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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