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Journal articles on the topic 'Entropic uncertainty'

Author: Grafiati

Published: 1 April 2023

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1

Li, Li-Juan, Fei Ming, Xue-Ke Song, Liu Ye, and Dong Wang. "Review on entropic uncertainty relations." Acta Physica Sinica 71, no. 7 (2022): 070302. http://dx.doi.org/10.7498/aps.71.20212197.

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The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.

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2

Majerník, V., and L. Richterek. "Entropic uncertainty relations." European Journal of Physics 18, no. 2 (March 1, 1997): 79–89. http://dx.doi.org/10.1088/0143-0807/18/2/005.

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3

Costa, Ana, Roope Uola, and Otfried Gühne. "Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems." Entropy 20, no. 10 (October 5, 2018): 763. http://dx.doi.org/10.3390/e20100763.

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The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

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4

Hsu, Li-Yi, Shoichi Kawamoto, and Wen-Yu Wen. "Entropic uncertainty relation based on generalized uncertainty principle." Modern Physics Letters A 32, no. 28 (September 4, 2017): 1750145. http://dx.doi.org/10.1142/s0217732317501450.

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We explore the modification of the entropic formulation of uncertainty principle in quantum mechanics which measures the incompatibility of measurements in terms of Shannon entropy. The deformation in question is the type so-called generalized uncertainty principle that is motivated by thought experiments in quantum gravity and string theory and is characterized by a parameter of Planck scale. The corrections are evaluated for small deformation parameters by use of the Gaussian wave function and numerical calculation. As the generalized uncertainty principle has proven to be useful in the study of the quantum nature of black holes, this study would be a step toward introducing an information theory viewpoint to black hole physics.

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5

SANTOS, M. A., and I. V. VANCEA. "ENTROPIC LAW OF FORCE, EMERGENT GRAVITY AND THE UNCERTAINTY PRINCIPLE." Modern Physics Letters A 27, no. 02 (January 20, 2012): 1250012. http://dx.doi.org/10.1142/s0217732312500125.

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The entropic formulation of the inertia and the gravity relies on quantum, geometrical and informational arguments. The fact that the results are completely classical is misleading. In this paper, we argue that the entropic formulation provides new insights into the quantum nature of the inertia and the gravity. We use the entropic postulate to determine the quantum uncertainty in the law of inertia and in the law of gravity in the Newtonian Mechanics, the Special Relativity and in the General Relativity. These results are obtained by considering the most general quantum property of the matter represented by the Uncertainty Principle and by postulating an expression for the uncertainty of the entropy such that: (i) it is the simplest quantum generalization of the postulate of the variation of the entropy and (ii) it reduces to the variation of the entropy in the absence of the uncertainty.

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6

Puchała, Zbigniew, Łukasz Rudnicki, and Karol Życzkowski. "Majorization entropic uncertainty relations." Journal of Physics A: Mathematical and Theoretical 46, no. 27 (June 21, 2013): 272002. http://dx.doi.org/10.1088/1751-8113/46/27/272002.

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7

Maassen, Hans, and J. B. M. Uffink. "Generalized entropic uncertainty relations." Physical Review Letters 60, no. 12 (March 21, 1988): 1103–6. http://dx.doi.org/10.1103/physrevlett.60.1103.

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8

Adamczak, Radosław, Rafał Latała, Zbigniew Puchała, and Karol Życzkowski. "Asymptotic entropic uncertainty relations." Journal of Mathematical Physics 57, no. 3 (March 2016): 032204. http://dx.doi.org/10.1063/1.4944425.

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9

Khedr, Ahmad N., Abdel-Baset A. Mohamed, Abdel-Haleem Abdel-Aty, Mahmoud Tammam, Mahmoud Abdel-Aty, and Hichem Eleuch. "Entropic Uncertainty for Two Coupled Dipole Spins Using Quantum Memory under the Dzyaloshinskii–Moriya Interaction." Entropy 23, no. 12 (November 28, 2021): 1595. http://dx.doi.org/10.3390/e23121595.

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In the thermodynamic equilibrium of dipolar-coupled spin systems under the influence of a Dzyaloshinskii–Moriya (D–M) interaction along the z-axis, the current study explores the quantum-memory-assisted entropic uncertainty relation (QMA-EUR), entropy mixedness and the concurrence two-spin entanglement. Quantum entanglement is reduced at increased temperature values, but inflation uncertainty and mixedness are enhanced. The considered quantum effects are stabilized to their stationary values at high temperatures. The two-spin entanglement is entirely repressed if the D–M interaction is disregarded, and the entropic uncertainty and entropy mixedness reach their maximum values for equal coupling rates. Rather than the concurrence, the entropy mixedness can be a proper indicator of the nature of the entropic uncertainty. The effect of model parameters (D–M coupling and dipole–dipole spin) on the quantum dynamic effects in thermal environment temperature is explored. The results reveal that the model parameters cause significant variations in the predicted QMA-EUR.

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10

Rudnicki, Łukasz. "Uncertainty-reality complementarity and entropic uncertainty relations." Journal of Physics A: Mathematical and Theoretical 51, no. 50 (November 20, 2018): 504001. http://dx.doi.org/10.1088/1751-8121/aaecf5.

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11

Schwonnek, René. "Additivity of entropic uncertainty relations." Quantum 2 (March 30, 2018): 59. http://dx.doi.org/10.22331/q-2018-03-30-59.

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We consider the uncertainty between two pairs of local projective measurements performed on a multipartite system. We show that the optimal bound in any linear uncertainty relation, formulated in terms of the Shannon entropy, is additive. This directly implies, against naive intuition, that the minimal entropic uncertainty can always be realized by fully separable states. Hence, in contradiction to proposals by other authors, no entanglement witness can be constructed solely by comparing the attainable uncertainties of entangled and separable states. However, our result gives rise to a huge simplification for computing global uncertainty bounds as they now can be deduced from local ones. Furthermore, we provide the natural generalization of the Maassen and Uffink inequality for linear uncertainty relations with arbitrary positive coefficients.

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12

Coffey, M. W. "Semiclassical position and momentum information entropy for sech2 and a family of rational potentials." Canadian Journal of Physics 85, no. 7 (July 1, 2007): 733–43. http://dx.doi.org/10.1139/p07-062.

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The classical and semiclassical position and momentum information entropies for the reflectionless sech2 potential and a family of rational potentials are obtained explicitly. The sum of these entropies is of interest for the entropic uncertainty principle that is stronger than the Heisenberg uncertainty relation. The analytic results relate the classical period of the motion, total energy, position and momentum entropy, and dependence upon the principal quantum number n. The logarithmic energy dependence of the entropies is presented. The potentials considered include as special cases the attractive delta function and square well. PACS Nos.: 03.67–a, 03.65.Sq, 03.65.Ge, 03.65.–w

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13

Rastegin, Alexey E. "Number-phase uncertainty relations in terms of generalized entropies." Quantum Information and Computation 12, no. 9&10 (September 2012): 743–62. http://dx.doi.org/10.26421/qic12.9-10-2.

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Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.

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14

Huang, Zhiming, Dongwu Wu, Tianqing Wang, Yungang Bian, and Wei Zhang. "Steering entropic uncertainty of qutrit system." Modern Physics Letters A 35, no. 16 (April 3, 2020): 2050127. http://dx.doi.org/10.1142/s0217732320501278.

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High-dimensional quantum system plays an important role in quantum information tasks. However, the interaction between quantum system and environment would give rise to decoherence. In this paper, we examine the quantum-memory-assisted entropic uncertainty relation under amplitude damping (AD) decoherence. It is found that entropic uncertainty first inflates and then reduces to a nonzero value with the growing decoherence strength. In addition, it is revealed that the mixedness is not closely associated with entropic uncertainty which is different from the previous result. Furthermore, we construct a remarkably effective filtering operator to steer and reduce the entropic uncertainty. Our exploration might offer fresh insights into the dynamics and manipulation of the entropic uncertainty in high-dimensional quantum system.

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15

Ambainis, Andris. "Limits on entropic uncertainty relations." Quantum Information and Computation 10, no. 9&10 (September 2010): 848–58. http://dx.doi.org/10.26421/qic10.9-10-10.

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We consider entropic uncertainty relations for outcomes of the measurements of a quantum state in 3 or more mutually unbiased bases (MUBs), chosen from the standard construction of MUBs in prime dimension. We show that, for any choice of 3 MUBs and at least one choice of a larger number of MUBs, the best possible entropic uncertainty relation can be only marginally better than the one that trivially follows from the relation by Maassen and Uffink for 2 bases.

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16

Berta, Mario, Stephanie Wehner, and Mark M. Wilde. "Entropic uncertainty and measurement reversibility." New Journal of Physics 18, no. 7 (July 6, 2016): 073004. http://dx.doi.org/10.1088/1367-2630/18/7/073004.

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17

Abdelkhalek, Kais, René Schwonnek, Hans Maassen, Fabian Furrer, Jörg Duhme, Philippe Raynal, Berthold-Georg Englert, and Reinhard F. Werner. "Optimality of entropic uncertainty relations." International Journal of Quantum Information 13, no. 06 (September 2015): 1550045. http://dx.doi.org/10.1142/s0219749915500458.

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The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be nonoptimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work, we establish optimal uncertainty relations by characterizing the optimal lower bound in scenarios similar to the Maassen–Uffink type. We disprove a conjecture by Englert et al. and generalize various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.

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18

Srinivas, M. D. "Entropic formulation of uncertainty relations." Pramana 25, no. 4 (October 1985): 369–75. http://dx.doi.org/10.1007/bf02846763.

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19

Hertz, Anaelle, and Nicolas J. Cerf. "Continuous-variable entropic uncertainty relations." Journal of Physics A: Mathematical and Theoretical 52, no. 17 (April 2, 2019): 173001. http://dx.doi.org/10.1088/1751-8121/ab03f3.

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20

Abe, Sumiyoshi, and Norikazu Suzuki. "Thermal information-entropic uncertainty relation." Physical Review A 41, no. 9 (May 1, 1990): 4608–13. http://dx.doi.org/10.1103/physreva.41.4608.

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21

Wehner, Stephanie, and Andreas Winter. "Entropic uncertainty relations—a survey." New Journal of Physics 12, no. 2 (February 26, 2010): 025009. http://dx.doi.org/10.1088/1367-2630/12/2/025009.

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22

Duan, Kai-Min, and Chuan-Feng Li. "Entanglement-assisted entropic uncertainty principle." Frontiers of Physics 7, no. 3 (January 9, 2012): 259–60. http://dx.doi.org/10.1007/s11467-011-0242-8.

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23

Karpat, Göktuğ. "Entropic uncertainty relation under correlated dephasing channels." Canadian Journal of Physics 96, no. 7 (July 2018): 700–704. http://dx.doi.org/10.1139/cjp-2017-0683.

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Uncertainty relations are a characteristic trait of quantum mechanics. Even though the traditional uncertainty relations are expressed in terms of the standard deviation of two observables, there exists another class of such relations based on entropic measures. Here we investigate the memory-assisted entropic uncertainty relation in an open quantum system scenario. We study the dynamics of the entropic uncertainty and its lower bound, related to two incompatible observables, when the system is affected by noise, which can be described by a correlated Pauli channel. In particular, we demonstrate how the entropic uncertainty for these two incompatible observables can be reduced as the correlations in the quantum channel grow stronger.

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24

Huang, Zhiming, Xiaobin Wang, Yiyong Ye, Xiaokui Sheng, Zhenbang Rong, and Dequan Ling. "Quantum-memory-assisted entropic uncertainty in fluctuating electromagnetic field with a boundary." Modern Physics Letters A 34, no. 17 (June 7, 2019): 1950099. http://dx.doi.org/10.1142/s0217732319500998.

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In this work, we investigate the dynamics of quantum-memory-assisted entropic uncertainty relation for a two-level atom coupled with fluctuating electromagnetic field in the presence of a perfectly reflecting plane boundary. The solution of the master equation that governs the system evolution is derived. We find that entropic uncertainty and mixedness increase to a stable value with evolution time, but quantum correlation reduces to zero with evolution time. That is, the mixedness is positively associated with entropic uncertainty, however, increasing quantum correlation can cause the decrease of the uncertainty. The tightness of entropic uncertainty grows at first and then declines to zero with evolution time. In addition, entropic uncertainty fluctuates to relatively stable values with increasing the atom’s distance from the boundary, especially for short evolution time, which suggests a possible way of testing the vacuum fluctuating and boundary effect. Finally, we propose an effective method to control the uncertainty via quantum weak measurement reversal.

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25

Rastegin, Alexey E. "On Uncertainty Relations and Entanglement Detection with Mutually Unbiased Measurements." Open Systems & Information Dynamics 22, no. 01 (March 2015): 1550005. http://dx.doi.org/10.1142/s1230161215500055.

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We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.

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26

Mohamed, Abdel-Baset A., Atta Ur Rahman, and Hichem Eleuch. "Measurement Uncertainty, Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect." Entropy 24, no. 4 (April 13, 2022): 545. http://dx.doi.org/10.3390/e24040545.

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In a system of two charge-qubits that are initially prepared in a maximally entangled Bell’s state, the dynamics of quantum memory-assisted entropic uncertainty, purity, and negative entanglement are investigated. Isolated external cavity fields are considered in two different configurations: coherent-even coherent and even coherent cavity fields. For different initial cavity configurations, the temporal evolution of the final state of qubits and cavities is solved analytically. The effects of intrinsic decoherence and detuning strength on the dynamics of bipartite entropic uncertainty, purity and entanglement are explored. Depending on the field parameters, nonclassical correlations can be preserved. Nonclassical correlations and revival aspects appear to be significantly inhibited when intrinsic decoherence increases. Nonclassical correlations stay longer and have greater revivals due to the high detuning of the two qubits and the coherence strength of the initial cavity fields. Quantum memory-assisted entropic uncertainty and entropy have similar dynamics while the negativity presents fewer revivals in contrast.

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27

Ricaud, Benjamin, and Bruno Torresani. "Refined Support and Entropic Uncertainty Inequalities." IEEE Transactions on Information Theory 59, no. 7 (July 2013): 4272–79. http://dx.doi.org/10.1109/tit.2013.2249655.

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28

Nicola, Sergio De, Renato Fedele, Margarita A. Man'ko, and Vladimir I. Man'ko. "Entropic uncertainty relations for electromagnetic beams." Physica Scripta T135 (July 2009): 014053. http://dx.doi.org/10.1088/0031-8949/2009/135/014053.

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29

Guanlei, Xu, Wang Xiaotong, and Xu Xiaogang. "Entropic uncertainty inequalities on sparse representation." IET Signal Processing 10, no. 4 (June 2016): 413–21. http://dx.doi.org/10.1049/iet-spr.2014.0072.

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30

Bialynicki-Birula, I., and J. L. Madajczyk. "Entropic uncertainty relations for angular distributions." Physics Letters A 108, no. 8 (April 1985): 384–86. http://dx.doi.org/10.1016/0375-9601(85)90277-4.

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31

Karthik, H. S., A. R. Usha Devi, J. Prabhu Tej, and A. K. Rajagopal. "Conditional entropic uncertainty and quantum correlations." Optics Communications 427 (November 2018): 635–40. http://dx.doi.org/10.1016/j.optcom.2018.07.006.

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32

Wang, Dong, Fei Ming, Ming‐Liang Hu, and Liu Ye. "Quantum‐Memory‐Assisted Entropic Uncertainty Relations." Annalen der Physik 531, no. 10 (August 8, 2019): 1900124. http://dx.doi.org/10.1002/andp.201900124.

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33

Liu, Zhengwei, and Jinsong Wu. "Non-commutative Rényi entropic uncertainty principles." Science China Mathematics 63, no. 11 (April 8, 2020): 2287–98. http://dx.doi.org/10.1007/s11425-019-9523-4.

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34

ABDEL-ATY, MAHMOUD, ISSA A. AL-KHAYAT, and SHOUKRY S. HASSAN. "SHANNON INFORMATION AND ENTROPY SQUEEZING OF A SINGLE-MODE CAVITY QED OF A RAMAN INTERACTION." International Journal of Quantum Information 04, no. 05 (October 2006): 807–14. http://dx.doi.org/10.1142/s021974990600216x.

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Entropy squeezing is examined in the framework of Shannon information entropy for a degenerate Raman process involving two degenerate Rydberg energy levels of an atom interacting with a single-mode cavity field. Quantum squeezing in entropy is exhibited via the entropic uncertainty relation.

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35

Zhang, Yanliang, Maofa Fang, Guodong Kang, and Qingping Zhou. "Reducing quantum-memory-assisted entropic uncertainty by weak measurement and weak measurement reversal." International Journal of Quantum Information 13, no. 05 (August 2015): 1550037. http://dx.doi.org/10.1142/s0219749915500379.

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We have investigated the dynamic features of the quantum-memory-assisted entropic uncertainty relation (QMA EUR) in the amplitude damping (AD) channel. The initial state of qubit system and quantum memory system shared between Alice and Bob is assumed as extended by Werner-like (EWL) state. To reduce the amount of entropic uncertainty of Pauli observables in this noisy channel, we presented a reduction scheme by means of weak measurements (WMs) and weak measurement reversals (WMRs) before and after the entangled system subjecting to the noisy channel. It is shown that the prior WM and poster WMR can effectively reduce quantity of entropic uncertainty, but the poster WM operation cannot played a positive role on reduction of quantity of entropic uncertainty. We hope that our proposal could be verified experimentally and might possibly have future applications in quantum information processing.

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36

Rastegin, Alexey E. "Renyi and Tsallis formulations of noise-disturbance trade-off relations." Quantum Information and Computation 16, no. 3&4 (March 2016): 313–31. http://dx.doi.org/10.26421/qic16.3-4-7.

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We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the R´enyi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional R´enyi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.

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37

Majumder, Barun. "The Effects of Minimal Length in Entropic Force Approach." Advances in High Energy Physics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/296836.

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With Verlinde’s recent proposal which says that gravity can be identified with an entropic force and considering the effects of generalized uncertainty principle in the black hole entropy-area relation we derive the modified equations for Newton’s law of gravitation, modified Newtonian dynamics, and Einstein’s general relativity. The corrections to the Newtonian potential is compared with the corrections that come from Randall-Sundrum II model and an effective field theoretical model of quantum general relativity. The effect of the generalized uncertainty principle introduces aareatype correction term in the entropy-area relation whose consequences in different scenarios are discussed.

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38

Ivancevic, Vladimir, Darryn Reid, and Peyam Pourbeik. "Tensor-Centric Warfare II: Entropic Uncertainty Modeling." Intelligent Control and Automation 09, no. 02 (2018): 30–51. http://dx.doi.org/10.4236/ica.2018.92003.

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Xu, Guanlei, Xiaotong Wang, Lijia Zhou, Limin Shao, and Xiaogang Xu. "Discrete Entropic Uncertainty Relations Associated with FRFT." Journal of Signal and Information Processing 04, no. 03 (2013): 120–24. http://dx.doi.org/10.4236/jsip.2013.43b021.

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40

Baek, Kyunghyun, and Wonmin Son. "Entropic Uncertainty Relations for Successive Generalized Measurements." Mathematics 4, no. 2 (June 7, 2016): 41. http://dx.doi.org/10.3390/math4020041.

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41

Rastegin, A. E. "Entropic uncertainty relations and quasi-Hermitian operators." Journal of Physics A: Mathematical and Theoretical 45, no. 44 (October 23, 2012): 444026. http://dx.doi.org/10.1088/1751-8113/45/44/444026.

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42

Tomamichel, Marco, and Esther Hänggi. "The link between entropic uncertainty and nonlocality." Journal of Physics A: Mathematical and Theoretical 46, no. 5 (January 17, 2013): 055301. http://dx.doi.org/10.1088/1751-8113/46/5/055301.

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43

Niekamp, Sönke, Matthias Kleinmann, and Otfried Gühne. "Entropic uncertainty relations and the stabilizer formalism." Journal of Mathematical Physics 53, no. 1 (January 2012): 012202. http://dx.doi.org/10.1063/1.3678200.

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44

Ion, D. B., and M. L. D. Ion. "Entropic uncertainty relations for nonextensive quantum scattering." Physics Letters B 466, no. 1 (October 1999): 27–32. http://dx.doi.org/10.1016/s0370-2693(99)01052-7.

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45

Feng, Jun, Yao-Zhong Zhang, Mark D. Gould, and Heng Fan. "Entropic uncertainty relations under the relativistic motion." Physics Letters B 726, no. 1-3 (October 2013): 527–32. http://dx.doi.org/10.1016/j.physletb.2013.08.069.

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46

Ji, Yinghua, Qiang Ke, and Juju Hu. "Control of entropic uncertainty in circuit QED." Physica E: Low-dimensional Systems and Nanostructures 110 (June 2019): 140–47. http://dx.doi.org/10.1016/j.physe.2019.02.023.

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47

Rojas González, A., John A. Vaccaro, and Stephen M. Barnett. "Entropic uncertainty relations for canonically conjugate operators." Physics Letters A 205, no. 4 (September 1995): 247–54. http://dx.doi.org/10.1016/0375-9601(95)00582-n.

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48

Karpat, Göktuğ, Jyrki Piilo, and Sabrina Maniscalco. "Controlling entropic uncertainty bound through memory effects." EPL (Europhysics Letters) 111, no. 5 (September 1, 2015): 50006. http://dx.doi.org/10.1209/0295-5075/111/50006.

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49

Rastegin, Alexey E., and Karol Życzkowski. "Majorization entropic uncertainty relations for quantum operations." Journal of Physics A: Mathematical and Theoretical 49, no. 35 (July 22, 2016): 355301. http://dx.doi.org/10.1088/1751-8113/49/35/355301.

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50

Majerník, V., and T. Opatrný. "Entropic uncertainty relations for a quantum oscillator." Journal of Physics A: Mathematical and General 29, no. 9 (May 7, 1996): 2187–97. http://dx.doi.org/10.1088/0305-4470/29/9/029.

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