Entanglement is a key concept in quantum theory because it is a resource of magical phenomena of quantum systems. For further development of quantum science, it is needed to investigate entanglement from both viewpoints of mathematics and physics in a deeper way. Therefore, this special issue is devoted to the theory of quantum entanglement in mathematics & physics, and its application to quantum information processing. Our focus is on the state discrimination and nonlocality, entanglement measure, separability problem, positive-partial-transpose (PPT) states, quantum computing, cryptography, mutually unbiased basis (MUBs), unextendible product basis (UPBs), and matrix theory, and properties of dynamical evolution and applications of entanglement in open systems. These structures can be applied to various types of quantum information processing including quantum computation and quantum secure protocols. As typical examples, we can list quantum teleportation, dense coding, one-way quantum computation, quantum secure multiparty computation, and quantum games.

Plenty of efforts have been devoted to the understanding of entanglement theory in the past decades, though a complete picture is still far away. For example, what can be explained about the separability criterion on the recent progress on two-qutrit PPT states? How much entanglement may be the least amount to distinguish UPBs assisted with local operations and classical communications? One may also study the differences and connections between entangled quantum correlations and the prediction of quantum phase transitions in open systems. Further, although various types of quantum information processing have been proposed based on entanglement, still the number of useful quantum information processing is very limited. Hence, its further developments are required.

The research topic contributors may address include (but not restricted to) the following:
• Entanglement measure;
• State discrimination and quantum nonlocality;
• State manipulation;
• Mutually unbiased basis;
• Unextendible product basis;
• Entangling power of multipartite unitary gates;
• Application to quantum computation;
• Application to quantum secure protocols;
• Quantum phase transitions are measured by entanglement;
• Dynamic evolution.

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