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arXiv:1907.10466 · Physical review. D/Physical review. D.

Measuring QED cross sections via entanglement

Jonas B. Araujo, B. Hiller, I. G. da Paz, Manoel M. Ferreira, Marcos Sampaio, H. A. S. Costa

Published Nov 19, 2019• 19 citations
arXivHTMLPDF

Abstract

We considered a QED scattering ($AB\ensuremath{\rightarrow}AB$), in which $B$ is initially entangled with a third particle ($C$) that does not participate directly in the scattering. The effect of the scattering over $C$'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 $AB$ scattering, besides being maximal if $BC$'s initial state is taken as a Bell basis. Furthermore, we calculated the initial and final mutual information ${I}_{AC}$ and ${I}_{BC}$, and noticed an increase (decrease) in ${I}_{AC}$ (${I}_{BC}$), which indicates that, after $AB$ interact, the total amount of correlations ($\mathrm{quantum}+\mathrm{classical}$) is distributed among the 3 subsystems.

DOI
10.1103/physrevd.100.105018

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We considered a QED scattering (AB\ensuremath→ABAB\ensuremath{\rightarrow}ABAB\ensuremath→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 ⟨\ensuremathσx⟩⟨{\ensuremath{\sigma}}_{x}⟩⟨\ensuremathσx​⟩ and ⟨\ensuremathσy⟩⟨{\ensuremath{\sigma}}_{y}⟩⟨\ensuremathσy​⟩ that are, in principle, measurable in a Stern-Gerlach apparatus. We chose a particular QED scattering (e+e\ensuremath−\ensuremath→\ensuremathμ+\ensuremathμ\ensuremath−{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}e+e\ensuremath−\ensuremath→\ensuremathμ+\ensuremathμ\ensuremath−) and found that ⟨\ensuremathσx⟩⟨{\ensuremath{\sigma}}_{x}⟩⟨\ensuremathσx​⟩ and ⟨\ensuremathσy⟩⟨{\ensuremath{\sigma}}_{y}⟩⟨\ensuremathσy​⟩ are proportional to the total cross section (\ensuremathσtotal{\ensuremath{\sigma}}_{\text{total}}\ensuremathσtotal​) 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.
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.
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.
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.
We analyze how entanglement is generated and distributed in a Bhabha scattering process (e−e+→e−e+)(e^-e^+\rightarrow e^-e^+)(e−e+→e−e+) at tree level. In our setup an electron AAA scatters with a positron BBB, which is initially entangled with another electron CCC (spectator), that does not participate directly to the process. We find that the QED scattering generates and distributes entanglement in a non-trivial way among the three particles: the correlations in the output channels ABABAB, ACACAC and BCBCBC are studied in detail as functions of the scattering parameters and of the initial entanglement weight. Although derived in a specific case, our results exhibit some general features of other similar QED scattering processes, for which the extension of the present analysis is straightforward.
We analyze how entanglement is generated and distributed in a Bhabha scattering process $({e}^{\ensuremath{-}}{e}^{+}\ensuremath{\rightarrow}{e}^{\ensuremath{-}}{e}^{+})$ at tree level. In our setup an electron AAA scatters with a positron BBB, which is initially entangled with another electron CCC (spectator), that does not participate directly to the process. We find that the QED scattering generates and distributes entanglement in a nontrivial way among the three particles; the correlations in the output channels ABABAB, ACACAC, and BCBCBC are studied in detail as functions of the scattering parameters and of the initial entanglement weight. Although derived in a specific case, our results exhibit some general features of other similar QED scattering processes, for which the extension of the present analysis is straightforward.
Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.