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

arXiv:2407.06599 · Physical Review A

Quantum coherence measures for generalized Gaussian wave packets under a Lorentz boost

Arnab Mukherjee, Soham Sen, Sunandan Gangopadhyay

Published Nov 12, 2024• 1 citation
arXivHTMLPDF

Abstract

In this paper, we consider a single-particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The effect of the relativistic boost on single-particle generalized Gaussian wave packets is studied. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general, the coherence decays with the increase in momentum of the state, as well as the boost applied to it. A more prominent loss of coherence due to relativistic boost is observed for a single-particle electron than that of a neutron. The analysis is carried out with a generalized Gaussian wave packet of the form $\mathcal{N}{p}^{n}exp(\ensuremath{-}\frac{{p}^{2}}{{\ensuremath{\sigma}}^{2}})$, with $n$ being the ``generalization parameter'' and $\mathcal{N}$ denoting the appropriate normalization constant. We also obtain a range for parameter $n$ appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence, ${l}_{1}$- and ${l}_{2}$-norm measure of coherence, and relative entropy of coherence for a $(1+1)$- and $(3+1)$-dimensional analysis. Corresponding to each of the cases, we observe that the ${l}_{1}$-norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function $\ensuremath{\sigma}$, boost parameter $\ensuremath{\beta}$, and generalization parameter $n$.

DOI
10.1103/physreva.110.052413

Related papers

Quantum coherence measures in entangled atomic systems

Arnab Mukherjee, Soham Sen, Sunandan Gangopadhyay

In this study, we investigate the effect of the Lorentz transformation on the measures of quantum coherence in an entangled atomic system. Here, we consider the effect of this relativistic boosts on two-particle entangled generalized Gaussian wave packets in two scenarios. In the first scenario, we consider that the relativistic boost affects the one particle and other remains unaffected while in the second scenario, we consider that both the particles are affected by the effect of the relativistic boost. The coherence of the wave function as measured by the boosted observer is studied as a function of the boost parameter and the width of the Gaussian wave packets. Using various formulations of coherence, it is shown that in general the coherence decays with increase in the width of the Gaussian wave packet, higher values of boost parameter, and the number of particles on which boost is applied.

quant-pharXiv:2511.20371•Nov 26, 2025•0 citations

Preservation of quantum coherence under Lorentz boost for narrow uncertainty wave packets

Riddhi Chatterjee, A. S. Majumdar

We consider the effect of relativistic boosts on single-particle Gaussian wave packets. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general the coherence decays with the increase of the momentum of the state, as well as the boost applied to it. Employing a basis-independent formulation, we show, however, that coherence may be preserved even for large boosts applied on narrow uncertainty wave packets. Our result is exemplified quantitatively for practically realizable neutron wave functions.

arXiv:1705.09265•Physical Review A•Nov 1, 2017•3 citations

Relativistic entanglement of single and two particle systems

Veiko Palge

One of the defining features of quantum theory is entanglement, the notion that quantum systems can display correlations that are impossible from the classical \npoint of view. In quantum information theory entanglement has come to be recognised as a physical resource that enables new technologies that perform information \nprocessing tasks which are beyond the limits of the classical realm. It has been realised only recently that entanglement is dependent on the frame of reference in \nboth inertial and accelerated systems. In this thesis, we investigate the relativistic entanglement of massive spin-1=2 particles in inertial frames by focussing on the \ndependence of entanglement on the geometry of the underlying boost scenario. We first explore the ‘qubit’ of the relativistic setting: a single particle with spin and momentum, with momentum given by a Gaussian distribution. We study the system in a variety of different boost scenarios, analysing the behaviour of entanglement \nfrom a geometric point of view. The spin-spin entanglement of two particle systems is then surveyed for many different discrete product and entangled momenta, with the spins in the Werner state. We also extend the analysis to continuous momentum states and study them in a variety of geometries. The results obtained from the analysis of discrete states are applied to continuous states, leading to a better \nunderstanding of the behaviour of entanglement. We lastly discuss the common view according to which Lorentz boosts leave the total entanglement of the state invariant.

•May 1, 2013•3 citations

Complete complementarity relations and their Lorentz invariance

Marcos L. W. Basso, Jonas Maziero

It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single-particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.

arXiv:2007.14480•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 1, 2021•12 citations

Relativistic Entanglements of Spin 1/2 Particles with General Momentum

Young Hoon Moon, Sung Woo Hwang, Doyeol Ahn

In this paper, the Lorentz transformations of entangled Bell states with general momentum not necessarily orthogonal to the boost direction and spin are studied. We extend quantum correlations and Bell's inequality to the relativistic regime by considering normalized relativistic observables. It is shown that quantum information along the direction perpendicular to the boost is eventually lost, and Bell's inequality is not always violated for entangled states in special relativity. This could impose restrictions on certain quantum information processing, such as quantum cryptography using massive particles.

arXiv:quant-ph/0304116•Progress of Theoretical Physics•Aug 1, 2004•21 citations

Side jump and anomalous Lorentz transformation of a coherent Dirac wave packet

Jin-Rui Wu, Shuai Y. F. Liu

Spin-orbit coupling makes the mean position of a spinning particle transform anomalously under a Lorentz boost, with an additional shift known as a side jump. In this work, we develop a trace method to derive an exact but compact transformation kernel for the probability center of a general three-dimensional Dirac wave packet under an arbitrary finite Lorentz boost. The expression includes the physics of finite momentum width, spin-momentum correlations, and coherent phase structure without a semi-classical expansion. We identify texture-dependent side jumps even at fixed initial mean momentum, spin, and orbital angular momentum, as well as a non-monotonic mass dependence for Gaussian packets at fixed momentum width. The explicit kernels further resolve the spin and orbital angular-momentum contributions for massive and massless fermions, providing a kinematic basis for covariant particle-based spin transport that retains wave-packet information.

nucl-thCross-listedarXiv:2610.36416•Oct 28, 2026•0 citations
In this paper, we consider a single-particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The effect of the relativistic boost on single-particle generalized Gaussian wave packets is studied. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general, the coherence decays with the increase in momentum of the state, as well as the boost applied to it. A more prominent loss of coherence due to relativistic boost is observed for a single-particle electron than that of a neutron. The analysis is carried out with a generalized Gaussian wave packet of the form Npnexp(\ensuremath−p2\ensuremathσ2)\mathcal{N}{p}^{n}exp(\ensuremath{-}\frac{{p}^{2}}{{\ensuremath{\sigma}}^{2}})Npnexp(\ensuremath−\ensuremathσ2p2​), with nnn being the ``generalization parameter'' and N\mathcal{N}N denoting the appropriate normalization constant. We also obtain a range for parameter nnn appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence, l1{l}_{1}l1​- and l2{l}_{2}l2​-norm measure of coherence, and relative entropy of coherence for a (1+1)(1+1)(1+1)- and (3+1)(3+1)(3+1)-dimensional analysis. Corresponding to each of the cases, we observe that the l1{l}_{1}l1​-norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function \ensuremathσ\ensuremath{\sigma}\ensuremathσ, boost parameter \ensuremathβ\ensuremath{\beta}\ensuremathβ, and generalization parameter nnn.
In this study, we investigate the effect of the Lorentz transformation on the measures of quantum coherence in an entangled atomic system. Here, we consider the effect of this relativistic boosts on two-particle entangled generalized Gaussian wave packets in two scenarios. In the first scenario, we consider that the relativistic boost affects the one particle and other remains unaffected while in the second scenario, we consider that both the particles are affected by the effect of the relativistic boost. The coherence of the wave function as measured by the boosted observer is studied as a function of the boost parameter and the width of the Gaussian wave packets. Using various formulations of coherence, it is shown that in general the coherence decays with increase in the width of the Gaussian wave packet, higher values of boost parameter, and the number of particles on which boost is applied.
We consider the effect of relativistic boosts on single-particle Gaussian wave packets. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general the coherence decays with the increase of the momentum of the state, as well as the boost applied to it. Employing a basis-independent formulation, we show, however, that coherence may be preserved even for large boosts applied on narrow uncertainty wave packets. Our result is exemplified quantitatively for practically realizable neutron wave functions.
One of the defining features of quantum theory is entanglement, the notion that quantum systems can display correlations that are impossible from the classical \npoint of view. In quantum information theory entanglement has come to be recognised as a physical resource that enables new technologies that perform information \nprocessing tasks which are beyond the limits of the classical realm. It has been realised only recently that entanglement is dependent on the frame of reference in \nboth inertial and accelerated systems. In this thesis, we investigate the relativistic entanglement of massive spin-1=2 particles in inertial frames by focussing on the \ndependence of entanglement on the geometry of the underlying boost scenario. We first explore the ‘qubit’ of the relativistic setting: a single particle with spin and momentum, with momentum given by a Gaussian distribution. We study the system in a variety of different boost scenarios, analysing the behaviour of entanglement \nfrom a geometric point of view. The spin-spin entanglement of two particle systems is then surveyed for many different discrete product and entangled momenta, with the spins in the Werner state. We also extend the analysis to continuous momentum states and study them in a variety of geometries. The results obtained from the analysis of discrete states are applied to continuous states, leading to a better \nunderstanding of the behaviour of entanglement. We lastly discuss the common view according to which Lorentz boosts leave the total entanglement of the state invariant.
It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single-particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.
In this paper, the Lorentz transformations of entangled Bell states with general momentum not necessarily orthogonal to the boost direction and spin are studied. We extend quantum correlations and Bell's inequality to the relativistic regime by considering normalized relativistic observables. It is shown that quantum information along the direction perpendicular to the boost is eventually lost, and Bell's inequality is not always violated for entangled states in special relativity. This could impose restrictions on certain quantum information processing, such as quantum cryptography using massive particles.
Spin-orbit coupling makes the mean position of a spinning particle transform anomalously under a Lorentz boost, with an additional shift known as a side jump. In this work, we develop a trace method to derive an exact but compact transformation kernel for the probability center of a general three-dimensional Dirac wave packet under an arbitrary finite Lorentz boost. The expression includes the physics of finite momentum width, spin-momentum correlations, and coherent phase structure without a semi-classical expansion. We identify texture-dependent side jumps even at fixed initial mean momentum, spin, and orbital angular momentum, as well as a non-monotonic mass dependence for Gaussian packets at fixed momentum width. The explicit kernels further resolve the spin and orbital angular-momentum contributions for massive and massless fermions, providing a kinematic basis for covariant particle-based spin transport that retains wave-packet information.