On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

Rastegin, Alexey E.

doi:10.1002/andp.201800466

Cited by 7 publications

(5 citation statements)

References 85 publications

(167 reference statements)

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“…Finally, let us remark that above, the Rényi and Tsallis functionals were considered in the position and momentum spaces, which are two non-commuting observables. In the last year or so, Rényi [74,75] and Tsallis [75] entropies were proposed in energy and time domains; in particular, corresponding uncertainty relations were derived [74,75]. Application of these measures and associated inequalities to the analysis of the QDs and QRs may present an interesting development of quantum information and quantum cryptography protocols.…”

Section: Discussionmentioning

confidence: 99%

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Olendski

2019

Entropy

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One-parameter functionals of the Rényi R ρ,γ (α) and Tsallis T ρ,γ (α) types are calculated both 1 in the position (subscript ρ) and momentum (γ) spaces for the azimuthally symmetric 2D nanoring that 2 is placed into the combination of the transverse uniform magnetic field B and the Aharonov-Bohm (AB) 3 flux φ AB and whose potential profile is modelled by the superposition of the quadratic and inverse 4 quadratic dependencies on the radius r. Position (momentum) Rényi entropy depends on the field B as 5 a negative (positive) logarithm of ω e f f ≡ ω 2 0 + ω 2 c /4 1/2 , where ω 0 determines the quadratic steepness 6 of the confining potential and ω c is a cyclotron frequency. This makes the sum R ρ nm (α)field-independent quantity that increases with the principal n and azimuthal m quantum numbers and 8 does satisfy corresponding uncertainty relation. In the limit α → 1 both entropies in either space tend to 9 their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary 10 of the semi-infinite range of the dimensionless coefficient α where the momentum entropies exist reveals 11 that it depends on the ring geometry, AB intensity and quantum number m. It is proved that there is the 12 only orbital for which both Rényi and Tsallis uncertainty relations turn into the identity at α = 1/2 and 13 which is not necessarily the lowest-energy level. At any coefficient α, the dependence of the position 14 Rényi entropy on the AB flux mimics the energy variation with φ AB what, under appropriate scaling, can 15 be used for the unique determination of the associated persistent current. Similarities and differences 16 between the two entropies and their uncertainty relations are discussed too. 17 Note that, as it follows from Eqs. (49a) and (49c), the sum of the entropies of the generalized Gaussian 111 approaches its edge values (which are equal to each other due to the fact that each item in it has the 112 same dependence on the Rényi parameter and, as a result, due to the condition from Eq. (18), at the rims 113 α and β simply interchange their places) from above and since the expression in the square brackets in 114 Eq. (49b) is always positive, left-hand side of Eq. (21) reaches its maximum at the Shannon entropy. Also, 115 the leading terms in all three cases are increasing functions of the magnetic index what means that at 116 the greater |m| the corresponding curve lies higher satisfying, of course, the uncertainty relation. As our 117 numerical results show, the same statement holds true at the fixed quantum number m and the increasing 118 principal index n. 119For the QR, a > 0, a comparison of Eqs. (25), (39) and (48) proves that the n = m = 0 orbital does 120 convert at α = 1/2 the Rényi uncertainty into the identity, as it did for the Tsallis inequality too. This is 121 also exemplified in Fig. 2(a), which shows that its sum R ρ (α) + R γ (β) at any parameter α is the smallest 122 one as compared to other levels. Dependence of the left-hand side of Eq. (21) on n and |m| is...

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Section: Discussionmentioning

confidence: 99%

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Olendski

2019

Entropy

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“…Similarly, Corollaries 5 and 6 imply the strong energy-time uncertainty relations

for the energy and canonical time observables of a periodic system with period

, where

denotes the canonical time observable corresponding to

[ 20 , 22 ], and

is the standard deviation about any reference time

. Note that the first of these relations, combined with the asymmetry bound in Theorem 3, implies and hence is stronger than the known Rényi entropic uncertainty relation

for the energy and time observables of periodic systems [ 49 ], analogous to Equation ( 4 ) for number and phase.…”

Section: Applications To Coherence Measures Rotations and Energy-time...mentioning

confidence: 99%

Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information

Hall

2022

Entropy

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An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.

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“…Besides, there are also some works concentrating on the energy-time uncertainty relations [25]. In particular, Rastegin [26] had derived the EUR for energy and time by means of the Pegg's approach [27].…”

Section: Uncertainty Relations Based On Rényi Entropymentioning

confidence: 99%

Quantum‐Memory‐Assisted Entropic Uncertainty Relations

Wang

Ming

et al. 2019

Annalen der Physik

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Uncertainty relations take a crucial and fundamental part in the frame of quantum theory, and are bringing on many marvelous applications in the emerging field of quantum information sciences. Especially, as entropy is imposed into the uncertainty principle, entropy-based uncertainty relations lead to a number of applications including quantum key distribution, entanglement witness, quantum steering, quantum metrology, and quantum teleportation. Herein, the history of the development of the uncertainty relations is discussed, especially focusing on the recent progress with regard to quantum-memory-assisted entropic uncertainty relations and dynamical characteristics of the measured uncertainty in some explicit physical systems. The aims are to help deepen the understanding of entropic uncertainty relations and prompt further explorations for versatile applications of the relations on achieving practical quantum tasks.

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On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

Cited by 7 publications

References 85 publications

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information

Quantum‐Memory‐Assisted Entropic Uncertainty Relations

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