Quantum Entropy and Correlations in Quantum Information

Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.