### Abstract

We show that the new quantum extension of Rényi’s α-relative entropies, introduced recently by Müller-Lennert et al. (J Math Phys 54:122203, 2013) and Wilde et al. (Commun Math Phys 331(2):593–622, 2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Rényi relative entropies depends on the parameter α: for α <1, the right choice seems to be the traditional definition (Formula presented.), whereas for α > 1 the right choice is the newly introduced version (Formula presented.). On the way to proving our main result, we show that the new Rényi α-relative entropies are asymptotically attainable by measurements for α > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.

Original language | English |
---|---|

Pages (from-to) | 1617-1648 |

Number of pages | 32 |

Journal | Communications in Mathematical Physics |

Volume | 334 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies.** / Mosonyi, Milán; Ogawa, Tomohiro.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 334, no. 3, pp. 1617-1648. https://doi.org/10.1007/s00220-014-2248-x

}

TY - JOUR

T1 - Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies

AU - Mosonyi, Milán

AU - Ogawa, Tomohiro

PY - 2014

Y1 - 2014

N2 - We show that the new quantum extension of Rényi’s α-relative entropies, introduced recently by Müller-Lennert et al. (J Math Phys 54:122203, 2013) and Wilde et al. (Commun Math Phys 331(2):593–622, 2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Rényi relative entropies depends on the parameter α: for α <1, the right choice seems to be the traditional definition (Formula presented.), whereas for α > 1 the right choice is the newly introduced version (Formula presented.). On the way to proving our main result, we show that the new Rényi α-relative entropies are asymptotically attainable by measurements for α > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.

AB - We show that the new quantum extension of Rényi’s α-relative entropies, introduced recently by Müller-Lennert et al. (J Math Phys 54:122203, 2013) and Wilde et al. (Commun Math Phys 331(2):593–622, 2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Rényi relative entropies depends on the parameter α: for α <1, the right choice seems to be the traditional definition (Formula presented.), whereas for α > 1 the right choice is the newly introduced version (Formula presented.). On the way to proving our main result, we show that the new Rényi α-relative entropies are asymptotically attainable by measurements for α > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.

UR - http://www.scopus.com/inward/record.url?scp=84923615213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923615213&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-2248-x

DO - 10.1007/s00220-014-2248-x

M3 - Article

VL - 334

SP - 1617

EP - 1648

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -