Mixed
States of Quantum Systems| Mixed
States of Quantum Systems |
In General > s.a. phase;
pilot wave.
$ 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 pk 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 e.g. 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 qp/06 [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 a0707-in; > s.a. distances, riemannian
geometry; states.
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].
@ 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).
@ Alternative representations: Kryszewski & Zachcial qp/06; Tucci qp/07 [as
Bayesian or Markov networks].
@ Related topics: Ajanapon AJP(87)
[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 cm/06 [density
matrix renormalization group].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008