Basic Concepts of Entangled States

Warit Asavanant, Akira Furusawa

Basic concepts of quantum optics are introduced in this chapter. In the beginning, we give a review of the basic quantum mechanics. Then, we perform quantization of an electromagnetic field, which naturally introduces the concept of mode and quadrature. After that, we introduce basic quantum states of light and their phase space representation. Quantum operations in the continuous-variable system and their universality are also explained. To obtain the values of the physical quantities, we have to measure our quantum systems. In the optical continuous-variable system, homodyne measurement and photon number measurement are among the most common measurements that are also discussed here. Finally, we introduce the concept of quantum entanglement, which is one of the central concepts in quantum computation.

Willi-Hans Steeb, Yorick Hardy

Jānis Ruža

E. Karlsson, Erkki Brändas

Modern studies of basic quantum concepts and phenomena, E.B. Karlsson and E. Brandas quantum entanglement - criteria and collective tests, A. Peres a gallery of quantum states -tomography of nonclassical light, J. Mlynek et al experimental and theoretical progress in interaction-free measurements, P.G. Kwia entanglement, mesoscopic superpositions and decoherence studies with atoms and photons in a cavity, S. Haroche quantum coherence with a single Cooper pair, V. Bouchiat et al mesoscopic physics and the fundamentals of quantum mechanics, Y. Imry decoherence, chaos, quantum-classical correspondence and the algorithmic arrow of time, W.H. Zurek quantum entanglement - a fundamental concept finding its applications, A. Zeilinger quantum communication in a quantum network, J.I. Cirac et al. (Part contents).

Shi-Ju Ran, Emanuele Tirrito, Cheng Peng +4 more

This chapter is to introduce some basic definitions and concepts of TN Tensor network (TN) . We will show that the TN can be used to represent quantum many-body states, where we explain MPS Matrix product state (MPS) in 1D and PEPS Projected entangled pair state (PEPS) in 2D systems, as well as the generalizations to thermal states and operators. The quantum entanglement properties of the TN states including the area law of entanglement entropy will also be discussed. Finally, we will present several special TNs that can be exactly contracted, and demonstrate the difficulty of contracting TNs in general cases.

Ph. Blanchard, Lech Jakóbczyk, Robert Olkiewicz