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

Physica Scripta

Entangled and disentangled multi-time quantum trajectories in relativistic systems

Yannick Noel Freitag, Julien Pinske, Jan Sperling

Published Oct 21, 2026• 0 citations
Publisher

Abstract

Abstract Quantum entanglement is a key resource for quantum technologies, including emerging ground-to-satellite quantum communication. In such a scenario, an important challenge to be overcome is to consider entanglement between two or more quantum particles in different inertial frames, potentially experiencing relativistic effects affecting quantum correlations. In this paper, we present a consistent framework that overcomes this challenge. To this end, we establish the notion of factorizable and entangled multi-time trajectories and derive a class of Euler-Lagrange equations under the constraint of a non-entangling behavior. Comparing this restricted evolution to the solutions of the unrestricted equations of motion allows us to investigate the trajectory-based entanglement of general systems. We solve our equations for interacting particles in a Klein-Gordon-type setting, thereby quantifying the dynamic and relativistic impact of entanglement in a self-consistent manner.

DOI
10.1088/1402-4896/aeb73a

Related papers

Entangled quantum trajectories in relativistic systems

Yannick Noel Freitag, Julien Pinske, Jan Sperling

Quantum entanglement is a key resource for quantum technologies, including emerging ground-to-satellite quantum communication. In such a scenario, an important challenge to be overcome is to consider entanglement between two or more quantum particles in different inertial frames, potentially experiencing relativistic effects affecting quantum correlations. In this paper, we present a consistent framework that overcomes this challenge. To this end, we establish the notion of factorizable and entangled multi-time trajectories and derive a class of Euler--Lagrange equations under the constraint of a non-entangling behavior. Comparing this restricted evolution to the solutions of the unrestricted equations of motion allows one to investigate the trajectory-based entanglement of general systems. We solve our equations for interacting particles in a Klein--Gordon-type setting, thereby quantifying the dynamic and relativistic impact of entanglement in a self-consistent manner.

arXiv:2410.05995•Oct 8, 2024•0 citations

Separable and Inseparable Quantum Trajectories

Jan Sperling, Ian A. Walmsley

The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schrödinger-type equations allow for comparing the trajectories in a separable configuration with the actual behavior of the system and, thereby, identifying inseparable and time-dependent quantum properties. As an example, we study bipartite discrete- and continuous-variable interacting systems. The generalization of our developed technique to multipartite scenarios is also provided.

arXiv:1708.04421•Physical Review Letters•Oct 24, 2017•9 citations

Trajectory-based Theory of Relativistic Quantum Particles

Bill Poirier

Recently, a self-contained trajectory-based formulation of non-relativistic quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4 (2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories that extremize a suitable action. Here, the trajectory-based approach is developed into a viable, generally covariant, relativistic quantum theory for single (spin-zero, massive) particles. Central to this development is the introduction of a new notion of global simultaneity for accelerated particles--together with basic postulates concerning probability conservation and causality. The latter postulate is found to be violated by the Klein-Gordon equation, leading to its well-known problems as a single-particle theory. Various examples are considered, including the time evolution of a relativistic Gaussian wavepacket.

arXiv:1208.6260•Aug 30, 2012•2 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

Restricted Monte Carlo wave-function method and Lindblad equation for identifying entangling open-quantum-system dynamics

Laura Ares, Julien Pinske, Benjamin Hinrichs +2 more

We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.

arXiv:2412.08735•Physical Review A•Dec 12, 2025•5 citations

On relativistic entanglement and localization of particles and on their comparison with the nonrelativistic theory

Horace W. Crater, Luca Lusanna

We make a critical comparison of relativistic and nonrelativistic classical and quantum mechanics of particles in inertial frames as well of the open problems in particle localization at both levels. The solution of the problems of the relativistic center-of-mass, of the clock synchronization convention needed to define relativistic 3-spaces and of the elimination of the relative times in the relativistic bound states leads to a description with a decoupled nonlocal (nonmeasurable) relativistic center-of-mass and with only relative variables for the particles (single particle subsystems do not exist). We analyze the implications for entanglement of this relativistic spatial nonseparability not existing in nonrelativistic entanglement. Then, we try to reconcile the two visions showing that also at the nonrelativistic level in real experiments only relative variables are measured with their directions determined by the effective mean classical trajectories of particle beams present in the experiment. The existing results about the nonrelativistic and relativistic localization of particles and atoms support the view that detectors only identify effective particles following this type of trajectories: these objects are the phenomenological emergent aspect of the notion of particle defined by means of the Fock spaces of quantum field theory.

arXiv:1306.6524•International Journal of Modern Physics A•Nov 11, 2014•23 citations
Abstract Quantum entanglement is a key resource for quantum technologies, including emerging ground-to-satellite quantum communication. In such a scenario, an important challenge to be overcome is to consider entanglement between two or more quantum particles in different inertial frames, potentially experiencing relativistic effects affecting quantum correlations. In this paper, we present a consistent framework that overcomes this challenge. To this end, we establish the notion of factorizable and entangled multi-time trajectories and derive a class of Euler-Lagrange equations under the constraint of a non-entangling behavior. Comparing this restricted evolution to the solutions of the unrestricted equations of motion allows us to investigate the trajectory-based entanglement of general systems. We solve our equations for interacting particles in a Klein-Gordon-type setting, thereby quantifying the dynamic and relativistic impact of entanglement in a self-consistent manner.
Quantum entanglement is a key resource for quantum technologies, including emerging ground-to-satellite quantum communication. In such a scenario, an important challenge to be overcome is to consider entanglement between two or more quantum particles in different inertial frames, potentially experiencing relativistic effects affecting quantum correlations. In this paper, we present a consistent framework that overcomes this challenge. To this end, we establish the notion of factorizable and entangled multi-time trajectories and derive a class of Euler--Lagrange equations under the constraint of a non-entangling behavior. Comparing this restricted evolution to the solutions of the unrestricted equations of motion allows one to investigate the trajectory-based entanglement of general systems. We solve our equations for interacting particles in a Klein--Gordon-type setting, thereby quantifying the dynamic and relativistic impact of entanglement in a self-consistent manner.
The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schrödinger-type equations allow for comparing the trajectories in a separable configuration with the actual behavior of the system and, thereby, identifying inseparable and time-dependent quantum properties. As an example, we study bipartite discrete- and continuous-variable interacting systems. The generalization of our developed technique to multipartite scenarios is also provided.
Recently, a self-contained trajectory-based formulation of non-relativistic quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4 (2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories that extremize a suitable action. Here, the trajectory-based approach is developed into a viable, generally covariant, relativistic quantum theory for single (spin-zero, massive) particles. Central to this development is the introduction of a new notion of global simultaneity for accelerated particles--together with basic postulates concerning probability conservation and causality. The latter postulate is found to be violated by the Klein-Gordon equation, leading to its well-known problems as a single-particle theory. Various examples are considered, including the time evolution of a relativistic Gaussian wavepacket.
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.
We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.
We make a critical comparison of relativistic and nonrelativistic classical and quantum mechanics of particles in inertial frames as well of the open problems in particle localization at both levels. The solution of the problems of the relativistic center-of-mass, of the clock synchronization convention needed to define relativistic 3-spaces and of the elimination of the relative times in the relativistic bound states leads to a description with a decoupled nonlocal (nonmeasurable) relativistic center-of-mass and with only relative variables for the particles (single particle subsystems do not exist). We analyze the implications for entanglement of this relativistic spatial nonseparability not existing in nonrelativistic entanglement. Then, we try to reconcile the two visions showing that also at the nonrelativistic level in real experiments only relative variables are measured with their directions determined by the effective mean classical trajectories of particle beams present in the experiment. The existing results about the nonrelativistic and relativistic localization of particles and atoms support the view that detectors only identify effective particles following this type of trajectories: these objects are the phenomenological emergent aspect of the notion of particle defined by means of the Fock spaces of quantum field theory.