Limits on entropic uncertainty relations

2010 ◽  
Vol 10 (9&10) ◽  
pp. 848-858
Author(s):  
Andris Ambainis
Keyword(s):  
Quantum State ◽  
Uncertainty Relation ◽  
Mutually Unbiased Bases ◽  
Uncertainty Relations ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Standard Construction ◽  
The One ◽  
Prime Dimension ◽  
Better Than

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.

scholarly journals Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 763 ◽  
Author(s):  
Ana Costa ◽  
Roope Uola ◽  
Otfried Gühne
Keyword(s):  
Relative Entropy ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
W States ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Local Measurements ◽  
Action At A Distance ◽  
Quantum Steering ◽  
Joint Convexity

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.

scholarly journals Optimality of entropic uncertainty relations

2015 ◽  
Vol 13 (06) ◽  
pp. 1550045 ◽  
Author(s):  
Kais Abdelkhalek ◽  
René Schwonnek ◽  
Hans Maassen ◽  
Fabian Furrer ◽  
Jörg Duhme ◽  
...  
Keyword(s):  
Lower Bound ◽  
Numerical Investigation ◽  
Uncertainty Relation ◽  
Small Dimension ◽  
Uncertainty Relations ◽  
Complete Understanding ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Analytical Results ◽  
Optimal Lower Bound

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.

scholarly journals Entropic Uncertainty Relations, Entanglement and Quantum Gravity Effects via the Generalised Uncertainty Principle

2018 ◽  
pp. 1-12
Author(s):  
Otto Gadea ◽  
Gardo Blado
Keyword(s):  
Shannon Entropy ◽  
Uncertainty Principle ◽  
Upper Bound ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Single System ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Gravity Effects ◽  
Entanglement Criterion

We apply the generalised uncertainty principle (GUP) to the entropic uncertainty relation conditions on quantum entanglement. In particular, we study the GUP corrections to the Shannon entropic uncertainty condition for entanglement. We combine previous work on the Shannon entropy entanglement criterion for bipartite systems and the GUP corrections to the Shannon entropy for a single system to calculate the GUP correction for an entangled bipartite system. As in an earlier paper of the second author, which dealt with variance relations, it is shown that there is an increase in the upper bound for the entanglement condition upon the application of the generalised uncertainty principle. Necessary fundamental concepts of the generalised uncertainty principle, entanglement and the entropic uncertainty relations are also discussed. This paper puts together the concepts of entanglement, entropic uncertainty relations and the generalised uncertainty principle all of which have been separately discussed in pedagogical papers by Schroeder, Majernik et al., Blado et al. and Sprenger.  

scholarly journals Entropic uncertainty relation in neutrino oscillations

2020 ◽  
Vol 80 (8) ◽  
Author(s):  
Dong Wang ◽  
Fei Ming ◽  
Xue-Ke Song ◽  
Liu Ye ◽  
Jing-Ling Chen
Keyword(s):  
Quantum Correlation ◽  
Neutrino Oscillations ◽  
Quantum Information Processing ◽  
Physical Phenomenon ◽  
Uncertainty Relations ◽  
Classical Counterpart ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Neutrino Flavor ◽  
Potential Applications

Abstract Neutrino oscillation is deemed as an interesting physical phenomenon and shows the nonclassical features made apparently by the Leggett–Garg inequality. The uncertainty principle is one of the fundamental features that distinguishes the quantum world to its classical counterpart. And the principle can be depicted in terms of entropy, which forms the so-called entropic uncertainty relations (EUR). In this work, the entropic uncertainty relations that are relevant to the neutrino-flavor states are investigated by comparing the experimental observation of neutrino oscillations to predictions. From two different neutrino sources, we analyze ensembles of reactor and accelerator neutrinos for different energies, including measurements performed by the Daya Bay collaboration using detectors at 0.5 and 1.6 km from their source, and by the MINOS collaboration using a detector with a 735km distance to the neutrino source. It is found that the entropy-based uncertainty conditions strengths exhibits non-monotonic evolutions as the energy increases. We also quantify the systemic quantumness measured by quantum correlation, and derive the intrinsic relationship between quantum correlation and EUR. Furthermore, we utilize EUR as a criterion to detect entanglement of neutrino-flavor state. Our results could illustrate the potential applications of neutrino oscillations on quantum information processing in the weak-interaction processes.

Unextendible mutually unbiased bases from Pauli C\classes

2014 ◽  
Vol 14 (9&10) ◽  
pp. 823-844
Author(s):  
Prabha Mandayam ◽  
Somshubhro Bandyopadhyay ◽  
Markus Grassl ◽  
William K. Wootters
Keyword(s):  
Uncertainty Relation ◽  
Higher Dimensions ◽  
Mutually Unbiased Bases ◽  
Entropic Uncertainty Relation ◽  
Interesting Connection

We provide a construction of sets of $d/2+1$ mutually unbiased bases (MUBs) in dimensions $d=4,8$ using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. Moreover, specific examples of sets of MUBs obtained using our construction are shown to be {\it strongly unextendible}; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions $d=2^{n} (n>3) $ as well.} {Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a $H_{2}$ entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in $d=4$.

scholarly journals Additivity of entropic uncertainty relations

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 59 ◽  
Author(s):  
René Schwonnek
Keyword(s):  
Uncertainty Relation ◽  
Natural Generalization ◽  
Uncertainty Relations ◽  
Entanglement Witness ◽  
Separable States ◽  
Entropic Uncertainty Relations ◽  
Multipartite System ◽  
Positive Coefficients ◽  
Global Uncertainty ◽  
Projective Measurements

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.

scholarly journals Entropic uncertainty relation under correlated dephasing channels

2018 ◽  
Vol 96 (7) ◽  
pp. 700-704 ◽  
Author(s):  
Göktuğ Karpat
Keyword(s):  
Quantum Mechanics ◽  
Standard Deviation ◽  
Lower Bound ◽  
Quantum System ◽  
Quantum Channel ◽  
Uncertainty Relation ◽  
Open Quantum System ◽  
Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Incompatible Observables

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.

Sign in / Sign up

Export Citation Format

Share Document