A Correlation Measure Based on Vector-Valued $L_p$-Norms

In this paper, we introduce a new measure of correlation for bipartite quantum states. This measure depends on a parameter $\alpha$, and is defined in terms of vector-valued $L_p$-norms. The measure is within a constant of the exponential of $\alpha$-R\'enyi mutual information, and reduces to the trace norm (total variation distance) for $\alpha=1$. We will prove some decoupling type theorems in terms of this measure of correlation, and present some applications in privacy amplification as well as in bounding the random coding exponents. In particular, we establish a bound on the secrecy exponent of the wiretap channel (under the total variation metric) in terms of the $\alpha$-R\'enyi mutual information according to \emph{Csisz\'ar's proposal}.