This chapter generalizes entanglement to multipartite systems, focusing on robustness and scalability across qudits.

This chapter introduces tensor networks as tools for describing multipartite systems and encoding correlations.

This chapter develops a non-asymptotic theory of random matrices. It starts with a quick refresher on linear algebra, including the perturbation theory for matrices and featuring a short proof of the Davis–Kahan inequality. Three key concepts are introduced – nets, covering numbers, and packing numbers – and linked to volume and error-correcting codes. Bounds on the operator norm and singular values of random matrices are established. Three applications are given: community detection in networks, covariance estimation, and spectral clustering. Exercises explore the power method to compute the top singular value, the Schur bound on the operator norm, Hermitian dilation,Walsh matrices, the Wedin theorem on matrix perturbations, a semidefinite relaxation of the cut norm, the volume of high-dimensional balls, and Gaussian mixture models.

This chapter introduces quantum entanglement in bipartite pure states, with measures like mutual information and entanglement entropy.

This chapter introduces measurement entropy, linking it to quantum coherence and the uncertainty principle across states.

This chapter explores entanglement dynamics under LOCC, discussing majorization and the no-cloning theorem.

This chapter explores methods of concentration that do not rely on independence. We introduce the isoperimetric approach and discuss concentration inequalities across a variety of metric measure spaces – including the sphere, Gaussian space, discrete and continuous cubes, the symmetric group, Riemannian manifolds, and the Grassmannian. As an application, we derive the Johnson–Lindenstrauss lemma, a fundamental result in dimensionality reduction for high-dimensional data. We then develop matrix concentration inequalities, with an emphasis on the matrix Bernstein inequality, which extends the classical Bernstein inequality to random matrices. Applications include community detection in sparse networks and covariance estimation for heavy-tailed distributions. Exercises explore binary dimension reduction, matrix calculus, additional matrix concentration results, and matrix sketching.

This chapter examines multipartite classical correlations through fragmentation, scale, and strength metrics.

This chapter explores multipartite structures using tensor networks and canonical decompositions for analyzing joint distributions.

This chapter analyzes quantum dynamics in closed systems, focusing on unitary transformations and quantum interference effects.

This chapter parallels classical relative entropy with quantum analogs, exploring fidelity and state purification concepts.