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| area law (quantum entanglement) | |
|---|---|
| Name | Area law (quantum entanglement) |
| Field | Quantum information theory, Condensed matter physics, Quantum field theory |
| Introduced | 1990s |
| Notable people | Mark Srednicki, John Preskill, Patrick Hayden, Alexei Kitaev, Matthew Hastings |
area law (quantum entanglement)
An area law describes how the entanglement entropy of a subsystem scales with the boundary area rather than the volume, a principle studied in contexts involving Mark Srednicki, Alexei Kitaev, John Preskill, Matthew Hastings, and Patrick Hayden that has connections to Stephen Hawking, Jacob Bekenstein, Gerard ’t Hooft, Leonard Susskind, and Juan Maldacena through insights from black hole thermodynamics, AdS/CFT correspondence, Bekenstein–Hawking entropy, holographic principle, and semiclassical gravity.
The area law is formalized by bounding the von Neumann entropy S(ρ_A) = −Tr(ρ_A log ρ_A) of a reduced density matrix ρ_A for a region A so that S(ρ_A) ∝ |∂A|, a statement proven or conjectured in proofs and theorems by researchers such as Matthew Hastings and discussed alongside methods from Tomita–Takesaki theory, Alain Connes, Lars Onsager, Elliott Lieb, Eugene Wigner, Paul Dirac, and techniques used in quantum Shannon theory and operator algebras; formulations often involve Rényi entropies, mutual information bounds, and entropy inequalities linked to Araki–Lieb inequality, Strong subadditivity of quantum entropy, Pinsker's inequality, Holevo bound, and Uhlmann's theorem.
Intuitively, ground states of local Hamiltonians like the Ising model, Heisenberg model, Hubbard model, Kitaev honeycomb model, and systems studied by Philip Anderson and Robert Laughlin exhibit short-range correlations so entanglement is localized near boundaries, a picture supported by arguments from renormalization group flows, Kenneth Wilson, Leo Kadanoff, Kenichi Fukui, Michael Fisher, and analyses invoking conformal field theory, Cardy formula, Calabrese–Cardy result, Sine–Gordon model, and examples in one dimension such as gapped chains with area-law scaling proved by arguments related to Matrix Product State approximations and results used by Frank Verstraete and Guifre Vidal.
Exceptions occur in critical systems and highly excited states: conformal critical points in 1+1 dimensions show logarithmic corrections described by Conformal field theory and central charge c as in works by Alexander Zamolodchikov and John Cardy, while thermal states and eigenstates satisfying the Eigenstate Thermalization Hypothesis connected to Mark Srednicki can obey a volume law; examples include chaotic dynamics studied in Sachdev–Ye–Kitaev model, quantum circuits examined by Scott Aaronson, entanglement growth in quenches related to Eugene Demler, and many-body localized phases investigated by David Basko, Igor Aleiner, and Dmitry Abanin which can produce intermediate scaling behavior.
Area-law behavior underpins efficient descriptions of ground states used in condensed matter approaches by Phillip Anderson, Walter Kohn, Steven White, and Subir Sachdev and influences holographic entanglement entropy calculations via the Ryu–Takayanagi formula in AdS/CFT correspondence research by Shinsei Ryu, Tadashi Takayanagi, and Juan Maldacena; implications extend to black hole entropy puzzles addressed by Stephen Hawking, Don Page, Samir Mathur, and Raphael Bousso and to computational complexity considerations linked to Scott Aaronson, Eleanor Rieffel, and Umesh Vazirani.
Area laws justify tensor network ansätze such as Matrix Product States, Projected Entangled Pair States, Multiscale Entanglement Renormalization Ansatz, and algorithms like Density Matrix Renormalization Group developed by Steven White and theoretical developments by Guifre Vidal, Frank Verstraete, Norbert Schuch, and Jutho Haegeman; these frameworks connect to numerical studies employing Quantum Monte Carlo, Exact diagonalization, Variational Monte Carlo, and implementations on platforms associated with IBM Quantum, Google Quantum AI, Rigetti Computing, and research groups at MIT and Caltech.
Experimental probes of entanglement scaling use cold atoms in optical lattices realized by groups at Max Planck Institute of Quantum Optics, Harvard University, MIT, and University of Cambridge employing quantum gas microscopes and protocols inspired by Immanuel Bloch, Wolfgang Ketterle, Martin Zwierlein, and Jakub Zakrzewski; proposals and measurements include swap-operator techniques, randomized measurements advanced by teams linked to Anton Zeilinger, Rainer Blatt, Monika Aidelsburger, and quantum simulators demonstrated by Greiner Lab, while future tests aim to connect laboratory observations to holographic expectations from Juan Maldacena and Tadashi Takayanagi.