Uncertainty relations in the framework of equalities

Ozawa, Tohru; Yuasa, Kazuya

doi:10.1016/j.jmaa.2016.08.023

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…For any symmetric operators A and B in L 2 with domains D ( A ) and D ( B ), respectively, a straightforward calculation (e.g. [ 16 ], Theorem 2.1) shows the equality

for f ∈ D ( A )∩ D ( B ) with Af ≢0 and Bf ≢0, which will be useful in our next proof.…”

Section: Main Results and Proofsmentioning

confidence: 95%

Uncertainty relations on nilpotent Lie groups

Ruzhansky

Сураган

2017

Proc. R. Soc. A.

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We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic double-struckRn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

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“…For any symmetric operators A and B in L 2 with domains D ( A ) and D ( B ), respectively, a straightforward calculation (e.g. [ 16 ], Theorem 2.1) shows the equality

for f ∈ D ( A )∩ D ( B ) with Af ≢0 and Bf ≢0, which will be useful in our next proof.…”

Section: Main Results and Proofsmentioning

confidence: 95%

Uncertainty relations on nilpotent Lie groups

Ruzhansky

Сураган

2017

Proc. R. Soc. A.

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“…Also, some other approaches were suggested by Ozawa and Yuasa in [6]. In fact, the above approaches for the definition of the uncertainty product do not deal with a new phenomenon, that appears in the multidimensional case, namely, the localization of a function along a particular direction.…”

Section: Basic Notations and Definitionsmentioning

confidence: 99%

“…where the matrix M is defined by (6). So, the statement of the theorem is a well-known fact of linear algebra.…”

Section: Consider the Following Estimatesmentioning

confidence: 99%

“…The corresponding uncertainty principle is valid automatically, the lower bound of the directional uncertainty product is equal to 1/4 and is attained on the class of functions, that are Gaussian exponentials up to a multiplication on arbitrary smooth functions. Our definition, in contrast to definitions given by Goh and Goodman in [4], Ozawa and Yuasa in [6], includes the directionality explicitly in a natural way.…”

Section: Introductionmentioning

confidence: 99%

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Directional Heisenberg uncertainty product

Krivoshein¹,

Lebedeva²,

Neiman³

et al. 2019

Mathematical Inequalities &Amp; Applications

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“…An especially important arena for ITEs in the past two decades has been quantum mechanics (QM) with applications ranging from quantum estimation and coding theory to quantum entanglement. The catalyst has been an infusion of new ideas from (quantum) information theory [ 12 , 13 , 14 , 15 ], functional analysis [ 16 , 17 ], condensed matter theory [ 18 , 19 ], and cosmology [ 20 , 21 ]. On the experimental front, the use of ITEs has been stimulated not only by new high-precision instrumentation [ 22 , 23 ] but also by, e.g., recent advances in stochastic thermodynamics [ 24 , 25 ] or observed violations of Heisenberg’s error-disturbance uncertainty relations [ 26 , 27 , 28 , 29 , 30 ].…”

Section: Introductionmentioning

confidence: 99%

From Rényi Entropy Power to Information Scan of Quantum States

Jizba

Dunningham

Prokš

2021

Entropy

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In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed.

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Uncertainty relations in the framework of equalities

Abstract: We study the Schrödinger-Robertson uncertainty relations in an algebraic framework. Moreover, we show that some specific commutation relations imply new equalities, which are regarded as equality versions of well-known inequalities such as Hardy's inequality.

Cited by 6 publications

References 26 publications

Uncertainty relations on nilpotent Lie groups

Uncertainty relations on nilpotent Lie groups

Directional Heisenberg uncertainty product

From Rényi Entropy Power to Information Scan of Quantum States

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