. and M.C. acknowledge support from the German Science Foundation (DFG) and the Swiss National Science Foundation. J.M.R. acknowledges the support of the Center for Advanced Security Research Darmstadt (CASED). R.C. and R.R. acknowledge support from the Swiss National Science Foundation.

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All authors contributed equally to this work.

Correspondence to Roger Colbeck.

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The authors declare no competing financial interests.

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Berta, M., Christandl, M., Colbeck, R. et al. The uncertainty principle in the presence of quantum memory. Nature Phys 6, 659–662 (2010) doi:10.1038/nphys1734

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