Majorization entropic uncertainty relations

Puchała, Zbigniew; Rudnicki, Łukasz; Życzkowski, Karol

doi:10.1088/1751-8113/46/27/272002

Cited by 160 publications

(277 citation statements)

References 29 publications

Supporting

Mentioning

271

Contrasting

Unclassified

Order By: Relevance

“…Note added: Soon after the first version of this manuscript appeared on the arXiv, Pucha la, Rudnicki andŻyczkowski submitted a similar result on the arXiv [27] (valid for orthonormal bases), now published in [28]. Their proof uses different techniques and we refer the interested reader to their paper for more details.…”

Section: Mulasmentioning

confidence: 99%

Universal Uncertainty Relations

2013

View full text Add to dashboard Cite

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A 84, 052117 (2011)]. Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.Uncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the noncommutative structure of the theory. In contrast to classical physics, where in principle any observable can be measured with arbitrary precision, quantum mechanics introduces severe restrictions on the allowed measurement results of two or more non-commuting observables. Uncertainty relations are not a manifestation of the experimentalists' (in)ability of performing precise measurements, but are inherently determined by the incompatibility of the measured observables.The first formulation of the uncertainty principle was provided by Heisenberg [1], who noted that more knowledge about the position of a single quantum particle implies less certainty about its momentum and vice-versa. He expressed the principle in terms of standard deviations of the momentum and position operators ∆X · ∆P 2 .Robertson [2] generalized Heisenberg's uncertainty principle to any two arbitrary observables A and B asA major drawback of Robertson's uncertainty principle is that it depends on the state |ψ of the system. In particular, when |ψ belongs to the null-space of the * Electronic address: friedlan@uic. Here H(A) is the Shannon entropy [5] of the probability distribution induced by measuring the state |ψ of the system in the eigenbasis {|a j } of the oservable A (and similarly for B). The bound on the right hand side c(A, B) := max m,n | a m |b n | represents the maximum overlap between the bases elements, and is independent of the state |ψ . Recently the study of uncertainty relations intensified [6,7] (see also [1,9] for recent surveys), and as a result various important applications have been discovered, ranging from security proofs for quantum cryptography [10][11][12], information locking ...

show abstract

Section: Mulasmentioning

confidence: 99%

Universal Uncertainty Relations

2013

View full text Add to dashboard Cite

show abstract

“…This kind of behavior is somehow typical in the majorization approach to entropic uncertainty relations. In the discrete case, the tensor-product EUR (weaker than the direct-sum EUR used in this paper) can outperform the Maassen-Uffink result in more than 98% of cases [9], even for a small dimension of the Hilbert space equal to 5. But the Maassen-Uffink lower bound [3] always dominates when U ij is sufficiently close to the Fourier matrix, so that both eigenbases of the observables A and B become mutually unbiased.…”

Section: Discussionmentioning

confidence: 86%

“…In the majorization approach one looks for the probability vectors Q (A, B) and W (A, B) which majorize the tensor product [9,10] and the direct sum [12] of the involved distributions (3):…”

Section: A Majorization Entropic Uncertainty Relationsmentioning

confidence: 99%

“…One decade later, entropic formulation of the uncertainty principle has as well been developed in the discrete settings [2,3]. Even though, the topic of entropic uncertainty relations (EURs) has a long history (for a detailed review see [4,5]), one can observe a recent increase of interest within the quantum information community leading to several improvements [6][7][8][9][10][11][12][13][14][15][16][17] or even a deep asymptotic analysis of different bounds [18]. This is quite understandable, because the entropic uncertainty relations have various applications, for example in entanglement detection [19][20][21][22][23], security of quantum protocols [24,25], quantum memory [26,27] or as an ingredient of Einstein-Podolsky-Rosen steering criteria [28,29].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Majorization approach to entropic uncertainty relations for coarse-grained observables

Rudnicki

2015

Phys. Rev. A

Self Cite

View full text Add to dashboard Cite

We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The obtained entropic inequalities involve two Rényi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.

show abstract

“…On the other hand, RE of any order are Schur-concave functions [38]. In fact, every function f (P) which is Schur concave can represent a reasonable measure of information, since it is maximized by a uniform probability distribution, while minimum is provided with concentrated distributions P = {p i = 1, p j i = 0}.…”

Section: Rény's Entropy: Entropy Of Multifractal Systemsmentioning

confidence: 99%

On q-non-extensive statistics with non-Tsallisian entropy

Jizba

Korbel

2016

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

We combine an axiomatics of Rényi with the q-deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W-function. We analyze the corresponding "high" and "lowtemperature" asymptotics and reveal a non-trivial structure of the parameter space.

show abstract

Majorization entropic uncertainty relations

Cited by 160 publications

References 29 publications

Universal Uncertainty Relations

Universal Uncertainty Relations

Majorization approach to entropic uncertainty relations for coarse-grained observables

On q-non-extensive statistics with non-Tsallisian entropy

Contact Info

Product

Resources

About