scholarly journals Unified and Exact Framework for Variance-Based Uncertainty Relations

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Xiao Zheng ◽  
Shao-Qiang Ma ◽  
Guo-Feng Zhang ◽  
Heng Fan ◽  
Wu-Ming Liu
Keyword(s):  
Uncertainty Relation ◽  
Hermitian Operator ◽  
Uncertainty Relations ◽  
Unified Framework ◽  
Incompatible Observables

AbstractWe provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework.

scholarly journals Mathematical Uncertainty Relations and their Generalization for Multiple Incompatible Observables

2017 ◽  
Vol 33 (1) ◽  
Author(s):  
Hung Quang Nguyen ◽  
Tu Quang Bui
Keyword(s):  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Generalized Uncertainty Principle ◽  
Schwarz Inequality ◽  
Uncertainty Relations ◽  
Heisenberg Uncertainty Relation ◽  
Generalized Uncertainty Relation ◽  
Heisenberg Uncertainty ◽  
Incompatible Observables

We show that the famous Heisenberg uncertainty relation for two incompatible observables can be generalized elegantly to the determinant form for N arbitrary observables. To achieve this purpose, we propose a generalization of the Cauchy-Schwarz inequality for two sets of vectors. Simple consequences of the N-ary uncertainty relation are also discussed. Keywords: Generalized uncertainty relation, Generalized uncertainty principle, Generalized Cauchy-Schwarz inequality.

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.

scholarly journals Tripartite entropic uncertainty relation under phase decoherence

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
R. A. Abdelghany ◽  
A.-B. A. Mohamed ◽  
M. Tammam ◽  
Watson Kuo ◽  
H. Eleuch
Keyword(s):  
Lower Bound ◽  
Uncertainty Relation ◽  
Nearest Neighbors ◽  
The Other ◽  
Uncertainty In Measurement ◽  
Entropic Uncertainty Relation ◽  
Specific Choice ◽  
Phase Decoherence ◽  
Time Invariant ◽  
Incompatible Observables

AbstractWe formulate the tripartite entropic uncertainty relation and predict its lower bound in a three-qubit Heisenberg XXZ spin chain when measuring an arbitrary pair of incompatible observables on one qubit while the other two are served as quantum memories. Our study reveals that the entanglement between the nearest neighbors plays an important role in reducing the uncertainty in measurement outcomes. In addition we have shown that the Dolatkhah’s lower bound (Phys Rev A 102(5):052227, 2020) is tighter than that of Ming (Phys Rev A 102(01):012206, 2020) and their dynamics under phase decoherence depends on the choice of the observable pair. In the absence of phase decoherence, Ming’s lower bound is time-invariant regardless the chosen observable pair, while Dolatkhah’s lower bound is perfectly identical with the tripartite uncertainty with a specific choice of pair.

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 Remarks on the uncertainty relations

2020 ◽  
Vol 35 (26) ◽  
pp. 2050219 ◽  
Author(s):  
Krzysztof Urbanowski
Keyword(s):  
Lower Bound ◽  
Characteristic Time ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Rigorous Analysis ◽  
Standard Deviations ◽  
Complete Sets

We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables [Formula: see text] and [Formula: see text] and such vectors that the lower bound for the product of standard deviations [Formula: see text] and [Formula: see text] calculated for these vectors is zero: [Formula: see text]. We also show that for some pairs of non-commuting observables the sets of vectors for which [Formula: see text] can be complete (total). The Heisenberg, [Formula: see text], and Mandelstam–Tamm (MT), [Formula: see text], time–energy uncertainty relations ([Formula: see text] is the characteristic time for the observable [Formula: see text]) are analyzed too. We show that the interpretation [Formula: see text] for eigenvectors of a Hamiltonian [Formula: see text] does not follow from the rigorous analysis of MT relation. We show also that contrary to the position–momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of [Formula: see text] and [Formula: see text].

Photon added coherent states of the parity deformed oscillator

2019 ◽  
Vol 34 (14) ◽  
pp. 1950104 ◽  
Author(s):  
A. Dehghani ◽  
B. Mojaveri ◽  
S. Amiri Faseghandis
Keyword(s):  
Coherent States ◽  
Uncertainty Relation ◽  
Annihilation Operator ◽  
Single Mode ◽  
Heisenberg Algebra ◽  
Uncertainty Relations ◽  
Deformed Oscillator ◽  
Cat States ◽  
Poissonian Statistics ◽  
Creation Operators

Using the parity deformed Heisenberg algebra (RDHA), we first establish associated coherent states (RDCSs) for a pseudo-harmonic oscillator (PHO) system that are defined as eigenstates of a deformed annihilation operator. Such states can be expressed as superposition of an even and odd Wigner cat states.[Formula: see text] The RDCSs minimize a corresponding uncertainty relation, and resolve an identity condition through a positive definite measure which is explicitly derived. We introduce a class of single-mode excited coherent states (PARDCS) of the PHO through “m” times application of deformed creation operators to RDCS. For the states thus constructed, we analyze their statistical properties such as squeezing and sub-Poissonian statistics as well as their uncertainty relations.

scholarly journals DEFORMED DENSITY MATRIX AND GENERALIZED UNCERTAINTY RELATION IN THERMODYNAMICS

2004 ◽  
Vol 19 (01) ◽  
pp. 71-81 ◽  
Author(s):  
A. E. SHALYT-MARGOLIN ◽  
A. YA. TREGUBOVICH
Keyword(s):  
Statistical Mechanics ◽  
Density Matrix ◽  
Uncertainty Relation ◽  
Planck Scale ◽  
Additional Term ◽  
Temperature Limit ◽  
Quantum Mechanical ◽  
Uncertainty Relations ◽  
Primary Object ◽  
Complete Analogy

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. In our opinion the approach proposed may lead to the proofs of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Planck scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Planck's. The associated deformation of a canonical Gibbs distribution is given explicitly.

scholarly journals BLACK HOLE UNCERTAINTIES

1993 ◽  
Vol 08 (20) ◽  
pp. 1925-1941
Author(s):  
ULF H. DANIELSSON
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Wave Functions ◽  
Uncertainty Relations ◽  
Two Dimensional ◽  
Black Hole Mass ◽  
Dilaton Black Holes

In this work the quantum theory of two-dimensional dilaton black holes is studied using the Wheeler-De Witt equation. The solutions correspond to wave functions of the black hole. It is found that for an observer inside the horizon, there are uncertainty relations for the black hole mass and a parameter in the metric determining the Hawking flux. Only for a particular value of this parameter can both be known with arbitrary accuracy. In the generic case there is instead a relation that is very similar to the so-called string uncertainty relation.

scholarly journals On the Space–Time Uncertainty Relations of Liouville Strings and D-Branes

1997 ◽  
Vol 12 (27) ◽  
pp. 2029-2035 ◽  
Author(s):  
G. Amelino-Camelia ◽  
N. E. Mavromatos ◽  
John Ellis ◽  
D. V. Nanopoulos
Keyword(s):  
String Theory ◽  
Commutation Relation ◽  
Uncertainty Relation ◽  
Space Time ◽  
Uncertainty Relations ◽  
Time Uncertainty ◽  
Space And Time

Within a Liouville approach to noncritical string theory, we argue for a nontrivial commutation relation between space and time observables, leading to a nonzero space–time uncertainty relation δx δt>0, which vanishes in the limit of weak string coupling.

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