### Abstract

In a thermodynamic model, Diósi [Int. J. Quantum Inf. 4, 99-104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.

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

Article number | 092102 |

Journal | Journal of Mathematical Physics |

Volume | 48 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*48*(9), [092102]. https://doi.org/10.1063/1.2779138

**Limit relation for quantum entropy and channel capacity per unit cost.** / Csiszár, I.; Hiai, Fumio; Petz, D.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 48, no. 9, 092102. https://doi.org/10.1063/1.2779138

}

TY - JOUR

T1 - Limit relation for quantum entropy and channel capacity per unit cost

AU - Csiszár, I.

AU - Hiai, Fumio

AU - Petz, D.

PY - 2007

Y1 - 2007

N2 - In a thermodynamic model, Diósi [Int. J. Quantum Inf. 4, 99-104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.

AB - In a thermodynamic model, Diósi [Int. J. Quantum Inf. 4, 99-104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.

UR - http://www.scopus.com/inward/record.url?scp=34848835721&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34848835721&partnerID=8YFLogxK

U2 - 10.1063/1.2779138

DO - 10.1063/1.2779138

M3 - Article

AN - SCOPUS:34848835721

VL - 48

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 092102

ER -