Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The European Physical Journal C

Variance-based uncertainty relations and entanglement amplification for particles constrained on a torus

Asma Bashir, Hafiz Muhammad Asif Javed, Muhammad Abdul Wasay, Muhammad Shabir Mahr, Muhammad Zafar Iqbal

Published Aug 14, 2024• 0 citations
Publisher

Abstract

Abstract We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field $$\mathcal {\vec {B}}$$ B → . We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that $$\mathcal {\vec {A}}$$ A → (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance ( $$\mathcal {POV}$$ POV ). Additionally, we give the critical range of $$\mathcal {\vec {A}}$$ A → and $$\mathcal {\vec {B}}$$ B → where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the $$\mathcal {POV}$$ POV of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to $$\mathcal {\vec {B}}$$ B → by saturating the angular momentum uncertainty relation and identify the critical coherence value $${\mathcal {B}}_c$$ B c such that when $${\mathcal {B}}\ne {\mathcal {B}}_c$$ B ≠ B c , the state decoheres.

DOI
10.1140/epjc/s10052-024-13182-x

Related papers

New uncertainty principle for a particle on a torus knot

Madhushri Roy Chowdhury, Subir Ghosh

This paper deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard–Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.

arXiv:2405.11322•International Journal of Geometric Methods in Modern Physics•May 30, 2025•0 citations

Dynamics and uncertainty for maximally entangled bipartite system constrained on a helicoid

Asma Bashir, Muhammad Abdul Wasay, Abdulaziz Al-Mogeeth +1 more

Abstract The classical and quantum dynamics of particles constrained on a right helicoid is discussed via Dirac approach. We show how the uncertainty in measurement for observables of maximally entangled system is affected by the total number of constrained particles $$\sigma $$ σ , external magnetic field $$\vec {\mathcal {B}}$$ B → ; as well as by geometric parameters like the pitch $$\rho $$ ρ and the radial position of particles. In doing so we also highlight numeric bounds on the external field strengths to tune both intraparticle and bipartite entanglement and we remark that the bipartite entanglement is more robust to changes in the fields than the intraparticle entanglement, in this framework. We also highlight specific parameter regimes which lead the uncertainty (in measurement) to achieve respective parameter independence and, for a particular subset of commutation relations, the system remains confined in the quantum regime even in the limit $$\sigma \rightarrow \infty $$ σ → ∞ . It is observed that the uncertainties are strongly influenced by the geometric parameters e.g., $$\rho $$ ρ , and the strength of bipartite as well as intraparticle entanglement might be controllable through $$\rho $$ ρ . The energy equation for this setup is obtained and the additional terms are discussed which arise due to quantum correlations, orbit–orbit interaction and the normal Zeeman effect, which leads to the splitting of the energy level into 11 non-degenerate levels. Finally we comment that, a linkage of this phenomenology with Aharonov–Bohm like effect might be possible by strictly confining $$\vec {\mathcal {B}}$$ B → along the central axis of the helicoid.

The European Physical Journal C•Sep 13, 2022•2 citations

Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices

Géza Tóth, Florian Fröwis

We present several inequalities related to the Robertson-Schr\"odinger uncertainty relation. In all these inequalities, we consider a decomposition of the density matrix into a mixture of states, and use the fact that the Robertson-Schr\"odinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in Fr\"owis et al. [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relation. We present a formulation of the Cram\'er-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schr\"odinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schr\"odinger uncertainty relation. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of the canonical position and momentum operators for two-mode continuous variable systems. We show that the violation of well-known entanglement conditions in these systems discussed in Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] implies that the state is more useful metrologically than certain relevant subsets of separable states. We present similar results concerning entanglement conditions with angular momentum operators for spin systems.

arXiv:2109.06893•Physical Review Research•Jan 31, 2022•31 citations

Near-Optimal Variance-Based Uncertainty Relations

Yunlong Xiao, Naihuan Jing, Bing Yu +2 more

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.

arXiv:1610.01692•Frontiers in Physics•Apr 1, 2022•8 citations

Constrained dynamics of maximally entangled bipartite system

Asma Bashir, Muhammad Abdul Wasay

Abstract The classical and quantum dynamics of two particles constrained on $$S^1$$ S1 is discussed via Dirac’s approach. We show that when state is maximally entangled between two subsystems, the product of dispersion in the measurement reduces. We also quantify the upper bound on the external field $$\vec {B}$$ B→ such that $$\vec {B}\ge \vec {B}_{upper }$$ B→≥B→upper implies no reduction in the product of dispersion pertaining to one subsystem. Further, we report on the cut-off value of the external field $$\vec {B}_{cutoff }$$ B→cutoff , above which the bipartite entanglement is lost and there exists a direct relationship between uncertainty of the composite system and the external field. We note that, in this framework it is possible to tune the external field for entanglement/unentanglement of a bipartite system. Finally, we show that the additional terms arising in the quantum Hamiltonian, due to the requirement of Hermiticity of operators, produce a shift in the energy of the system.

arXiv:2105.00096•The European Physical Journal C•Apr 1, 2021•2 citations

Multipartite entanglement criterion from uncertainty relations

Jérémie Gillet, Thierry Bastin, G. S. Agarwal

We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schr\"odinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\ifmmode\times\else\texttimes\fi{}2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.

arXiv:0810.1480•Physical Review A•Nov 11, 2008•19 citations
Abstract We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field B⃗\mathcal {\vec {B}}B B → . We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that A⃗\mathcal {\vec {A}}A A → (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance ( POV\mathcal {POV}POV POV ). Additionally, we give the critical range of A⃗\mathcal {\vec {A}}A A → and B⃗\mathcal {\vec {B}}B B → where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the POV\mathcal {POV}POV POV of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to B⃗\mathcal {\vec {B}}B B → by saturating the angular momentum uncertainty relation and identify the critical coherence value Bc{\mathcal {B}}_cBc​ B c such that when B≠Bc{\mathcal {B}}\ne {\mathcal {B}}_cB=Bc​ B ≠ B c , the state decoheres.
This paper deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard–Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.
Abstract The classical and quantum dynamics of particles constrained on a right helicoid is discussed via Dirac approach. We show how the uncertainty in measurement for observables of maximally entangled system is affected by the total number of constrained particles σ\sigma σ σ , external magnetic field B⃗\vec {\mathcal {B}}B B → ; as well as by geometric parameters like the pitch ρ\rho ρ ρ and the radial position of particles. In doing so we also highlight numeric bounds on the external field strengths to tune both intraparticle and bipartite entanglement and we remark that the bipartite entanglement is more robust to changes in the fields than the intraparticle entanglement, in this framework. We also highlight specific parameter regimes which lead the uncertainty (in measurement) to achieve respective parameter independence and, for a particular subset of commutation relations, the system remains confined in the quantum regime even in the limit σ→∞\sigma \rightarrow \infty σ→∞ σ → ∞ . It is observed that the uncertainties are strongly influenced by the geometric parameters e.g., ρ\rho ρ ρ , and the strength of bipartite as well as intraparticle entanglement might be controllable through ρ\rho ρ ρ . The energy equation for this setup is obtained and the additional terms are discussed which arise due to quantum correlations, orbit–orbit interaction and the normal Zeeman effect, which leads to the splitting of the energy level into 11 non-degenerate levels. Finally we comment that, a linkage of this phenomenology with Aharonov–Bohm like effect might be possible by strictly confining B⃗\vec {\mathcal {B}}B B → along the central axis of the helicoid.
We present several inequalities related to the Robertson-Schr\"odinger uncertainty relation. In all these inequalities, we consider a decomposition of the density matrix into a mixture of states, and use the fact that the Robertson-Schr\"odinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in Fr\"owis et al. [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relation. We present a formulation of the Cram\'er-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schr\"odinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schr\"odinger uncertainty relation. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of the canonical position and momentum operators for two-mode continuous variable systems. We show that the violation of well-known entanglement conditions in these systems discussed in Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] implies that the state is more useful metrologically than certain relevant subsets of separable states. We present similar results concerning entanglement conditions with angular momentum operators for spin systems.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
Abstract The classical and quantum dynamics of two particles constrained on S1S^1S1 S1 is discussed via Dirac’s approach. We show that when state is maximally entangled between two subsystems, the product of dispersion in the measurement reduces. We also quantify the upper bound on the external field B⃗\vec {B}B B→ such that B⃗≥B⃗upper\vec {B}\ge \vec {B}_{upper }B≥Bupper​ B→≥B→upper implies no reduction in the product of dispersion pertaining to one subsystem. Further, we report on the cut-off value of the external field B⃗cutoff\vec {B}_{cutoff }Bcutoff​ B→cutoff , above which the bipartite entanglement is lost and there exists a direct relationship between uncertainty of the composite system and the external field. We note that, in this framework it is possible to tune the external field for entanglement/unentanglement of a bipartite system. Finally, we show that the additional terms arising in the quantum Hamiltonian, due to the requirement of Hermiticity of operators, produce a shift in the energy of the system.
We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schr\"odinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\ifmmode\times\else\texttimes\fi{}2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.