| entanglement entropy | |
|---|---|
| Name | Entanglement entropy |
| Field | Quantum mechanics |
| Introduced | 1970s |
| Notable exponent | von Neumann |
entanglement entropy
Entanglement entropy is a quantitative measure of quantum correlations between subsystems of a composite quantum system. It captures how information is nonlocally shared in states such as Bell states or ground states of interacting many-body systems, and underpins results in quantum information theory, condensed matter physics, and quantum field theory.
Entanglement entropy is defined for a bipartition of a pure quantum state on a Hilbert space H = H_A ⊗ H_B by constructing the reduced density matrix ρ_A = Tr_B ρ and computing an entropy functional. It quantifies nonclassical correlations distinct from classical statistical correlations and is invariant under local unitary operations. Physically, entanglement entropy governs constraints on state preparation, decoherence rates in decoherence processes, and limits on information transfer in protocols such as quantum teleportation and superdense coding. In many contexts it signals phase structure: area-law scaling versus volume-law scaling distinguishes ground states of local Hamiltonians like the Heisenberg model and critical systems described by conformal field theory.
The standard measure is the von Neumann entropy S(ρ_A) = −Tr(ρ_A log ρ_A), named for von Neumann. Related Rényi entropies S_α(ρ_A) = (1−α)^{-1} log Tr ρ_A^α interpolate between min- and max-entropies and are used in both mathematical analysis and experiments. For mixed states, measures such as entanglement of formation, entanglement negativity, and concurrence provide alternate quantifications; these are applied in studies involving Werner states or thermal ensembles. In quantum field theory the entanglement entropy of spatial regions diverges and is regularized by cutoffs, leading to universal terms connected to central charges and anomalies in conformal field theory (e.g., results by Holzhey, Cardy, and Calabrese) and to geometric quantities in semiclassical gravity via the Ryu–Takayanagi formula.
Practical computation employs analytic and numerical techniques. For 1D critical systems, methods from conformal field theory yield logarithmic scaling S ∼ (c/3) log L with central charge c. Exact diagonalization and density matrix renormalization group (DMRG) compute entanglement in lattice models such as the transverse field Ising model and the XXZ model. Tensor network ansätze — including matrix product states and projected entangled pair states — efficiently represent low-entanglement states and are connected to area-law behavior. In higher dimensions, quantum Monte Carlo with replica-trick estimators, and series expansion methods, estimate Rényi entropies. Continuum field-theory computations use heat-kernel and replica manifold techniques; holographic computations use extremal surface prescriptions in AdS/CFT correspondence contexts. Benchmark examples include entanglement in Bell states, bipartite splitting of Bose–Einstein condensates, and topological entanglement entropy in fractional quantum Hall effect states and Kitaev model phases.
Entanglement entropy organizes understanding of ground-state complexity in many-body localization and thermalization, distinguishing area-law entangled ground states from volume-law entangled highly excited states. It provides diagnostics for quantum phase transitions, symmetry-protected topological order, and long-range entangled phases such as in the toric code and topological order examples. In quantum field theory, entanglement across regions yields insight into vacuum structure, renormalization group flow (cf. entropic c-theorems), and universal coefficients associated with anomalies. In gravitational settings, entanglement entropy plays a central role in black hole thermodynamics through the Bekenstein–Hawking entropy correspondence and in developments relating spacetime geometry to entanglement as advocated in holographic duality research led by figures such as Juan Maldacena.
Entanglement entropy is fundamental to quantum information theory tasks: it bounds distillable entanglement, guides error-correcting codes such as stabilizer codes, and underlies resource theories of entanglement. In quantum thermodynamics, it relates to work extraction limits, typicality in statistical mechanics, and the emergence of thermodynamic entropy from entanglement with environments in closed quantum systems. Concepts like entanglement Hamiltonian (modular Hamiltonian) connect modular flow to notions of temperature and the Unruh effect; these ideas link to experimental platforms studied by institutions such as IBM and Google in noisy intermediate-scale quantum devices.
Experimental access uses full state tomography for few-qubit systems (implemented in platforms by groups at MIT and Harvard), randomized measurement protocols for Rényi entropies implemented on trapped-ion and cold-atom platforms (e.g., experiments by the MIT-Harvard Center and groups like those of Immanuel Bloch and Christopher Monroe), and interferometric schemes for twin-copy measurements. Solid-state implementations probe entanglement via shot-noise, quantum point contacts, and entanglement witnesses in superconducting qubit arrays (pursued by teams at Yale University and Caltech). Advances in quantum simulators, optical lattice experiments, and cryogenic architectures continue to improve resolution of entropic measures, enabling comparison with theoretical predictions from DMRG, tensor networks, and holographic models.
Category:Quantum mechanics Category:Quantum information science