Truncation of basis states is a vital step in the tensor network renormalization. This chapter introduces the concept of reduced density matrices and discusses the criterion of judging which state should be retained and which not in the basis truncation. In a Hermitian system, the reduced density matrix of a quantum state is semi-positive definite, and its eigenvalues measure the probabilities of the corresponding eigenvectors. Therefore, we should do the truncation according to the eigenvalues of the reduced density matrix. This criterion is equivalent to taking a Schmidt decomposition for the wave function of the quantum state and truncating the basis states according to their singular values. It is also equivalent to maximizing the fidelity of the targeted state before and after truncation. We also introduce the edge and bond density matrices and show that they have the same eigen-spectra as the reduced density matrix.

Chapter devoted to the basic quantum properties of entanglement and separability. Introduces the Schmidt decomposition for pure states and the positive partial transpose criterion for mixed states as entanglement witnesses. Introduces the famous Einstein–Podolsky–Rosen paradox and its implementation in terms of qubits, then the Bell inequality, quickly reviewing the experimental demonstrations that quantum mechanics violates this inequality. Gives examples of the use of entanglement in a quantum algorithm to accelerate an information task, namely a database search (Grover algorithm) and the possibility of teleportation of a quantum state.