Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems

TL;DR

The book bridges quantum information science and quantum many-body physics by recasting correlations and entanglement as central tools for understanding condensed-matter phases. It develops a rigorous framework: classical information notions (entropy, mutual information) extend to quantum systems via density matrices, Schmidt decomposition, and the stabilizer formalism, while locality and ground-state properties are analyzed through local Hamiltonians and complexity theory. Key contributions include a detailed treatment of entanglement area laws, the role of topological order, and tensor-network approaches, as well as connections between quantum error correction and topological phases (e.g., toric code) that illuminate robust ground-state degeneracies. The work argues that many-body entanglement patterns—and their area-law scaling and irreducible multipartite components—provide universal diagnostics for gapped and topologically ordered phases, offering a unified information-theoretic lens on condensed-matter phenomena with practical implications for quantum simulation and materials science. The text also highlights fundamental limits (local Hamiltonian problem, N-representability) that shape what is computable in practice, guiding future directions in combining quantum information with many-body physics.

Abstract

This is the draft version of a textbook, which aims to introduce the quantum information science viewpoints on condensed matter physics to graduate students in physics (or interested researchers). We keep the writing in a self-consistent way, requiring minimum background in quantum information science. Basic knowledge in undergraduate quantum physics and condensed matter physics is assumed. We start slowly from the basic ideas in quantum information theory, but wish to eventually bring the readers to the frontiers of research in condensed matter physics, including topological phases of matter, tensor networks, and symmetry-protected topological phases.

Figures (143)

• Figure 1: Binary entropy function $H(p)$
• Figure 2: Mutual information: $H(X)$ and $H(Y)$ are plotted as the regions inside two circles, and the mutual information $I(X{:}Y)$ is just their overlap. The quantities $H(X,Y)$, $H(X|Y)$ and $H(Y|X)$ are also illustrated.
• Figure 3: Bloch sphere.
• Figure 4: An inituitive distribution of correlations in a tripartite quantum state $\rho_{ABC}$. The true three-body correlation is the overlap of $A,B,C$, and the two-body correlations between $A$ and $B$, between $A$ and $C$, between $B$ and $C$ are represented by the mutual information $I(A:B)$, $I(A:C)$, $I(B:C)$ respectively.
• Figure 5: Circuit diagram with the number of qubits $N=5$ and the number of layers $M=3$.