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Uncertainty relation for symmetric Petz-Rényi relative entropy.

Domingos S P Salazar1

  • 1Unidade de Educação a Distância e Tecnologia, <a href="https://ror.org/02ksmb993">Universidade Federal Rural de Pernambuco</a>, 52171-900 Recife, Pernambuco, Brazil.

Physical Review. E
|June 22, 2024
PubMed
Summary

Holevo

Area of Science:

  • Quantum Information Theory
  • Quantum Thermodynamics

Background:

  • Holevo introduced a symmetric quantum fidelity, comparable to trace distance for assessing quantum state similarity.
  • This fidelity is linked to trace distance via Holevo's inequality.
  • Holevo's fidelity is a member of a broader family of symmetric Petz-Rényi relative entropies.

Purpose of the Study:

  • To analyze the relationship between symmetric Petz-Rényi relative entropies and trace distance.
  • To improve upon existing inequalities, specifically Pinsker's inequality, in this context.
  • To connect these findings to quantum and stochastic thermodynamics via uncertainty relations.

Main Methods:

  • Investigating the mathematical properties of symmetric Petz-Rényi relative entropies.
  • Deriving new inequalities relating these entropies to trace distance.

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  • Establishing a connection between these entropic inequalities and uncertainty relations.

    • Main Results:

    • Symmetric Petz-Rényi relative entropies satisfy a tight inequality with respect to trace distance.
    • This new inequality improves upon the existing Pinsker's inequality.
    • Holevo's inequality is shown to be a specific case of the newly derived tight inequality.
    • The results are linked to a symmetric Petz-Rényi uncertainty relation.

    Conclusions:

    • A novel, tight inequality relating symmetric Petz-Rényi relative entropies and trace distance has been established.
    • This work refines our understanding of quantum state comparison and information-theoretic inequalities.
    • The findings have broader implications, unifying concepts in quantum thermodynamics and stochastic thermodynamics.