### Abstract

The general quantum χ^{2}-divergence has recently been introduced by Temme et al. [Temme K, Kastoryano M, Ruskai M, Wolf M, Verstrate F (2010) J Math Phys 51:122201] and applied to quantum channels (quantum Markov processes). The quantum χ^{2}-divergence is not unique, as opposed to the classical χ^{2}-divergence, but depends on the choice of quantum statistics. It was noticed that the elements in a particular one-parameter family χ_{α}^{2}(ρ,σ) of quantum χ^{2}-divergences are convex functions in the density matrices (ρ,σ), thus mirroring the convexity of the classical χ^{2}(p,q)-divergence in probability distributions (p,q). We prove that any quantum χ^{2}-divergence is a convex function in its two arguments.

Original language | English |
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Pages (from-to) | 10078-10080 |

Number of pages | 3 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 108 |

Issue number | 25 |

DOIs | |

Publication status | Published - 2011 Jun 21 |

### Keywords

- Monotone metric
- Operator monotone function
- Quantum mechanics

### ASJC Scopus subject areas

- General

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## Cite this

^{2}-divergence.

*Proceedings of the National Academy of Sciences of the United States of America*,

*108*(25), 10078-10080. https://doi.org/10.1073/pnas.1106423108