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Symmetry of large physical systems implies independence of subsystems

Nature Physics volume 3, pages 645–649 (2007)Cite this article

Abstract

Composite systems consisting of a large number of similar subsystems play an important role in many areas of physics as well as in information theory. Their analysis, however, often relies on the assumption that the subsystems are mutually independent (or only weakly correlated). Here, we show that this assumption is generally justified for quantum systems that are symmetric, that is, invariant under permutations of the subsystems. Because symmetry is often implied by natural properties, for example, the indistinguishability of identical particles, the result has a wide range of consequences. In particular, it implies that global properties of a large composite system can be estimated by measurements applied to a limited number of (randomly chosen) sample subsystems, a fact that is important for the interpretation of experimental data. Moreover, it generalizes statements in quantum information theory and cryptography, which previously have only been known to hold under certain independence assumptions.

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Figure 1: Tomography.
Figure 2: Symmetry.
Figure 3: Relation between symmetry and i.i.d.
Figure 4: The effect of symmetrization.

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Acknowledgements

I thank C. Bennett, M. Christandl, R. Colbeck, A. Ekert, R. König, U. Maurer and G. Mitchison for their helpful comments on earlier versions of the manuscript. This research was supported by HP Labs Bristol and by the EU through the project SECOQC. Parts of this work have been accomplished while the author was at ETH Zurich.

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  1. Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Cambridge CB3 OWA, UK

    Renato Renner

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Correspondence to Renato Renner.

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Renner, R. Symmetry of large physical systems implies independence of subsystems. Nature Phys 3, 645–649 (2007). https://doi.org/10.1038/nphys684

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