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State-Independent Uncertainty Relations and Entanglement Detection in Noisy Systems

René Schwonnek, Lars Dammeier, and Reinhard F. Werner
Phys. Rev. Lett. 119, 170404 – Published 27 October 2017
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  • Introduction.—
  • Methods.—
  • Geometry of outer approximations.—
  • Basic algorithm.—
  • Generalization to POVMs.—
  • Application to entanglement detection.—
  • Entanglement detection with noisy…
  • Conclusions and outlook.—
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.

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    • Received 31 May 2017

    DOI:https://doi.org/10.1103/PhysRevLett.119.170404

    © 2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Entanglement detectionNonlocalityQuantum entanglementQuantum foundationsQuantum measurementsQuantum metrology
    General PhysicsQuantum Information, Science & Technology

    Authors & Affiliations

    René Schwonnek*, Lars Dammeier†, and Reinhard F. Werner‡

    • Leibniz Universität Hannover—Institut für Theoretische Physik, Hannover 30167, Germany

    • *rene.schwonnek@itp.uni-hannover.de
    • †lars.dammeier@itp.uni-hannover.de
    • ‡reinhard.werner@itp.uni-hannover.de

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    Issue

    Vol. 119, Iss. 17 — 27 October 2017

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    Images

    • Figure 1
      Figure 1

      Minimizing the sum of the variances of two observables A and B can be expressed entirely in terms of the set C of possible triples (⟨A⟩ρ,⟨B⟩ρ,⟨A2+B2⟩ρ) (red solid convex body), namely, as finding that vertical displacement of the surface z=x2+y2 (green paraboloid) which just touches C from below. We successively approximate C by polytopes (blue edges, boxed vertices) from the outside, and perform the minimization on this polytope. This gives a converging sequence of correct state-independent uncertainty relations.

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    • Figure 2
      Figure 2

      Two dimensional sketch of geometry and the basic algorithm: The set C (red) with its outer approximation P(R) (blue and blue dashed lines) and the extremal points E(R) (white squares). By adding the direction r′, the polyhedral approximation is refined and the lower bound c−(R) is improved from μ(v*) (dashed green parabola) to μ(v**) (green parabola).

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    • Figure 3
      Figure 3

      Improving the outer approximation of C (red convex body) by adding more directions to the set R. Every direction r∈R gives a face of P(R) (blue polytope). New directions are chosen such that the vertex with the lowest value of μ will be cut off. Example generated from randomly chosen A,B∈R10×10.

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    • Figure 4
      Figure 4

      Uncertainty regions for entangled and separable states. Superposition of the graphs for different noise levels α: green=0, blue=0.2, red=0.5. In this example we consider local measurements of orthogonal spin-1 components, i.e., Mi=LiA+LiB.

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