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The theory of quantum entanglement is still under construction and has generated great interest over the last thirty years. Indeed, determining whether a given quantum state is entangled or not is still an open problem today in physics, known as the separability problem. In this manuscript, we aim at giving a selective but up to date review of the separability problem. We focus on the theoretical perspective of the problem. Theoretically, quantum entanglement is defined by a mathematical property of quantum states that are described by density operators acting on Hilbert spaces. We first explicit this definition and introduce some basis notions of the entanglement theory such as the Bloch representation of quantum states. Then, we present and analyse several separability criteria. We begin with the most important ones developed from 1996 to 2003, that is the PPT criterion, criteria based on entanglement witnesses, on entanglement measures (such as negativity or concurrences) and the CCNR criterion. We then move on to more recent criteria, developed from 2009 to 2020. Most of them make use of the Bloch representation. We finally conclude by comparing the exposed criteria.

Quantum entanglement --- Separability problem --- Separability criteria --- Bloch representation --- Intrication quantique --- Problème de séparabilité --- Critères de séparabilité --- Représentation de Bloch --- Physique, chimie, mathématiques & sciences de la terre > Physique