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How to Quantify a Dynamical Quantum Resource.

Gilad Gour1, Andreas Winter2

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The generalization of relative entropy for quantum channels is not unique, with at least six versions identified. Two of these are crucial for quantum information theory, appearing in quantum Stein's lemma.

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Area of Science:

  • Quantum Information Theory
  • Quantum Thermodynamics
  • Mathematical Physics

Background:

  • The relative entropy is a fundamental measure in quantum information theory, quantifying distinguishability between quantum states.
  • Generalizing concepts from quantum states to quantum channels is a key challenge in the field.
  • Understanding resource theories for quantum channels requires appropriate entropic quantities.

Purpose of the Study:

  • To investigate the non-unique generalizations of the relative entropy of a resource from quantum states to quantum channels.
  • To identify which generalizations possess desirable properties for applications in quantum information theory.
  • To explore the connection between these generalized entropies and fundamental theorems like quantum Stein's lemma.

Main Methods:

  • Development of a novel "liberal smoothing" technique applicable to functions of quantum channels.
  • Analysis of at least six distinct generalizations of the relative entropy of a resource for channels.
  • Investigation of asymptotic continuity and the asymptotic equipartition property for these generalizations.

Main Results:

  • Demonstration that at least six generalizations of the relative entropy of a resource for channels exist.
  • Identification of two specific generalizations that are asymptotically continuous and satisfy the asymptotic equipartition property.
  • Showing that regularizations of these two generalizations appear in the exponent of channel versions of quantum Stein's lemma.
  • Expressing the diamond norm as a max relative entropy distance to the set of quantum channels.

Conclusions:

  • The choice of relative entropy generalization for quantum channels significantly impacts their properties and applications.
  • The identified generalizations and their properties are crucial for understanding channel capacities and information transmission limits.
  • Liberal smoothing provides a powerful new tool for analyzing functions of quantum channels.