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arXiv:2112.04254 · Physical review. D/Physical review. D.

Entanglement entropy and monotones in scattering process

Jinbo Fan, Gao-Ming Deng, Xi-Jun Ren

Published Dec 23, 2021• 12 citations
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Abstract

In this paper, we study the entanglement property of a four-particle system. In this system, two initially entangled electrons A and C are scattered by two uncorrelated positrons B and D, respectively. We calculate the entanglements among the particles both before and after the double QED scattering ($AB\ensuremath{\rightarrow}AB,CD\ensuremath{\rightarrow}CD$). We find that the change of entanglement entropy between subsystems A and B during the scattering process is proportional to the total cross section, ${\ensuremath{\sigma}}_{\mathrm{tot}}={\ensuremath{\sigma}}_{AB}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{CD}$. Even though there is no direct interaction between subsystems A and C (or B and D), the scattering process induces entanglement change among them which is also proportional to ${\ensuremath{\sigma}}_{\mathrm{tot}}$. This result shows some kind of entanglement sharing property in multipartite system. In order to further investigate the entanglement sharing, we calculate the entanglement monotones that quantify the genuine multipartite entanglement in a multipartite system. For our chosen scattering process, ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$, however, we find that the outgoing state is a W-type four-partite entangled state that has no genuine four-partite entanglement.

DOI
10.1103/physrevd.104.116021

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In this paper, we study the entanglement property of a four-particle system. In this system, two initially entangled electrons A and C are scattered by two uncorrelated positrons B and D, respectively. We calculate the entanglements among the particles both before and after the double QED scattering (AB\ensuremath→AB,CD\ensuremath→CDAB\ensuremath{\rightarrow}AB,CD\ensuremath{\rightarrow}CDAB\ensuremath→AB,CD\ensuremath→CD). We find that the change of entanglement entropy between subsystems A and B during the scattering process is proportional to the total cross section, \ensuremathσtot=\ensuremathσAB\ifmmode×\else\texttimes\fi\ensuremathσCD{\ensuremath{\sigma}}_{\mathrm{tot}}={\ensuremath{\sigma}}_{AB}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\sigma}}_{CD}\ensuremathσtot​=\ensuremathσAB​\ifmmode×\else\texttimes\fi\ensuremathσCD​. Even though there is no direct interaction between subsystems A and C (or B and D), the scattering process induces entanglement change among them which is also proportional to \ensuremathσtot{\ensuremath{\sigma}}_{\mathrm{tot}}\ensuremathσtot​. This result shows some kind of entanglement sharing property in multipartite system. In order to further investigate the entanglement sharing, we calculate the entanglement monotones that quantify the genuine multipartite entanglement in a multipartite system. For our chosen scattering process, e+e\ensuremath−\ensuremath→\ensuremathμ+\ensuremathμ\ensuremath−{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}e+e\ensuremath−\ensuremath→\ensuremathμ+\ensuremathμ\ensuremath−, however, we find that the outgoing state is a W-type four-partite entangled state that has no genuine four-partite entanglement.
We study the behavior of entanglement between different degrees of freedom of scattering fermions, based on an exemplary QED scattering process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. The variation of entanglement entropy between two fermions from an initial state to the final state was computed, with respect to different entanglement between the ingoing particles. This variation of entanglement entropy is found to be proportional to an area quantity, the total cross section. We also study the spin-momentum and helicity-momentum entanglements within one particle in the aforementioned scattering process. The calculations of the relevant variations of mutual information in the same inertial frame reveals that, for a maximally entangled initial state, the scattering between the particles does not affect the degree of both of these entanglements of one particle in the final state. It is also found that the increasing degree of entanglement between two ingoing particles would restrict the generation of entanglement between spin (helicity) and momentum of one outgoing particle. And the entanglement between spin and momentum within one particle in the final state is shown to always be stronger than that for helicity-momentum for a general initial entanglement state, implying significantly distinct properties of entanglement for the helicity and spin perceived by an inertial observer.
We considered a QED scattering ($AB\ensuremath{\rightarrow}AB$), in which BBB is initially entangled with a third particle (CCC) that does not participate directly in the scattering. The effect of the scattering over CCC's final state was evaluated and we noted coherence (off-diagonal) terms were created, which led to non-null values for $⟨{\ensuremath{\sigma}}_{x}⟩$ and $⟨{\ensuremath{\sigma}}_{y}⟩$ that are, in principle, measurable in a Stern-Gerlach apparatus. We chose a particular QED scattering (${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$) and found that $⟨{\ensuremath{\sigma}}_{x}⟩$ and $⟨{\ensuremath{\sigma}}_{y}⟩$ are proportional to the total cross section (${\ensuremath{\sigma}}_{\text{total}}$) of the ABABAB scattering, besides being maximal if BCBCBC's initial state is taken as a Bell basis. Furthermore, we calculated the initial and final mutual information IAC{I}_{AC}IAC​ and IBC{I}_{BC}IBC​, and noticed an increase (decrease) in IAC{I}_{AC}IAC​ (IBC{I}_{BC}IBC​), which indicates that, after ABABAB interact, the total amount of correlations (quantum+classical\mathrm{quantum}+\mathrm{classical}quantum+classical) is distributed among the 3 subsystems.
We study the entanglement entropy between the two outgoing particles in an elastic scattering process. It is formulated within an S-matrix formalism using the partial wave expansion of two-body states, which plays a significant role in our computation. As a result, we obtain a novel formula that expresses the entanglement entropy in a high energy scattering by the use of physical observables, namely the elastic and total cross sections and a physical bound on the impact parameter range, related to the elastic differential cross-section.
We consider a general quantum field relativistic scattering involving two half-spin fermions, AAA and BBB, which are initially entangled with another fermion CCC that does not participate in the scattering dynamics. We construct general expressions for the reduced spin matrices for the out-state considering a general tripartite spin-entangled state. In particular we study an inelastic QED process at tree-level, namely ${e}^{\ensuremath{-}}{e}^{+}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}$ and a half-spin fermion CCC as a spectator particle which can be entangled to the ABABAB system in the following ways: W state, GHZ state, $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩\ensuremath{\bigotimes}|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}{⟩}_{\mathrm{BC}}$, and $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩\ensuremath{\bigotimes}|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}{⟩}_{\mathrm{BC}}$, where ${|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}⟩,|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}⟩}$ are the Bell basis states and $|{\mathrm{A}}^{\ensuremath{\alpha}}⟩$ is a spin superposition state of system AAA. We calculate the von Neumann entropy variation before and after the scattering for the particle CCC and show that spin measurements in CCC contain numerical information about the total cross section of the process. We compare the initial states W and GHZ as well as study the role played by the parameter $\ensuremath{\alpha}$ in the evaluation of the entropy variations and the cross section encoded in the spectator particle.
From the perspective of quantum information science, we investigate tree-level Bhabha scattering between an incident electron AAA and a positron BBB, where BBB is initially entangled with a spectator electron CCC, which does not participate in the scattering interaction. We find that the quantum electrodynamics (QED) scattering between AAA and BBB can drive the global ABCABCABC system into a genuine tripartite entangled (GTE) state. Using four canonical tripartite entanglement metrics, we systematically characterize and quantify the GTE of the composite system, and demonstrate that the scattering momentum of the AAA-BBB pair and the initial BBB-CCC entanglement are the key resources governing GTE generation. We further analyze the monogamy of quantum correlations, which imposes fundamental constraints on the shareability of quantum resources in multipartite systems. Specifically, we systematically study the monogamy relations for the squared entanglement of formation and squared quantum discord in our scattering model, and find that monogamy constraints are markedly relaxed in the non-relativistic regime, enabling enhanced shareability of quantum correlations across the three particles. This work uncovers novel quantum correlation properties of fundamental QED scattering processes, and provides direct theoretical guidance for the development of QED-based quantum information processing protocols.
We construct the entanglement entropy of a general scattering process between one particle species (denoted as AAA) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle AAA's Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle AAA, SAS_ASA​, in terms of scattering probabilities and differential cross sections, assuming no more than one particle AAA is produced in the final state. SAS_ASA​ contains the information of both the total scattering probabilities (or cross sections) and the differential distributions --- the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution (S∣i⟩S|i\rangleS∣i⟩) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle AAA is subleading. We also obtain the n≥2n\geq2n≥2 Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.