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, B ∈ M
S(A) + S(B) ≥ S(A \(\lor\) B) + S(A ∧ B) , S(A) + S(B) ≥ S(A \(\lor\) B ⊥) + S(B ∧ A⊥) .
@ 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 quantum field theories:
Sorkin et al GRG(81) [radiation];
Narnhofer CQG(11) [in curved spacetime];
Yoshida PRA(20)-a1909 [Boltzmann entropy];
Longo & Xu a1911 [von Neumann entropy].
@ Other systems: 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];
Civitarese & Gadella Ent(18)-a1803 [unstable systems].
> 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 MPAG(16)-a1511 [elementary proof].
@ Entropy production: Aschbacher & Spohn LMP(06)mp/05 [positivity];
Deffner EPL(13)-a1307 [non-equilibrium];
Kaneko et al PRE(17)-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 relative and 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 et al a1110
[geometrical framework, morphogenetic calculus];
Hansen a1604 [from first principles];
Ansari a1605-FdP [entropy flow, and black holes];
Majewski & Labuschagne a1804 [general approach];
Facchi et al a2104 [for states of an algebra of observables].
@ 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];
Boes et al a1807 [operational characterization];
Minello et al a1809 [of a graph];
Sheridan a2007 [hist];
Parzygnat a2009 [functorial characterization].
@ 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].
@ Spacetime form: Sorkin JPCS(14)-a1205 [in terms of correlation functions];
Chen et al a2002 [for interacting theories];
Surya et al a2008 [causal set de Sitter horizons].
@ 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 CMP(16)-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 LMP(17)-a1512 [variational expression];
Xu CMP(19)-a1810 [for free fermion quantum field theory].
@ 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 QIP(16)-a1506,
Bizet & Obregón a1507 [generalized].
main page
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