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### Abstract

A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.

I am grateful to V. Eremeev, G. Ferrini, A. Joye, M. Orszag, and A. Smerzi for interesting discussions and to V. Jakšić for pointing out to me the works of Refs. 114 and 60. I acknowledge support from the ANR project no. ANR-13-JS01-0005-01.

I. INTRODUCTION II. QUANTUM STATES A. Quantum states and observables B. The Schmidt decomposition C. Purifications and pure state decompositions of mixed states D. Entangled and separable states III. QUANTUM MEASUREMENTS A. Physical realization of a measurement process B. Quantum operations C. Generalized measurements D. Connections between POVMs, quantum operations, and state ensembles IV. TRANSPOSE OPERATION AND LEAST SQUARE MEASUREMENT A. Recovery operation in quantum error correction B. Transpose operation as an approximate reverse operation C. Least square measurement V. QUANTUM STATE DISCRIMINATION A. Discriminating quantum states drawn from a given ensemble B. Ambiguous and unambiguous discriminations of two states 1. Ambiguous discrimination 2. Unambiguous discrimination of two pure states 3. Unambiguous discrimination of two mixed states C. Discrimination with least square measurements D. General results on ambiguous discrimination E. Bounds on the maximal success probability F. The Holevo bound VI. QUANTUM ENTROPIES A. The von Neumann entropy B. Relative entropy C. Quantum relative Rényi entropies 1. Definitions 2. Main properties 3. Monotonicity in α VII. THE BURES DISTANCE AND UHLMANN FIDELITY A. Contractive and convex distances B. The Bures distance C. Bures distance and statistical distance in classical probability D. Comparison of the Bures and trace distances E. Bures and quantum Hellinger metrics, quantum Fisher information F. Characterization of the Riemannian contractive distances VIII. STATE DISCRIMINATION AND PARAMETER ESTIMATION IN LARGE SYSTEMS A. Quantum hypothesis testing: Discriminating two states from many identical copies B. Parameter estimation in quantum metrology 1. Phase estimation in Mach-Zehnder interferometers 2. Quantum Cramér-Rao bound 3. Interferometer precision and inter-particle entanglement IX. MEASURES OF ENTANGLEMENT IN BIPARTITE SYSTEMS A. Entanglement as correlations between local measurements B. LOCC operations C. Axioms on entanglement measures D. Entanglement of formation 1. Entanglement of formation for pure states 2. Convex roof constructions 3. The Wootters formula for two qubits E. Maximally entangled states X. THE QUANTUM DISCORD A. Definition of the quantum discord B. The -classical states C. Properties of the quantum discord 1. Invariance and monotonicity properties 2. States with the highest discord 3. Monotonicity when disregarding a part of the measured subsystem D. Monogamy relation XI. DISTANCE AND ENTROPIC MEASURES OF QUANTUM CORRELATIONS A. Geometric and relative-entropy measures of entanglement 1. Definition and main properties 2. Relation between the geometric measure of entanglement and convex roof constructions 3. Geometric measure of entanglement for two qubits B. Geometric quantum discord 1. Discord-like measures of quantum correlations 2. Geometric discord for pure states 3. Geometric discord for mixed states and quantum state discrimination 4. The qubit case 5. States with the highest geometric discord 6. Geometric discord and least square measurements

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