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Covariance matrices and the separability problem.

O Gühne1, P Hyllus, O Gittsovich

  • 1Institut für Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria.

Physical Review Letters
|October 13, 2007
PubMed
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We introduce a new method using covariance matrices to solve the separability problem. This approach provides strong entanglement criteria for detecting bound entangled states in higher dimensions and is definitive for two qubits.

Area of Science:

  • Quantum Information Theory
  • Quantum Entanglement
  • Quantum Many-Body Systems

Background:

  • The separability problem is a fundamental challenge in quantum information theory, concerning the identification of entangled quantum states.
  • Existing entanglement criteria have limitations in detecting certain classes of entangled states, particularly in higher dimensions.

Purpose of the Study:

  • To develop a unifying approach to the quantum separability problem.
  • To establish strong and universally applicable entanglement criteria.
  • To gain fundamental insights into the relationships and limitations of various entanglement criteria.

Main Methods:

  • Utilizing covariance matrices of locally measurable observables.
  • Developing novel entanglement criteria based on these matrices.

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  • Testing the criteria against known entanglement measures and bound entangled states.

    • Main Results:

    • The proposed approach offers a unifying framework for the separability problem.
    • Strong entanglement criteria are derived, capable of detecting entanglement in many bound entangled states in higher dimensions.
    • For two-qubit systems, these criteria are proven to be both necessary and sufficient for detecting entanglement.

    Conclusions:

    • The covariance matrix approach provides a powerful tool for entanglement detection and characterization.
    • This work clarifies the connections and limitations of existing entanglement criteria, including the computable cross-norm and local uncertainty criteria.
    • The findings have implications for both theoretical understanding and practical applications in quantum information processing.