States of Quantum Systems

In General > s.a. [quantum theory]; fractals; schrödinger equation [solutions]; state of a system.
* Idea: A state can be defined either as a square-integrable, complex-valued function on a suitably defined quantum configuration space , or as a linear functional on an algebra of observables.
@ General references: Park AJP(68)mar [classical vs quantum states]; Mohrhoff qp/01; Marchildon CJP(06)qp/05-conf, Spekkens PRA(07) [epistemic view]; Brunet IJTP(09)-a0804 [partial description]; Strauss et al IJTP-a1101 [transition decomposition]; Paris a1110/EPJST [tutorial].
@ State recovery / estimation: Moroz & Perelomov TMP(94)mp/01 [from distribution functions]; Leonhardt et al PRA(95) [tomography]; Gill in(01)m.ST/04 [spin-1/2, asymptotics]; Rehácek & Hradil PRL(02) [invariant information]; D'Alessandro JPA(03)qp; Jones & Linden PRA(05)qp/04 [from parts of states]; Adamson et al PRL(07)qp/06 [tomography]; Liu & Sun LMP(06) [from complete set of quantum correlations]; Leifer & Spekkens a1110 [compatible state assignments]; Chen et al a1212 [on uniqueness given a set of measurement results].
@ Construction: Isidro MPLA(04)qp [from tangent vectors to phase space].
@ And probabilities: Page a0808 [insufficiency for deducing observational probabilities]; Page a1203 [observational probabilities from non-normalizable states].
@ Alternative representations: Palmer qp/01 [R, and computability]; Dias & Prata AP(04)ht [phase space, admissible states]; Battilotti a1102-in [characterization in predicative logic]; Fedorova & Zeitlin a1109-proc, a1109-proc [quantum states as sheaves]; Adhikari et al a1205 [graph representation]; de Gosson a1208 [as surfaces in the position-momentum phase plane]; Tosiek & Brzykcy AP(13)-a1210 [Hilbert-space and phase-space formulations]; > s.a. wigner functions.
> Related topics: see formalism of quantum theory; interpretations of quantum theory; Quantum Carpet; quantum foundations.

Evolution and Transformations > s.a. collapse; geometric phase; quantum effects [including speed of evolution, Margolus-Levitin theorem].
@ Canonical transformations: Anderson PLB(93), PLB(93)ht, AP(94); Schuch & Moshinsky Sigma(08)-a0807 [and Wigner functions]; Dereli et al IJMPA-a0904 [in the Weyl-Wigner-Groenewold-Moyal formalism]; Błaszak & Domański AP(13)-a1208 [general theory].
@ Other transformations: Doebner & Goldin PRA(96), Czachor PRA(98) [gauge, non-linear]; Luís PRA(04) [in phase space and Hilbert space]; García-Mata et al PRA(05) [and phase space contraction]; Brody & Hook JPA(06) [adapted Hamiltonian]; Weimer et al EPL(08)-a0708 [local effective dynamics, work and heat]; Huang et al JMP(12) [physical]; > s.a. fock space.
@ Measurement: Embacher & Narnhofer AP(04) [strategies]; > s.a. quantum measurement.
@ Manipulation: Caban et al JPA(02) [destruction]; Lloyd & Viola PRA(02) [engineering]; Herbert a0802 [states cannot be merged]; > s.a. quantum technology [teleportation, no-cloning, etc].
@ Jumps, transitions: Macomber AJP(77)jun; Wiseman & Gambetta PRL-a1110 + news physorg(12)jun [objective or observer-dependent?].
@ Evolution: Aharonov & Albert PRD(84) [relativistic]; Styer AJP(90)aug, Weigert PRL(00)qp/99 [in terms of expectation values and uncertainties]; Mohrhoff FP(04)qp/03 [and Pondicherry interpretation]; D'Alessandro & Romano JMP(06)qp [and entanglement]; García Quijas & Arévalo Aguilar PS(07)qp/06 [factorization]; Vaidman qp/06/JPA [backward]; Schuch & Moshinsky PRA(06) [Ermakov invariant]; Mohseni & Lidar PRL(06)qp [direct characterization of quantum dynamics]; Kryukov FP(07)-a0704 [as geodesic motion on space of states]; Andrews AJP(08)dec-a0801 [free wave packets]; Tkachuk a1006 [curvature and torsion of evolution]; Bernatska & Messina a1006 [reconstruction of Hamiltonian]; McClean et al a1301 [discrete, and Feynman's clock]; > s.a. mixed states [including thermalization].
@ From pure to mixed states: Horwitz et al qp/96-proc [and Lax-Phillips theory]; Svec a0708 [in pion creation]; > s.a. arrow of time; decoherence; Semigroup.

Other Properties > s.a. classical-quantum relationship; fluctuations; fock space [number operator]; Phase.
* Inequalities: For any two states ψ and ψ', and any self-adjoint A and B, if cos θ = |ψ|ψ'⟩|,

(ΔA)_ψ (ΔB)_ψ ≤  | [A,B]_ψ | ,      (⟨A⟩_ψ' – ⟨A⟩_ψ) / (ΔA)_ψ' – (ΔA)_ψ) ≤ tan θ .

@ Invariants: Albeverio et al PLA(05)qp [mixed states, under unitary transformations].
@ Inequalities: Fleming phy/01; Cabello a1210 [maximum quantum violation of the simplest non-contextuality inequality]; > s.a. bell inequalities; uncertainty relations.
@ Related topics: Vigoureux PRA(94) [composition of amplitudes]; Zurek Nat(01)qp/02 [sub-Planck structure]; Schirmer et al JPA(02), JPA(02) [controllability-reachability]; Poulin & Blume-Kohout PRA(03) [compatibility]; Arrighi & Patricot AP(04)qp/03 [and quantum operations]; Bartlett et al IJQI(06)qp/05 [coherence, dialogue]; > s.a. quantum effects [state diffusion].

Space of Quantum States > s.a. distance types; mixed states; types of states.
* Metric: Many metrics have been proposed; As a criterion for the choice of an appropriate one, Chentsov has proposed the concept of statistical monotonicity, or contraction under coarse graining, and Petz has classified the metrics with this property.
@ General references: Brody & Hughston JMP(00)qp/99 [density function]; Grasselli AIP(01)mp/00 [norm, connection]; Walck qp/02-conf [decomposition]; Barnett et al PLA(03)qp/02 [comparisons]; Gibilisco & Isola JMP(03) [Wigner-Yanase information]; Brody JPA(04)qp/03 [and sets of points]; Marmo et al IJGMP(05)qp [structure, bi-Hamiltonian systems]; Vicary IJTP(08)-a0706 [categorical framework, applied to the simple harmonic oscillator]; Shirokov Izv-a0804 [and entanglement monotones]; Biamonte et al AIP(11)-a1012 [tensor network factorization]; Ma & Zhu JMP(11).
@ Geometry: Życzkowski & Słomczyński JPA(01)qp/00; Gibilisco & Isola JMP(05) [information-geometry curvature]; Andai JPA(06) [volume]; Aalok IJTP(07)qp [metric]; Isidro et al AIP(10)-a0912 [positive Ricci curvature and emergent quantum theory]; Mielnik a1202 [the limits of our knowledge]; > s.a. coherent states; metric types; poisson structure; topology and physics.
@ Subsets: Narnhofer mp/02-conf, IJTP(03) [leaves, from entanglement or conditional entropy]; Shirokov qp/05 [Holevo capacity]; Bengtsson AIP(07)-a0707 [maximally entangled].
@ State distinguishability: Barndorff-Nielsen & Gill JPA(00) [and Fisher information]; Jozsa & Schlienz PRA(00) [and von Neumann entropy]; Greentree et al PRL(04) [finite number]; Herzog & Bergou PRA(04)qp, Majtey et al PRA(05)qp [mixed states]; Ericsson JPA(05)qp.
> Related topics: see Born Rule [discrete state space]; entanglement; non-standard analysis.

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