Geometric or Entanglement Entropy |
In General
> s.a. entanglement; quantum entropy
[including spacetime form]; Density Matrix [reduced].
* Idea: A quantity which characterizes the
correlations between subsystems A and B in a larger quantum-mechanical
system; It is calculated as the von Neumann entropy of the reduced density matrix
ρA for subsystem A,
obtained by ignoring the correlations between A and B in a (pure)
state of the whole system,
SA = −k tr ρA ln ρA ;
If the subsystem is defined by a domain A in space, then
the entanglement entropy is called geometric entropy.
* Result: In many cases,
the leading term of the entanglement entropy of A for the vacuum
state is proportional to the area of the boundary ∂A.
@ Intros and reviews: Calabrese & Cardy IJQI(06)qp/05-conf [intro];
Casini & Huerta JPA(09)-a0905;
Solodukhin LRR(11)-a1106;
Nishioka RMP(18)-a1801 [holography and renormalization group];
Headrick a1907-ln [and holography].
@ General references:
Sorkin in(83)-a1402;
Callan & Wilczek PLB(94) [flat space];
Benedict & Pi AP(96)ht/95;
Popescu & Rohrlich PRA(97)qp/96;
Mukohyama PRD(98)gq,
et al gq/98-conf;
Hammer in(04)ht/98;
Giraldi & Grigolini PRA(01);
Brustein & Yarom PRD(04)ht/03 [subsets of Minkowski space];
Casini CQG(04)ht/03;
Calabrese & Cardy JSM(04)ht [in quantum field theory, see oct(08) revision];
Solodukhin PLB(08)-a0802 [and extrinsic geometry];
Calabrese et al JPA(09)#50 [and issue articles];
Koksma et al AP(10)-a1002 [entropy and correlators];
Padmanabhan PRD(10)-a1007 [and quantum of area];
Hertzberg & Wilczek PRL(11)-a1007 [calculable contributions];
Ercolessi et al PRB(12)-a1201 [correlation length and corrections];
Bianchi & Myers a1212 [as a probe of the architecture of spacetime];
Neiman PRL(13)-a1310 [and non-additive imaginary term in the action];
Rosenhaus & Smolkin JHEP(14)-a1403 [perturbative calculation];
Astaneh et al PRD(14)-a1407 [maximizing surface];
Asplund & Berenstein AP(16)-a1503 [and classical entropy];
Lin & Radičević a1808 [precise path integral expressions];
Pontello a2005-PhD [in algebraic quantum field theory].
@ Operator algebra approach: Balachandran et al PRA(13)-a1301;
Bianchi & Satz PRD(19)-a1901 [entropy of a state restricted to a subalgebra];
Longo & Xu a1911 [rigorous].
@ Area law: Buniy & Hsu PLB(07)ht/05 [faster-than-A scaling, correlations, holography];
Riera & Latorre PRA(06)qp;
Pakman & Parnachev JHEP(08) [and holography];
Eisert et al RMP(10)-a0808 [rev];
Brandão & Horodecki CMP(15)-a1206 [and exponential decay of correlations];
Collins et al JPA(13) [for random graph states];
Mariën et al a1411 [stability];
Chandran et al Ent(16)-a1511 [intuitive explanation, and logarithmic prefactor];
Leichenauer et al JHEP(16)-a1604 [evolution of the universal area law];
> s.a. entanglement phenomenology [rate of production];
statistical-mechanical systems [lattice systems].
Related Topics > s.a. Subsystems.
@ Infrared divergences: Mallayya et al PRD(14)-a1404;
Yazdi JHEP(17)-a1608 [and zero modes].
@ Holographic: Ryu & Takayanagi PRL(06)ht,
Solodukhin PRL(06) [and AdS-cft];
Hubeny et al JHEP(07)-a0705 [covariant];
Nishioka et al JPA(09)-a0905;
Casini et al JHEP(11)-a1102 [derivation for spherical entangling surfaces];
Takayanagi CQG(12),
GRG(14);
Headrick et al JHEP(14)-a1408 [and causality];
Bueno & Myers JHEP(15)-a1505 [corner contributions];
Rangamani & Takayanagi a1609
& book 17 [rev];
Sorce JHEP(19)-a1908 [cutoff covariance];
Fareghbal & Shalamzari PRD(19)-a1908 [first law];
Jankowski a2005 [Hamilton-Jacobi formulation];
> s.a. holography.
@ Dynamics: Calabrese & Cardy JSM(05)cm [1D];
Avery & Paulos PRL(14)-a1407 [bounds on time rate of change];
Zhang et al PRE(15)-a1501 [near equilibrium, and relaxation];
Casini et al JHEP(16)-a1509 [time evolution and causality constraints];
> s.a. Kolmogorov-Sinai Entropy.
@ Measures of entanglement: Vedral et al PRL(97)qp;
Henderson & Vedral PRL(00)qp/99 [entropy];
Wu & Zhang PRA(01)qp/00;
Brandão & Vianna PRL(04)qp;
Miranowicz & Ishizaka PRA(08)-a0805.
@ Surfaces, shape dependence: Balachandran et al IJMPA(97)ht/95 [and edge states];
Bhattacharyya & Sinha IJMPD(13)-a1305-GRF [from surface terms in general relativity];
Carmi JHEP(15)-a1506 [shape dependence];
Dorn PLB(16)-a1608 [regions with conical singularities];
Kusuki et al JHEP(17)-a1703 [Lorentz-boosted subsystems];
Witczak-Krempa PRB(19)-a1810
[deformations of the entangling surface, entanglement susceptibilities].
@ Other topics: Wódkiewicz PRA(95)qp [EPR];
Piani PRL(09),
Friedland & Gour JMP(11)-a1007 [relative entropy of entanglement];
Jacobson IJMPD(12)-a1204-GRF [finiteness, and existence of gravity];
Klebanov et al JHEP(12),
JHEP(12)-a1207 [and the renormalization group];
Kalogeropoulos EPJB(14)-a1403 [extensivity];
Nishioka & Yarom JHEP(16)-a1509 [with gauge and gravitational anomalies];
> s.a. temperature.
Variations and Related Topics
> s.a. entanglement entropy for various types of systems;
Matter-Gravity Entanglement Hypothesis.
@ Rényi entanglement entropy:
Drut & Porter PRB(15)-a1506 [interacting fermions];
Barghathi et al PRL(18)-a1804.
@ Topological entanglement entropy:
Kitaev & Preskill PRL(06);
Brown et al PRL(13)-a1303.
@ Other variations and special cases:
Das & Shankaranarayanan PRD(06)gq/05 [field not in the ground state];
Fursaev PRD(06)ht [critical phenomena and analog quantum gravity];
Casini & Huerta JHEP(09)-a0812 [for disconnected regions];
Solodukhin JHEP(10)-a0909 [non-relativistic field theory];
Solodukhin JPA(12)-a1112 [with fluctuating entangling surfaces];
Liu & Mezei JHEP(13)-a1202 [refinement,"renormalized"];
Kumar et al PRD(14)-a1401 [area scaling in genus-1 geometries];
Ibnouhsein et al PRD(14)-a1407 [renormalized];
Maxfield JHEP(15)-a1412 [3D gravity];
Taylor JHEP(16)-a1507 [generalized].
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