Entropy in Quantum Theory  

In General > s.a. H Theorem; quantum information; quantum chaos.
* Von Neumann entropy: For a quantum state represented by a density matrix ,

H = – k tr ( ln ) .

* Interpretation: The von Neumann entropy and the subentropy of a mixed quantum state are upper and lower bounds, respectively, on the accessible information of any ensemble consistent with the given mixed state.
* Properties: Strong subadditivity (proved in 1973 by Lieb & Ruskai); Quantum entropy is not increasing with the size of the subsystem, but it is concave, i.e., for all i such that 1 + 2 = 1 it satisfies S(1 1 + 2 2) > 1 S(1) + 2 S(2).
* Objection: The Von Neumann entropy is a convenient quantification of information, but entropy and information are not synonymous, one can change while the other is conserved [@ Shenker; rebuttal Henderson BJPS(03)].
* Remark: Unlike in classical (Shannon) information theory, quantum (von Neumann) conditional entropies can be negative when considering quantum entangled systems, a fact related to quantum nonseparability – entangled particles can carry negative (virtual) information [??? see below, and @ in Casini CQG(04)ht/03].
* Wehrl entropy: Gives a basis-independent measure of the localization of quantum states in phase space; Can be generalized to Rényi-Wehrl entropies for pure states of spin systems, which according to Lieb's conjecture (unproven) are minimized by the spin coherent states.

For Quantum Fields > s.a. cmb; relativistic cosmology.
$ Relativistic entropy: A positive function on causally closed sets in Minkowski space, invariant under Poincaré transformations, and satisfying for commuting pairs of subsets A, B M

S(A) + S(B) S(A B) + S(A B) ,   S(A) + S(B) S(A B perp) + S(B A perp) .

@ General references: Cacciatori et al PRD(09)-a0803 [with different localization scheme].
@ Cosmology: Castagnino et al GRG(96)gq/00 [particle production]; Brustein PRL(00); Randall et al JHEP(02)ht [and area].
@ Quantum gravity: Major & Setter CQG(01)gq [and area].

Geometric / Entanglement Entropy > s.a. black holes and information; entanglement; gravitational thermodynamics.
* Idea: The von Neumann entropy obtained using the reduced density matrix A for a subsystem A; Corresponds to ignoring the correlations between A and the rest of the system in a (pure) state of a quantum field.
@ Black holes: 't Hooft NPB(85); Bombelli et al PRD(86); Srednicki PRL(93); Holzhey et al NPB(94); Donnelly PRD(08)-a0802 [in lqg].
@ Other systems: Han et al cm/97 [coupled oscillators]; Cramer et al PRA(06)qp/05 [bosonic lattices, scaling]; Korepin a0906 [XY spin chain].
@ More general: 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-in; Brustein & Yarom PRD(04)ht/03 [subsets of Minkowski]; Casini CQG(04)ht/03; Calabrese & Cardy JSM(04)ht [in quantum field theory, see oct(08) revision], JSM(05)cm [1D, evolution], IJQI(06)qp/05-in [intro]; Das & Shankaranarayanan PRD(06)gq/05 [field not in ground state]; Fursaev PRD(06)ht [critical phenomena and analog quantum gravity]; Berger & Buniy JHEP(08)-a0801 [scalar field in arbitrary curved spacetime]; Solodukhin PLB(08)-a0802 [and extrinsic geometry]; Casini & Huerta a0905 [rev].
@ 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-a0808 [rev].
@ And holography: Ryu & Takayanagi PRL(06)ht, Solodukhin PRL(06) [and AdS-cft]; Hubeny et al JHEP(07) [covariant]; Nishioka et al a0905-JPA.
@ Measures of entanglement: Vedral et al PRL(97)qp; Henderson & Vedral PRL(00)qp/99 [entropy]; Wu & Zhang PRA(01)qp/00; Brandao & Vianna PRL(04)qp; Miranowicz & Ishizaka PRA(08)-a0805.
@ Matter-gravity entanglement: Kay ht/98, CQG(98)ht [and decoherence].
@ For fermions: Larsen & Wilczek AP(95); Lévay et al PRA(05)qp; Gioev & Klich PRL(06)qp/05 [various dimensions, scaling and Widom conjecture]; Wolf PRL(06)qp/05, Barthel et al PRL(06) [scaling]; Cramer et al PRL(07)qp/06; Swingle a0908 [and the Fermi surface].
@ Related topics: Wodkiewicz PRA(95)qp [EPR]; Bal et al IJMPA(97)ht/95 [and edge states]; Kitaev & Preskill PRL(06) [topological]; Casini & Huerta NPB(07) [2+1]; Hubeny & Rangamani JHEP(08)-a0711 [1+1, disjoint intervals]; Casini & Huerta JHEP(09)-a0812 [for disconnected regions]; Piani PRL(09) [relative entropy of entanglement].

Other Systems > s.a. coherent states; thermodynamics.
@ References: Sorkin et al GRG(81) [radiation]; Kandrup IJTP(88), IJTP(89) [N interacting particles]; Page PRL(93)gq, Sen PRL(96)ht [subsystem]; Wu & Cai gq/99/PRD [gas in curved spacetime]; Caticha FP(00)qp/98 [array entropy]; Peres et al PRL(02) [spin-1/2 particle].
> Gravity-related: see gravitational thermodynamics; quantum black holes; regge calculus.

References > s.a. causality [information causality]; Coarse-Graining; Gibbs Paradox; particle statistics [identical particles].
@ General: Lieb BAMS(75); Wehrl RMP(78); Schiffer GRG(93) [and quantum gravity]; Mirback & Korsch PRL(95) [phase space entropy and chaotic systems]; Elze qp/97-in [open systems]; Caticha qp/98-in, FP(00)qp/98; Ruelle CMP(01)mp [non-equilibrium spin system]; Gyftopoulos qp/05.
@ von Neumann entropy: in Von Neumann; Fujikawa JPSJ(02)cm/00 [vs Shannon]; Petz mp/01-in; Hemmo & Shenker PhSc(06) [and thermodyamics]; Farkas & Zimboras a0706 [scaling, d-dimensional fermionic systems]; Ostapchuk et al a0707 [geometric interpretation]; Hoerhammer & Buettner a0710 [and thermodynamics, quantum Brownian motion]; Shirokov a0904 [continuity].
@ Information entropy: Isham & Linden PRA(97)qp/96 [and consistent histories]; Orlowski PRA(97) [and squeezing of fluctuations]; Brody & Hughston JMP(00); Stotland et al EPL(04)qp; Kak IJTP(07)qp/06; Hwang a0806-in [vs physical, objective entropy].
@ Entropy vs information: Shenker BJPS(99); Shafiee & Karimi qp/06; Hörhammer & Büttner JSP(08) [for brownian motion].
@ Shannon & von Neumann: Brukner & Zeilinger PRA(01)qp/00; Hall qp/00; Linden & Winter CMP(05)qp/04 [new inequality].
@ Wehrl entropy: Gnutzmann & Zyczkowski JPA(01) [Rényi-Wehrl entropy]; Abdel-Khalek PS(09) [trapped ion interacting with laser field].
@ And entanglement: Hammer ht/98-in; Giraldi & Grigolini PRA(01).
@ And measurement: Grigolini et al PLA(01) [entropy production]; Alicki & Fannes RPMP(05) [rev].
@ Strong subadditivity: Robinson & Ruelle CMP(67); Lanford & Robinson JMP(68); Petz RPMP(86), Nielsen & Petz qp/04/QIC [proof]; Lieb & Seiringer PRA(05)mp/04 [stronger]; Ruskai RPMP(07) [new short proof].
@ Subentropy: Nichols & Wootters qp/02 [intermediate quantities].
@ In phase space representation: Manfredi & Feix PRE(00)qp/02 [based on Wigner functions]; Wlodarz IJTP(03).
@ Dynamical entropy: Fannes & Haegeman RPMP(03)mp/02 [stochastic systems]; Miyadera & Ohya RPMP(05)qp/03 [spin systems].
@ Related topics: Hudetz JMP(94) [topological entropy]; Sen PRL(96)ht [subsystems, average entropy]; Zapatrin qp/04 [a priori/posteriori relative entropy]; Zecca IJTP(04) [state superposition and decomposition]; Liao & Fang PhyA(04) [entropy squeezing]; Aschbacher & Spohn mp/05 [positivity of S production]; De Nicola et al EPJB-qp/06 [tomographic entropy]; Campisi PRE(08)-a0803, comment Sadri a0803 [and entropy increase].

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