### Abstract

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger (ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

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

Article number | 6670246 |

Pages (from-to) | 8014-8026 |

Number of pages | 13 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2013 |

### Fingerprint

### Keywords

- Capacity
- hypothesis testing
- quantum channels
- smooth max-relative entropy
- strong converse

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Cite this

*IEEE Transactions on Information Theory*,

*59*(12), 8014-8026. [6670246]. https://doi.org/10.1109/TIT.2013.2282160

**A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels.** / Datta, Nilanjana; Mosonyi, M.; Hsieh, Min Hsiu; Brandao, Fernando G S L.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 59, no. 12, 6670246, pp. 8014-8026. https://doi.org/10.1109/TIT.2013.2282160

}

TY - JOUR

T1 - A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

AU - Datta, Nilanjana

AU - Mosonyi, M.

AU - Hsieh, Min Hsiu

AU - Brandao, Fernando G S L

PY - 2013/12

Y1 - 2013/12

N2 - We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger (ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

AB - We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger (ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

KW - Capacity

KW - hypothesis testing

KW - quantum channels

KW - smooth max-relative entropy

KW - strong converse

UR - http://www.scopus.com/inward/record.url?scp=84889594285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84889594285&partnerID=8YFLogxK

U2 - 10.1109/TIT.2013.2282160

DO - 10.1109/TIT.2013.2282160

M3 - Article

AN - SCOPUS:84889594285

VL - 59

SP - 8014

EP - 8026

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 12

M1 - 6670246

ER -