Exponents in smoothing the max-relative entropy and of randomness extraction against quantum side information

Li，Ke1; Yao，Yongsheng2,3; Hayashi，Masahito3,4,5

10.1109/ISIT50566.2022.9834595

2022

2157-8095

978-1-6654-2160-7

IEEE International Symposium on Information Theory - Proceedings

2022-June

1862-1867

26 June-1 July 2022

Espoo, Finland

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award.The smooth max-relative entropy is a basic tool in quantum information theory and cryptography. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information and obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general.Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order between 1 and 2, which has not been addressed previously in the quantum setting.

Privacy Upper bound Smoothing methods Quantum state Entropy Data mining Cryptography

Others

English

EI

National Natural Science Foundation of China[12031004];National Natural Science Foundation of China[12071099];National Natural Science Foundation of China[61871156];National Natural Science Foundation of China[61871156];National Natural Science Foundation of China[62171212];

20223512624603

Extraction ; Information theory ; Quantum optics ; Random processes

Thermodynamics:641.1 ; Information Theory and Signal Processing:716.1 ; Light/Optics:741.1 ; Chemical Operations:802.3 ; Probability Theory:922.1 ; Quantum Theory; Quantum Mechanics:931.4

2-s2.0-85136292681

Scopus

1.Harbin Institute of Technology,Institute for Advanced Study in Mathematics,Harbin,150001,China
2.Harbin Institute of Technology,Institute for Advanced Study in Mathematics,School of Mathematics,Harbin,150001,China
3.Shenzhen Institute for Quantum Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
4.International Quantum Academy (SIQA),Shenzhen,518048,China
5.Graduate School of Mathematics,Nagoya University,Nagoya,Furo-cho, Chikusa-ku,464-8602,Japan

Li，Ke,Yao，Yongsheng,Hayashi，Masahito. Exponents in smoothing the max-relative entropy and of randomness extraction against quantum side information[C],2022:1862-1867.