Modified Types of Quantum Uncertainty Relations| Modified Types of Quantum Uncertainty Relations |
In General
> s.a. correlations; deformed uncertainty
relations; phenomenology of quantum uncertainties [including relativistic particles].
* Arbitrary self-adjoint
operators: Applying the Schwarz inequality to \(f := A - \langle A \rangle\)
and \(g:= B - \langle B \rangle\), one can derive that
\[ \Delta A\, \Delta B \ge {\textstyle{1\over2}}|\langle[A, B]\rangle|\;.\]
* Robertson uncertainty principle: If A1, ..., AN are complex self-adjoint matrices and ρ a density matrix, then the quantum generalized covariance is bounded in terms of the commutators [Ah, Aj] by
det (covρ(Ah, Aj)) ≥ det (− \({\rm i}\over2\)tr (ρ [Ah, Aj])) .
@ General references: Belavkin TMP(76)qp/04 [and efficient measurements];
Braunstein et al AP(96)qp/95 [generalized measurement];
Mensky PLA(96) [continuous measurements];
Cirelli et al JGP(99);
Chisolm AJP(01)mar-qp/00 [covariance];
Trifonov EPJB(02)qp/01 [rev];
García Díaz et al NJP(05) [local and non-local observables];
Serafini PRL(06)qp [multimode];
Machluf a0807 ["Landau-Pollak-Slepian" principle];
Huang PRA(12)-a1012 [variance-based, for arbitrary observables];
Luis PRA(11)-a1104 [contradictory sets of uncertainty relations?];
Zozor et al PRA(11) [based on moments of arbitrary order];
Hasse PRA(12)-a1210 [for multi-particle states of indistinguishable particles];
Campisi NJP(13)-a1306
[without quantum collapse, using work as change in expectation value of the Hamiltonian];
Dias & Prata JPA(19)-a1905 [non-canonical phase-space algebra];
Li & Qiao AdP-a2003 [generalized to non-linear correlations];
Qian et al a2004
[localized quantum fields, accelerated observers];
Barrow & Magueijo a2006 [cosmology and contextual Planck parameter].
@ For higher-order moments:
Santhanam JPA(00);
Brizuela PRD(14)-a1410.
@ Exact uncertainty relations: Hall & Reginatto JPA(02)qp/01;
Hall qp/01,
PRA(01)qp;
> s.a. foundations of quantum mechanics.
@ Number-angle / phase relations:
Susskind & Glogower Physics(64);
Lahti & Maczynski IJTP(98);
Rastegin QIC(12)-a1112 [in terms of generalized entropies];
Busch et al a1604.
@ Number-annihilation operator relations:
Urizar-Lanz & Tóth PRA(10)-a0907;
Rastegin PS(11)-a1007 [entropic approach].
@ Robertson uncertainty principle: Ozawa LNP(01)-a1505;
Lovas & Andai IJQI-a1311;
Ozawa & Yuasa JMAA(17)-a1606 [equality versions].
@ More than two observables: Kechrimparis & Weigert PRA(14)-a1407 [Heisenberg-type uncertainty relation for the product of three standard deviations];
Hou & He a1601,
Song et al sRep(17)-a1701;
Dodonov IJQI-a1711 [two observables entangled with a third one],
PRA(18)-a1711 [3 and 4 observables];
Park & Jung QIP-a1909 [sum rule, N-oscillator systems].
@ Other types: Wolsky AJP(74)sep [kinetic energy–size];
Pati & Sahu PLA(07) [for sums of observables];
Zizzi IJTP(13) [for quantum propositions];
Pezzé PRA(13) [sub-Heisenberg phase uncertainties];
de Gosson a1310
[quantum indeterminacy, polar duality, and symplectic capacities];
Rastegin QIP(15)-a1402 [fine-grained uncertainty relations for several quantum measurements];
Maccone & Pati PRL(14)-a1407,
Bannur a1503 [stronger versions];
Song & Qiao PLA(16)-a1504;
Herdegen & Ziobro LMP(16)-a1511 [using two state vectors];
Bera et al a1706 [based on the median rather than the mean];
Awasthi et al PRA(18)-a1707 [non-ergodicity and loss of information];
Popławski a1712 [for momentum, with torsion];
Lake a2008-in [without modified commutation relations];
> s.a. relativistic quantum theory [in special relativity].
In Terms of Entropy / Information > s.a. phenomenology
of uncertainty relations; tomographic representation.
* 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.
@ Reviews: Białynicki-Birula & Rudnicki in(11)-a1001;
Jizba et al AP(15)-a1406 [and role];
Coles et al RMP(17)-a1511 [and applications].
@ Entropic uncertainty principle:
Białynicki-Birula & Mycielski CMP(75) [information entropy];
Rojas et al PLA(95) [canonically conjugate operators];
Santhanam PRA(04)qp/03 [ground state of a coupled system];
de Vicente & Sánchez-Ruiz PRA(08)-a0709 [improved bounds];
Zozor et al PhyA(08);
Wehner & Winter JMP(08) [higher number of measurements];
Rastegin a0810 [re paper by Massar];
Schürmann JRLR(12)-a1006 [for position and momentum];
Coles & Piani PRA(14);
Korzekwa et al PRA(14)-a1402 [quantum and classical];
Adamczak et al JMP(16)-a1412 [asymptotic];
Abdelkhalek et al a1509 [optimality];
Berta et al NJP(16)-a1511 [and measurement reversibility];
Li & Qiao JPA-a1512 [equivalence between formulations];
Rastegin AdP(16)-a1604 [successive measurements of canonically conjugate observables];
Hsu et al MPLA(17)-a1605 [based on gup];
> s.a. uncertainty principle [time-energy].
@ Information uncertainty principle:
Garrett & Gull PLA(90) [numerical];
Gibilisco et al JSP(07)-a0707 [Robertson-type];
Schneeloch et al PRA(14)-a1404 [for mutual information];
Xiao et al a1908 [complementary information principle].
@ With quantum information / memory: Berta et al nPhys(10)-a0909;
Tomamichel & Renner PRL(11)-a1009 [for smooth entropies];
Prevedel et al nPhys(11)-a1012 [experimental];
Xiao et al a1807
[unified, quasi-fine-grained uncertainty relations].
@ Generalizations: Renes & Boileau PRL(09)-a0806;
Huang PRA(11)-a1101 [in multidimensional position and momentum spaces];
Feng et al PLB(15)-a1501 [in Schwarzschild spacetime];
Jizba et al PRE(16)-a1606
[one-parameter class of uncertainty relations based on entropy power];
Rastegin AP(17)-a1607 [with minimal length];
Hertz & Cerf JPA(19)-a1809 [continuous-variable].
@ And different entropies: Białynicki-Birula PRA(06)qp [in terms of Rényi entropies];
Wilk & Włodarczyk PRA(09)-a0806,
comment Białynicki-Birula & Rudnicki a1001 [in terms of Tsallis non-extensive entropy];
Barchielli et al Ent(17)-a1705 [in terms of relative entropy];
De Palma LMP(18)-a1709 [with quantum memory for the Wehrl entropy];
> s.a. deformed uncertainty relations;
quantum states [state revivals].
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 \(\Delta U\,\Delta(1/T) \ge k_{_{\rm B}}\) ;
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];
Barato et al a1810 [unifying picture];
Hasegawa a2003 [for open quantum systems].
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