| entanglement entropy | |
|---|---|
| Name | Entanglement entropy |
| Dimension | Information (bits or nats) |
| Used in | Quantum mechanics, Quantum information theory |
entanglement entropy
Entanglement entropy is a measure of quantum correlations between subsystems of a composite quantum system. It quantifies the degree to which the state of one subsystem is non‑separable from the rest, and is central to studies of quantum entanglement, quantum information theory, and many‑body physics. Entanglement entropy plays a key role in characterizing phases of matter, in understanding the black hole information problem, and in algorithms for simulating quantum systems.
Entanglement entropy is defined for a bipartition of a pure state of a composite system into parts A and B. For a pure global state |ψ⟩ in the Hilbert space H_A ⊗ H_B, the reduced density matrix ρ_A = Tr_B(|ψ⟩⟨ψ|) yields the von Neumann entropy S(ρ_A) = −Tr(ρ_A log ρ_A), commonly called the entanglement entropy of subsystem A. This quantity vanishes for separable (product) states and is maximal for maximally entangled states such as the Bell state or GHZ state when subsystem dimensions match. Physically, entanglement entropy measures resources for protocols in quantum teleportation and entanglement distillation, and serves as an order parameter for non‑local correlations across quantum phase transitions studied in models like the Ising model and the Heisenberg model.
Standard formulations use the von Neumann entropy S(ρ) or the family of Rényi entropies S_α(ρ) = (1−α)^{-1} log Tr(ρ^α) for α>0. For mixed global states one often considers the entanglement of formation, entanglement negativity, or mutual information I(A:B)=S(ρ_A)+S(ρ_B)−S(ρ_{AB}). In lattice systems (e.g., spin chains), continuum limits connect entanglement entropy to conformal data via the replica trick and results from conformal field theory (CFT). In finite dimensions the Schmidt decomposition provides a direct link between the Schmidt coefficients {λ_i} of |ψ⟩ and S = −∑_i λ_i^2 log λ_i^2. For fermionic systems one constructs correlation matrices and uses their eigenvalues to compute entropy, as in studies of the Kitaev chain and Hubbard model.
Analytic results arise for solvable models: 1+1D critical systems described by Virasoro algebra CFTs produce S_A = (c/3) log(ℓ/ε)+const for an interval of length ℓ, where c is the central charge and ε is a short‑distance cutoff. For free bosons and fermions, correlation function methods and mode decompositions yield entropies; notable examples include the harmonic chain and free Dirac fields. Numerical techniques include Density Matrix Renormalization Group (DMRG), Matrix Product State (MPS) ansatzes, and Quantum Monte Carlo with replica schemes. Tensor network methods such as Projected Entangled Pair States (PEPS) and the Multi-scale Entanglement Renormalization Ansatz (MERA) exploit entanglement scaling—area versus volume laws—to efficiently represent ground states of local Hamiltonians like those of the AKLT model or 2D Heisenberg systems.
In quantum information theory, entanglement entropy quantifies entanglement as a resource for tasks like quantum communication and error correction; it underlies capacities of quantum channels and bounds on distillable entanglement. In many‑body physics, scaling laws—area law versus volume law—distinguish ground states of local gapped Hamiltonians from highly excited or thermal states (e.g., eigenstate thermalization hypothesis). Entanglement spectra (eigenvalues of ρ_A) reveal topological order and were used to characterize fractional quantum Hall states and symmetry‑protected topological phases studied by groups at Princeton University, Harvard University, and Max Planck Institute for the Physics of Complex Systems.
Entanglement entropy in quantum field theory (QFT) links ultraviolet divergences to geometric cutoffs and plays a central role in understanding the thermodynamic nature of horizons. The Ryu–Takayanagi formula in the AdS/CFT correspondence relates entanglement entropy of a boundary region to the area of a minimal surface in an anti-de Sitter bulk, connecting to the Bekenstein–Hawking entropy and the black hole information paradox debated by researchers like Stephen Hawking and Juan Maldacena. Holographic investigations have motivated concepts such as entanglement wedge reconstruction and complexity proposals (e.g., "complexity = volume/action") pursued at institutions including Institute for Advanced Study and Perimeter Institute.
Experimental access to entanglement entropy is challenging but feasible in engineered systems. Cold atom experiments in optical lattices (e.g., at MIT and MPQ) have measured Rényi entropies using many‑body interference and quantum gas microscopy. Superconducting qubit processors (e.g., from IBM and Google), trapped ion platforms (e.g., IonQ, University of Innsbruck), and photonic setups have demonstrated entanglement benchmarking and tomography for small subsystems. Practical applications include quantum error correcting codes such as the surface code where entanglement structure impacts robustness, and entanglement exploitation in quantum metrology and sensing protocols developed at national labs like NIST.
Category:Quantum information theory Category:Quantum mechanics