Mixed States of Quantum Systems  

In General > s.a. Gleason's Theorem; pilot-wave interpretation; quantum states [Maximum Entropy estimation (MaxEnt)]; schrödinger equation.
$ Def: A quantum state s such that there exist two distinct states s1 and s2 and positive c1 and c2 such that s = c1s1 + c2s2; Alternatively, one given by a density matrix ρ that cannot be written in factorized form ρ = |ψ\(\rangle\)\(\langle\)ψ|.
* As ensembles: A mixed state can be represented (in infinitely many ways) as an ensemble, ρ = k |ψk\(\rangle\) pk \(\langle\)ψk|; The Hughston-Jozsa-Wootters theorem entails that any finite ensemble compatible with a given density operator can be prepared from a fixed initial state by operations on a spacelike separated system.
* Preferred ensemble fallacy: Extrapolating the interpretation of an experiment in terms of one ensemble to other ensembles without justification (some systems do have preferred ensembles, e.g., open ones).
* Use: The expectation value of any observable in such a state is \(\langle A \rangle\) = tr().
@ General references: Fano RMP(57); Wichmann JMP(63) [from incomplete measurement]; Grossman PhSc(74)dec [ignorance interpretation]; Ghirardi et al NCB(75), comment Newton NCB(76); Belinfante pr(79); Gibbons JGP(92); Blum 96; Aharonov & Anandan qp/98-conf, comment d'Espagnat qp/98 [including single system]; Mermin qp/01-ch [meaning]; Poulin & Blume-Kohout PRA(03)qp/02 [compatibility]; Reimann JSP(08) [random sampling by pure states]; Lan IJTP(08) [superposition does not imply mixture]; in Haigh et al OC(10)-a0907 [coherent superposition vs statistical mixture]; Luo et al TMP(11); Hasse PRA(12)-a1206 [limits to the observability of operations]; in Bendersky et al PRL(16)-a1407; de Gosson a1703 [rev, mathematical]; Bardyn et al PRX(18) [topological properties]; Anaya-Contreras a1809 [quantifying mixedness]; Barata et al BJP(21)-a1909 [rev]; de Ronde & Massri FS-a1902 [vs pure states]; Gyamfi EJP(20)-a2003 [Liouville space].
@ Relationship with probability distributions: Nielsen PRA(00)qp/99, PRA(00)qp; Montina PRL(06); Warmuth & Kuzmin a0901; Temporão & von der Weid a1109 [interpretation, and polarized light]; Nenashev a1601.
@ Phase-space representation: in Ozorio de Almeida & Brodier JPA(16)-a1507 [and evolution]; > s.a. wigner function.
@ Other representations: Kryszewski & Zachciał qp/06/JPA; Tucci qp/07, a1204 [as Bayesian or Markov networks]; Fratini & Hayrapetyan PS(11)-a1108, a1204 [limitations of the density-matrix formalism]; Shen et al SRep(16)-a1608 [Bloch representation, separability criteria]; Anza & Crutchfield a2008 [geometric quantum states], a2008 [geometric quantum thermodynamics]; > s.a. Density Matrix [including meaning].

Space of Mixed States > s.a. geometric phase; quantum states; riemannian geometry.
* Geometry: A metric on this space can be used to represent distinguishability of states, and one may want it to have the property that states which are more mixed are less distinguishable than those which are less mixed; Petz argued that the scalar curvature of a statistically relevant (monotone) metric can be interpreted as an average statistical uncertainty.
@ General references: Życzkowski & Sommers JPA(01)qp/00 [measures], JPA(03) [Hilbert-Schmidt volume]; Kuah & Sudarshan qp/03 [density matrices]; Man'ko et al a0705-in [positive maps and tomograms]; Clemente-Gallardo & Marmo in(07)-a0707; Oreshkov & Calsamiglia PRA(09)-a0812 [distinguishability measures]; Puchała & Miszczak JPA(11)-a1107 [probability measure generated by superfidelity].
@ Geometry: Slater qp/97 [metrics and volume elements]; Man'ko et al RPMP(05)qp, Grabowski et al JPA(05)mp; Boya & Dixit PRA(08)-a0808; Brody JPA(11)-a1009 [Riemannian metric, from Fisher-Rao information measure]; Bengtsson et al proc(12)-a1112 [apophatic approach]; Gimeno & Sotoca JMP(13)-a1302 [dynamics, fiber-bundle approach]; Andersson & Heydari a1302 [Hamiltonian dynamics, Riemannian structures on phase space, and time-energy uncertainty relation]; Andersson & Heydari PS(14)-a1312; Heydari JMP-a1409, FP(15)-a1504-conf [geometric framework based on a Kähler structure]; Contreras et al JMP(16)-a1501 [and the Fisher information tensor]; in Lashkari & Van Raamsdonk JHEP(16)-a1508 [distance]; > s.a. types of distances.
@ Bures metric: Dittmann LMP(98); Sommers & Życzkowski JPA(03)qp [Bures volume]; Bengtsson qp/05-conf; > s.a. critical phenomena; riemannian geometry.

Specific Types and Systems > s.a. photon; quantum systems; spin models; statistical mechanics.
@ Semiclassical, Gaussian: Oszmaniec & Kuś JPA(12)-a1111; Link & Strunz a1503 [Hilbert-Schmidt measure on mixed Gaussian states].
@ Types of mixed states: Ruetsche SHPMP(04) [intrinsically mixed]; Garola & Sozzo TMP(07)qp/06, Masillo et al TMP(09)-a0901 [partial traces and improper mixtures]; Brüning et al JMO(12)-a1103 [characterization and different parametrizations]; Mazziotti PRL(12)-a1112 [fermionic]; Singh et al PRA(15)-a1503 [maximally coherent]; Hahn & Fine PRE(16)-a1601 [stability under local measurements].
@ Separability: Horodecki et al PLA(96) [nsc]; Rudolph JPA(00)qp; Gao & Hong EPJD(11)-a1006 [criteria]; Li & Qiao SRep-a1607 [nsc].
@ For composite systems: Belokurov et al qp/02 [conditional density matrix]; Schilling & Schilling JPA(14)-a1408 [duality of reduced density matrices]; Grusdt PRB(17)-a1609 [many-body systems, topological order]; > s.a. entanglement.
@ For quantum field theory: Brustein & Yarom PRD(04)ht/03 [tracing over volumes]; > s.a. black-hole entropy; diffeomorphisms.

Related Concepts > s.a. bell inequalities; entropy in quantum theory; Purity; quantum effects [speed of evolution]; Superposition; uncertainty relations.
* Quantum state tomography: The ability to deduce the density matrix of a quantum system from measured data.
@ Decomposition: Sanpera et al PRA(98); Ellinas & Floratos JPA(99)qp/98; Bengtsson & Ericsson PRA(03)qp/02; Roa et al PRA(11)-a1010; Fan et al a1711 [mixed states and partitions of unity].
@ Discrimination: Rudolph et al PRA(03)qp; Tolar & Hájíček PLA(06)qp/03; Albeverio et al PLA(05) [invariants], JPA(07); Zhang et al PLA(06) [unambiguous], PRA(07)qp/06 [pure vs mixed]; Cramer & Plenio a1002; > s.a. quantum states.
@ Transition from pure to mixed states: Reznik PRL(96)qp/95 [unitary]; Gerber PRL(98) [search, in \(K^0\bar K{}^0\) systems]; Brody & Hughston JMP(99)qp/97 [thermalization]; Horodecki et al PRA(03)qp/02 [reversible]; Hari Dass et al IJMPA(03) [self-thermalization]; > s.a. quantum-gravity phenomenology.
@ And non-linear evolution: Brody et al JPA(10) [from constraints]; Cavalcanti & Menicucci a1006 [and failure of density-matrix formalism].
@ Preferred ensembles: Wiseman & Vaccaro PRL(01) [open systems].
@ Random density matrices: Sommers & Życzkowski JPA(04)qp [statistical properties]; Życzkowski JMP(11) [generating ensembles].
@ Related topics: Ajanapon AJP(87)feb [classical limit]; Brun et al PRA(02)qp/01; Sudarshan & Shaji JPA(03)qp/02 [stochastic maps of ρs]; Halvorson JMP(04)qp/03 [generalized HJW theorem]; Mosonyi & Petz LMP(04)qp/03 [coarse-grainings]; De Chiara et al JCTN(08)cm/06 [density matrix renormalization group]; Gamel & James PRA(10)-a1010 [effective Hamiltonian for averaged density matrix]; Wootton JPA(12) [topological entropy]; Lee a1309 [degenerate density matrices, parametrization]; Schmied JMO(16)-a1407 [tomography, overview of methods, for a single qubit].
@ Generalizations: Bona qp/99-proc [in non-linear quantum mechanics]; Heunen et al EPTCS(14)-a1405 [in higher categories]; Popescu a1811 [states intermediate between pure states and density matrices]; > s.a. category theory in physics; causality in quantum theory [pseudo-density matrix].

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