Uncertainties
in Quantum Theory| Uncertainties
in Quantum Theory |
In General > s.a. equivalence
principle; fluctuations;
Reference Frame.
* Idea: Some observables can have no uncertainty, but not all of them,
and any change in the expectation value of an observable quantity must be associated
with some degree of uncertainty.
@ General references: DeWitt JMP(62)
[and commutation relations]; Hilgevoord & Uffink FP(91)
[in prediction and inference]; Franson PRA(96)
[and changes of expectation values]; Hilgevoord AJP(02)oct
[and standard deviation]; Trifonov JPA(03)
[position on the circle]; Busch et al PRP(07)qp/06 [rev,
conceptual]; Marburger AJP(08)jun
[re early derivation].
@ Quantum vs classical: Cini & Serva PLA(92); Luo TMP(05);
Beretta PhD(81)qp/05 [quantum
thermodynamics]; > s.a. diffusion, fluctuations.
@ Origin: Anderson & Halliwell PRD(93)gq [quantum
+ thermal fluctuations]; Wesson GRG(04)gq/03 [from
higher dimensions]; Arbatsky qp/06 [derivation
from "certainty principle"].
@ Systems: Barros e Sá JMP(01)
[SU(2)-invariant, 
2j 
2j(j+1)];
> s.a. spinors in field theory.
@ Tests: Soucek qp/04 [with t 
10–12 s].
Configuration-Momentum Uncertainty Relations > s.a. histories; optics; time.
* Idea: In any preparation
of a system, uncertainties are constrained to satisfy q p 
/2;
In the usual approach to quantum theory, the bound can be traced back to the
[q, p] commutation
relations.
* In terms of creation-annihilation: Expressed by 
a
a
a a.
@ General references: Heisenberg ZP(27);
Schrödinger SPAW(30),
translation qp/99;
Gamow SA(58)jan; Lévy-Leblond AJP(72)jun
[non-simultaneous measurements]; Fefferman
BAMS(83);
Landsberg FP(88)
[and classical and quantum mechanics]; Chisolm AJP(01)mar-qp/00 [restrictions];
D'Ariano FdP(03)qp/02;
Rigolin a0709 [derivation];
Schürmann & Hoffmann FP(09)-a0811.
@ And phase space geometry: Curtright & Zachos MPLA(01)ht;
Anastopoulos & Savvidou AP(03)qp;
de Gosson PLA(03)
[phase space quantization], qp/04 ["quantum
blobs"], mp/06 [and
symplectic non-squeezing]; de Gosson & Luef PRP(09) [symplectic capacity].
@ And complementarity: Uffink & Hilgevoord PhyB(88)qp/99;
Björk et al PRA(99)qp.
@ Special types of states: Trifonov PW(01)phy [minimizing];
Park JMP(05)mp/04,
Luo PRA(05)
[mixed];
> s.a. coherent states.
@ System preparation / measurement: Park & Band FP(92);
Krenn et al PRA(00)qp/95;
de Muynck FP(00)qp/99;
Werner qp/04-in
[joint measurement].
@ Violations, ways to beat it: Ozawa PLA(03)qp/02;
Hall PRA(04)qp/03 [with
prior information]; Kitano a0803 [not really violated].
@ In terms of information: Gibilisco & Isola mp/05,
JMP(07)mp,
Chakrabarty APS(04)qp/05 [and
Fischer information].
@ And disturbance: Brown & Redhead FP(81); Hofmann PRA(03)qp/02;
Wulleman PE(03)qp/06, CoP(06).
@ Related topics: Landsberg Mind(47)
[philosophy]; Yu PLA(96);
Rigolin FPL(02)qp/00,
qp/01 [entanglement];
Kowalski & Rembielinski
JPA(03)qp [on
S1]; Hewitt-Horsman qp/03 [and
many worlds]; Sarris et al PLA(04)
[as
invariant of motion]; Franke-Arnold et al NJP(04),
Dürr pw(04)oct
[for
angular momentum]; de Gosson & Luef PLA(07)-qp/07
[and density matrices]; Busch & Pearson JMP(07)
[for error bar widths]; Kryukov PLA(07)-a0710 [geometric]; Görlich
et al a0812 [experimental test]; > s.a. Gerbe [reformulation].
Time-Energy Uncertainty Relation > s.a. time
in quantum mechanics.
@ General references: Aharonov & Bohm PR(61);
Kijowski RPMP(76);
Sorkin FP(79);
Busch FP(90), FP(90);
Kobe & Aguilera-Navarro
PRA(94);
Pfeifer & Fröhlich RMP(95);
Hilgevoord AJP(96)dec, AJP(98)may;
Busch in(02)qp/01 [types,
history]; Brunetti & Fredenhagen RVMP(02)qp [rigorous
derivation].
@ Time-mass: Kudaka & Matsumoto JMP(99)qp [
and m as
operators];
Ram mp/02.
@ Related topics: Pegg PRA(98)
[operator conjugate to H]; Aharonov et al PRA(02)qp/01 [and
estimating the Hamiltonian]; Gillies & Allison FPL(05)
[time-temperature]; Karkuszewski qp/05 [upper
bound on uncertainties].
Generalized Versions > s.a. correlations; modified
uncertainty relations; QED [fluxes]; modified
versions of QED.
* Arbitrary self-adjoint
operators: Applying the Schwarz inequality to f:= A – A and g:= B – B,
one can derive that
A B | [A,B] /
2i | .
* Entropic or information
uncertainty principle:
A reformulation from the information-theoretic point of view; A lower bound
on the sum of the Shannon information entropies of two operators over all
wave functions.
* Robertson uncertainty
principle:
If A1, ..., AN are
complex self-adjoint matrices and 
a
density matrix, then the quantum generalized covariance is bounded in terms
of the commutators [Ah,
Aj] by
det (covrho(Ah,
Aj)) det
(– i/2 tr ( [Ah, Aj]))
.
@ General references: Wolsky AJP(74)sep
[kinetic energy–size]; Belavkin TMP(76)qp/04 [and
efficient measurements]; Braunstein et al AP(96)qp/95 [generalized
measurement]; Mensky PLA(96)
[continuous measurements]; Brody & Meister JPA(99)
[discrete]; Cirelli et al JGP(99);
Santhanam JPA(00)
[higher-order moments]; Chisolm AJP(01)mar-qp/00 [covariance];
Trifonov JOSA(00)qp [coherent-squeezed
states], EPJB(02)qp/01 [rev];
Golovnev & Prokhorov JPA(04)qp/03 [in
curved spacetime]; García Díaz et al NJP(05)
[local and non-local observables]; Serafini PRL(06)qp [multimode];
Pati & Sahu PLA(07)
[uncertainties for sums of observables]; Machluf a0807 ["Landau-Pollak-Slepian" principle].
@ Exact uncertainty relations: Hall & Reginatto JPA(02)qp/01;
Hall qp/01, PRA(01)qp; > s.a. foundations
of quantum mechanics.
@ Relations involving the number operator: Lahti & Maczynski IJTP(98)
[number-phase]; Urizar & Toth a0907 [number-annihilation].
@ Entropic / information version: Bialynicki-Birula & Mycielski CMP(75);
Deutsch PRL(83);
Garrett & Gull PLA(90)
[numerical]; Rojas et al PLA(95);
Sánchez-Ruíz PRA(98)
[single- and double-slit];
Santhanam PRA(04)qp/03;
Bialynicki-Birula PRA(06)qp [in
terms of
Renyi entropies]; Gibilisco et al JSP(07)-a0707 [Robertson-type];
de Vicente & Sánchez-Ruiz PRA(08)-a0709 [improved
bounds]; Zozor et al PhyA(08);
Wilk & Wlodarczyk PRA(09)-a0806 [in
terms of Tsallis non-extensive entropy]; Renes & Boileau PRL(09)-a0806 [generalization];
Wehner & Winter JMP(08)
[higher number of measurements]; Rastegin a0810 [re
paper by Massar]; Berta et al a0909 [with quantum side information].
Thermodynamic Uncertainty Relation
* Idea:
A definite temperature can be attributed only to a system submerged in a heat
bath, in which case energy fluctuations are unavoidable,
while a
definite energy can be assigned only to systems in thermal isolation,
thus
excluding the simultaneous determination of its temperature; In general,
the
situation is intermediate.
* History: Bohr and Heisenberg
suggested that T and U are
complementary in the same way as position and momentum in quantum mechanics;
Rosenfeld extended this analogy
and obtained a quantitative uncertainty relation in the form U (1/T)
kB;
The two extreme cases of this relation would then characterize the complementarity
between
isolation (U definite)
and contact with a heat bath (T definite); Other formulations of
the thermodynamical
uncertainty relations were proposed by Mandelbrot (1956,
1989),
Lindhard (1986), and Lavenda (1987, 1991).
@ References: Uffink & van Lith FP(99);
Pennini et al PLA(02)
[non-fundamental].
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