Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification

Li，Ke1; Yao，Yongsheng2; Hayashi，Masahito3

2022

10.1109/TIT.2022.3217671

IEEE TRANSACTIONS ON INFORMATION THEORY   Impact Factor & Quartile

0018-9448

1557-9654

The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic 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. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order s ϵ (1, 2], which has not been addressed previously in the quantum setting.

Behavioral sciences Entropy equivocation exponent max-relative entropy Privacy quantum privacy amplification Quantum state sandwiched Rényi divergence Security Smoothing methods Testing

English

Others

COMPUTER SCIENCE

2-s2.0-85141479117

Scopus

1.Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Nangang District, Harbin, China
2.Institute for Advanced Study in Mathematics, School of Mathematics, Harbin Institute of Technology, Nangang District, Harbin, China
3.Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Nanshan District, Shenzhen, China

Li，Ke,Yao，Yongsheng,Hayashi，Masahito. Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification[J]. IEEE TRANSACTIONS ON INFORMATION THEORY,2022,PP(99):1-1.

Li，Ke,Yao，Yongsheng,&Hayashi，Masahito.(2022).Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification.IEEE TRANSACTIONS ON INFORMATION THEORY,PP(99),1-1.

Li，Ke,et al."Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification".IEEE TRANSACTIONS ON INFORMATION THEORY PP.99(2022):1-1.