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Paper   IPM / M / 17403
School of Mathematics
  Title:   Operator-valued Schatten spaces and quantum entropies
1.  Salman Beigi
2.  Milad Moazami Goodarzi
  Status:   Published
  Journal: Letters in Mathematical Physics
  Vol.:  113
  Year:  2023
  Pages:   1-54
  Supported by:  IPM
Operator-valued Schatten spaces were introduced by G. Pisier as a noncommutative counterpart of vector-valued $\ell_p$-spaces. This family of operator spaces forms an interpolation scale which makes it a powerful and convenient tool in a variety of applications. In particular, as the norms coming from this family naturally appear in the definition of certain entropic quantities in Quantum Information Theory (QIT), one may apply Pisier's theory to establish some features of those quantities. Nevertheless, it could be quite challenging to follow the proofs of the main results of this theory from the existing literature. In this article, we attempt to fill this gap by presenting the underlying concepts and ideas of Pisier's theory in a self-contained way which we hope to be more accessible, especially for the QIT community at large. Furthermore, we describe some applications of this theory in QIT. In particular, we prove a new uniform continuity bound for the quantum conditional R\'{e}nyi entropy.

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