Mixed States of Quantum Systems

In General > s.a. phase; pilot-wave interpretation.
$ Def: One given by a density matrix (an operator on a Hilbert space satisfying = and tr = 1), which cannot be written in factorized form = | |.
* As ensembles: A mixed state can be represented (in infinitely many ways) as an ensemble, = _k |_k p_{k k}|; 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 A = tr(A).

Specific Systems and Theories > s.a. photon; quantum systems; spin models; statistical mechanics.
@ For composite systems: Belokurov et al qp/02 [conditional density matrix]; > s.a. entanglement.
@ For quantum field theory: Brustein & Yarom PRD(04)ht/03 [tracing over volumes]; > s.a. black-hole entropy.

Related Concepts > s.a. Density Matrix.
* Purity: The quantity = tr ², that can be used for example to quantify entropy increase in decoherence.
* Space of mixed states: 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.
@ Separability: Horodecki et al PLA(96) [nsc]; Rudolph JPA(00)qp.
@ Decomposition: Sanpera et al PRA(98); Ellinas & Floratos JPA(99)qp/98; Bengtsson & Ericsson PRA(03)qp/02.
@ Discrimination: Rudolph et al PRA(03)qp; Tolar & Hájícek 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]; > s.a. quantum 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].
@ Bures metric: Dittmann LMP(98); Sommers & Zyczkowski JPA(03)qp [Bures volume]; Bengtsson qp/05-in; > s.a. critical phenomena; riemannian geometry.
@ Space of mixed states: Slater qp/97 [metrics and volume elements]; Zyczkowski & Sommers JPA(01)qp/00 [measures], JPA(03) [Hilbert-Schmidt volume]; Kuah & Sudarshan qp/03 [density matrices]; Man'ko et al RPMP(05)qp, Grabowski et al JPA(05)mp [geometry]; Man'ko et al a0705-in [positive maps and tomograms]; Clemente-Gallardo & Marmo in(07)-a0707; Boya & Dixit PRA(08)-a0808 [geometry]; Oreshkov & Calsamiglia a0812 [distinguishability measures]; > s.a. types of distances; quantum states; riemannian geometry.

References > s.a. bell inequalities; quantum effects [speed of evolution].
@ General: Fano RMP(57); Wichmann JMP(63) [from incomplete measurement]; Grossman PhSc(74) [ignorance interpretation]; Ghirardi et al NCB(75), comment Newton NCB(76); Belinfante pr(79); Gibbons JGP(92); Blum 96; Aharonov & Anandan qp/98-in, comment d'Espagnat qp/98 [including single system]; Anandan & Aharonov FPL(99) [meaning of density matrix]; Mermin qp/01-in [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].
@ Transition pure to mixed: Reznik PRL(96)qp/95 [unitary]; Gerber PRL(98) [search, in K⁰-bar K⁰]; 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.
@ Preferred ensembles: Wiseman & Vaccaro PRL(01) [open systems].
@ Random density matrices: Sommers & Zyczkowski JPA(04)qp [statistical properties].
@ And probability distributions: Nielsen PRA(00)qp/99, PRA(00)qp; Montina PRL(06); Warmuth & Kuzmin a0901.
@ Alternative representations: Kryszewski & Zachcial qp/06; Tucci qp/07 [as Bayesian or Markov networks].
@ Related topics: Ajanapon AJP(87)feb [classical limit]; Bona qp/99-in [non-linear quantum mechanics]; Brun et al PRA(02)qp/01; Ercolessi et al qp/01 [geometric phases]; Sudarshan & Shaji JPA(03)qp/02 [stochastic maps of 's]; 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]; in Haigh et al a0907 [coherent superposition vs statistical mixture].

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