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

Ke Li,1, ∗ Yongsheng Yao,1, 2, 3, † and Masahito Hayashi3, 4, 5, 6, ‡ Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China School of Mathematics, Harbin Institute of Technology, Harbin 150001, China Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China International Quantum Academy (SIQA), Shenzhen 518048, China Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan