Entropy in Quantum Theory  

In General > s.a. entanglement entropy; H Theorem; quantum information; quantum chaos.
* Von Neumann entropy: A quantum version of the Shannon entropy; For a quantum state represented by a density matrix ρ,

H = – k tr (ρ ln ρ) .

* 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.
* Remark: 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; This is related to quantum non-separability and negative (virtual) information of entangled particles [??? see below, and @ in Casini CQG(04)ht/03].
* Wehrl entropy: It gives a basis-independent measure of the localization of quantum states in phase space; It 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.
> Online resources: see Wikipedia page.

For Specific Types of Systems > s.a. cmb; coherent states; relativistic cosmology; thermodynamics.
$ Relativistic entropy: A positive function on causally closed sets in Minkowski space, invariant under Poincaré transformations, and satisfying for commuting pairs of subsets A, BM

S(A) + S(B) ≥ S(A \(\lor\) B) + S(AB) ,   S(A) + S(B) ≥ S(A \(\lor\) B) + S(BA) .

@ 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].
@ Other systems: Sorkin et al GRG(81) [radiation]; Kandrup IJTP(88), IJTP(89) [N interacting particles]; Page PRL(93)gq, Sen PRL(96)ht [subsystem]; Elze qp/97-proc [open systems]; Wu & Cai gq/99/PRD [gas in curved spacetime]; Caticha FP(00)qp/98 [array entropy]; Ruelle CMP(01)mp [non-equilibrium spin system]; Peres et al PRL(02) [spin-1/2 particle]; Narnhofer CQG(11) [quantum fields in curved spacetime].
> Gravity-related: see gravitational thermodynamics; particle effects [particle creation]; quantum black holes; regge calculus.

Properties and Related Topics > s.a. causality [information causality]; Coarse-Graining; Gibbs Paradox; Subentropy.
* 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 λi such that λ1 + λ2 = 1 it satisfies S(λ1 ρ1 + λ2 ρ2) > λ1 S(ρ1) + λ2 S(ρ2).
* Ambiguities: For a given state on an algebra of observables there may be many associated density matrices, with different values of the entropy; This ambiguity can often be traced to a gauge symmetry emergent from the non-trivial topological character of the configuration space of the underlying system, and can also happen in finite-dimensional matrix models.
@ And measurement: Grigolini et al PLA(01) [entropy production]; Alicki & Fannes RPMP(05) [rev]; Shirokov JMP(11)-a1011 [entropy reduction].
@ 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]; Hansen a1511 [elementary proof].
@ Entropy production: Aschbacher & Spohn LMP(06)mp/05 [positivity]; Deffner EPL(13)-a1307 [non-equilibrium]; Kaneko et al a1706 [saturation, in many-body systems].
@ Related topics: Sen PRL(96)ht [subsystems, average entropy]; Zecca IJTP(04) [state superposition and decomposition]; Liao & Fang PhyA(04) [entropy squeezing]; Campisi PRE(08)-a0803, comment Sadri a0803 [and entropy increase]; Casini JSM(10)-a1004 [infinite sequence of inequalities]; Jakšić et al a1106 [entropic fluctuations]; Berta et al JMP(16)-a1107 [smooth entropy formalism]; Balachandran et al a1212, PRD(13)-a1302 [ambiguities]; Hansen JSP(14)-a1305 [convexity of the residual entropy]; Kim & Ruskai JMP(14)-a1404 [upper and lower bounds on the concavity].

References > s.a. types of entropies [including Rényi entropy]; non-extensive statistics; 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]; Caticha qp/98-conf, FP(00)qp/98; Gyftopoulos qp/05; Rastegin JSP(11)-a1012 [general properties of entropies]; Frank & Lieb a1109 [and the uncertainty principle]; Resconi a1110 [geometrical framework, morphogenetic calculus]; Sorkin JPCS(14)-a1205 [in terms of spacetime correlation functions]; Hansen a1604 [from first principles]; Ansari a1605-FdP [entropy flow, and black holes].
@ Von Neumann entropy: in von Neumann; Fujikawa JPSJ(02)cm/00 [vs Shannon]; Petz in(01)mp; Hemmo & Shenker PhSc(06)apr [and thermodyamics]; Farkas & Zimboras JMP(07)-a0706 [scaling, d-dimensional fermionic systems]; Ostapchuk et al a0707 [geometric interpretation]; Hörhammer & Büttner JSP(08)-a0710 [and thermodynamics, quantum Brownian motion]; Shirokov CMP(10)-a0904 [continuity]; del Río et al Nat(11)jun-a1009 [negative entropy, thermodynamic meaning]; Maziero RBEP(15)-a1502 [physical meaning].
@ 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 [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 & Życzkowski JPA(01) [Rényi-Wehrl entropy]; Abdel-Khalek PS(09) [trapped ion interacting with laser field].
@ Wehrl's entropy conjecture: Luo JPA(00) [proof]; Lieb & Solovej a1506 [extension to SU(N) and proof].
@ In the phase-space representation: Manfredi & Feix PRE(00)qp/02 [based on Wigner functions]; Włodarz IJTP(03).
@ Dynamical entropy: Fannes & Haegeman RPMP(03)mp/02 [stochastic systems]; Miyadera & Ohya RPMP(05)qp/03 [spin systems].
@ Relative entropy: Zapatrin qp/04 [a priori/posteriori relative entropy]; Lewin & Sabin LMP(14) [monotonicity]; Berta et al a1512 [variational expression].
@ Other entropies: De Nicola et al EPJB(06)qp [tomographic]; Hansen JSP(07), Seiringer LMP(07)-a0704 [Wigner-Yanase entropy, not subadditive]; Demarie & Terno CQG(13)-a1209 [in polymer quantization]; Dupuis et al in(13)-a1211, Bosyk et al a1506, Bizet & Obregón a1507 [generalized].


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