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In this paper, we introduce a new measure of correlation for bipartite quantum states. This measure depends on a parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, and is defined in terms of vector-valued <inline-formula> <tex-math notation="LaTeX">$\textit {L}_{\textit {p}}$ </tex-math></inline-formula>-norms. The measure is within a constant of the exponential of <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-R&#x00E9;nyi mutual information, and reduces to the trace norm (total variation distance) for <inline-formula> <tex-math notation="LaTeX">$\alpha =1$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-R&#x00E9;nyi mutual information according to <italic>Csisz&#x00E1;r&#x2019;s proposal</italic>.

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