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Concentration of Measure for Quantum States with a Fixed Expectation Value

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Abstract

Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.

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Authors and Affiliations

  1. Institute of Mathematics, Technical University of Berlin, 10623, Berlin, Germany

    Markus P. Müller

  2. Institute of Physics and Astronomy, University of Potsdam, 14476, Potsdam, Germany

    Markus P. Müller & Jens Eisert

  3. Institute for Theoretical Physics, Leibniz University Hannover, 30167, Hannover, Germany

    David Gross

  4. Institute for Theoretical Physics, ETH Zürich, 8093, Zürich, Switzerland

    David Gross

  5. Institute for Advanced Study Berlin, 14193, Berlin, Germany

    Jens Eisert

Authors
  1. Markus P. Müller
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  2. David Gross
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  3. Jens Eisert
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Correspondence to David Gross.

Additional information

Communicated by M.B. Ruskai

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Müller, M.P., Gross, D. & Eisert, J. Concentration of Measure for Quantum States with a Fixed Expectation Value. Commun. Math. Phys. 303, 785–824 (2011). https://doi.org/10.1007/s00220-011-1205-1

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  • DOI: https://doi.org/10.1007/s00220-011-1205-1

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