States of Quantum Systems  

In General > s.a. quantum theory / fractals; schrödinger equation [solutions]; representations; 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 \(\cal C\), or as a linear functional on an algebra \(\cal A\) of observables.
* State recovery / estimation: When some data are available on a quantum state but not the results of a complete set of observations, the state cannot be fully determined and the best one can do is estimate it by assigning a mixed state; The Maximum Entropy estimation (MaxEnt) and Variational Quantum Tomography (VQT) are methods to choose, among all density matrices compatible with the available data, one which makes few assumptions about the missing information.
@ 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(11)-a1101 [transition decomposition]; Paris EPJST(12)-a1110 [tutorial]; Iwata JPCS(14)-a1407 [coexistence of states]; Gogioso EPTCS(15)-a1506 [and the category fRel of finite sets and relations]; Inamori a1810 [the initial state of the experimenter]; Foreman a1907, a1907 [quantum states as equivalence classes of density operators].
@ 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 JPA(14)-a1110 [compatible state assignments]; Chen et al PRA(13)-a1212 [on uniqueness given a set of measurement results]; Gonçalves et al PRA(13)-a1306 [modified VQT formulation].
@ Construction: Isidro MPLA(04)qp [from tangent vectors to phase space]; Girolami PRL(19)-a1808 [state preparation].
@ And probabilities: Page a0808 [insufficiency for deducing observational probabilities]; Page a1203 [observational probabilities from non-normalizable states].
@ Graph representation: Adhikari et al QIP-a1205; Dutta et al QIP(16)-a1502 [and unitary operations].
@ Alternative representations: Palmer qp/01 [\(\mathbb R\), and computability]; Dias & Prata AP(04)ht [phase space, admissible states]; Battilotti IJTP(11)-a1102-in [characterization in predicative logic]; Fedorova & Zeitlin SPIE(11)-a1109, SPIE(11)-a1109 [quantum states as sheaves]; 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.

State Transformations > s.a. collapse; geometric phase; quantum effects; state evolution [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(09)-a0904 [in the Weyl-Wigner-Groenewold-Moyal formalism]; Błaszak & Domański AP(13)-a1208 [general theory]; > s.a. canonical quantum 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].

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 = |\(\langle\)ψ|ψ'\(\rangle\)|,

A)ψB)ψ ≤ \(1\over2\)| \(\langle\)[A, B]\(\rangle\)ψ | ,      (\(\langle A \rangle\)ψ' − \(\langle A \rangle\)ψ) / (ΔA)ψ' − (ΔA)ψ) ≤ tan θ .

@ Invariants: Albeverio et al PLA(05)qp [mixed states, under unitary transformations].
@ Inequalities: Fleming phy/01; Cabello PRL(13)-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.
* More general distinguishability measures: A measure of distinguishability for quantum states is relative entropy.
@ 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(10)-a0804 [and entanglement monotones]; Ma & Zhu JMP(11); Cariñena et al a1705 [tensorial dynamics]; Okołów a1911 [over the set of all metrics of arbitrary fixed signature on a manifold].
@ 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]; Man'ko et al JPA(17)-a1612 [from relative entropy]; Avron & Kenneth RVMP(20)-a1901 [introduction with picture book]; > 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 IzvM(06)-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; Spedalieri et al JPA(13)-a1204 [limit formula for quantum fidelity]; Spehner JMP(14) [and correlations].
> Related topics: see Born Rule [discrete state space]; entanglement; non-standard analysis.

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