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Bound entangled Gaussian states.

R F Werner1, M M Wolf

  • 1Institut für Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany. r.werner@tu-bs.de

Physical Review Letters
|May 1, 2001
PubMed
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We establish criteria for Gaussian Wigner function entanglement. Positivity of the partial transpose implies separability for single-oscillator systems, but this fails for two-oscillator systems, revealing bound entangled Gaussian states.

Area of Science:

  • Quantum Information Theory
  • Quantum Optics
  • Mathematical Physics

Background:

  • Bipartite quantum states are fundamental to quantum information processing.
  • Gaussian Wigner functions provide a phase-space representation for quantum states.
  • Entanglement, separability, and the positivity of the partial transpose are key concepts in quantifying quantum correlations.

Purpose of the Study:

  • To establish necessary and sufficient criteria for separability and positivity of the partial transpose of bipartite Gaussian states.
  • To investigate the relationship between these criteria in systems with varying numbers of oscillators.
  • To identify conditions under which positivity of the partial transpose does not imply separability.

Main Methods:

  • Utilizing the covariance matrix of Gaussian states to derive analytical criteria.

Related Experiment Videos

  • Analyzing the properties of bipartite systems with one oscillator for Alice and an arbitrary number for Bob.
  • Constructing a family of bound entangled Gaussian states for systems with two oscillators per party.

    • Main Results:

    • Explicit criteria for separability and positivity of the partial transpose were derived in terms of the covariance matrix.
    • For systems with one oscillator per party, positivity of the partial transpose was shown to be equivalent to separability.
    • A five-parameter family of bound entangled Gaussian states was constructed for two-oscillator systems, demonstrating a failure of the implication.

    Conclusions:

    • The covariance matrix provides a complete description for entanglement properties of bipartite Gaussian states.
    • The equivalence between positivity of the partial transpose and separability holds for certain system configurations but not universally.
    • The discovery of bound entangled Gaussian states highlights the complexity of entanglement in multipartite systems.