  We treat secret key extraction when the eavesdropper has correlated quantum
states. We propose quantum privacy amplification theorems different from
Renner's, which are based on quantum conditional R\'{e}nyi entropy of order
1+s. Using those theorems, we derive an exponential decreasing rate for leaked
information and the asymptotic equivocation rate, which have not been derived
hitherto in the quantum setting.
