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arXiv:0810.0922

Soft-Photons and the entanglement of charged qubtis

Eduardo Martín-Martínez

Published Oct 6, 2008• 0 citations
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Abstract

In this work, I consider the effect of the unavoidable presence of soft-photons on the spin entanglement of charged qubits, arguing that the spin entanglement is not modified if we consider the effects of the infrared structure of QED on it.
DOI
10.48550/arxiv.0810.0922

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We consider soft-photon effects (ir structure of QED) on the construction of physical qubits. Soft photons appear when we build charged qubits from the asymptotic states of QED. This construction is necessary in order to include the effect of soft photons on entanglement measures. The nonexistence of free charged particles (due to the long range of QED interactions) leads us to question the sense of the very concept of free charged qubit. In this work, using the ``dressing'' formalism, we build physical charged qubits from dressed fields which have the correct asymptotic behavior, are gauge invariant, have propagators with a particle pole structure, and are free from infrared divergences. Finally, we discuss the impact of the soft corrections on the entanglement measures.
We consider scattering of Faddeev-Kulish electrons in QED and study the entanglement between the hard and soft particles in the final state at the perturbative level. The soft photon spectrum naturally splits into two parts: (i) soft photons with energies less than a characteristic infrared scale Ed{E}_{d}Ed​ present in the clouds accompanying the asymptotic charged particles, and (ii) sufficiently low energy photons with energies greater than Ed{E}_{d}Ed​, comprising the soft part of the emitted radiation. We construct the density matrix associated with tracing over the radiative soft photons and calculate the entanglement entropy perturbatively. We find that the entanglement entropy is free of any infrared divergences order by order in perturbation theory. On the other hand, infrared divergences in the perturbative expansion for the entanglement entropy appear upon tracing over the entire spectrum of soft photons, including those in the clouds. To leading order the entanglement entropy is set by the square of the Fock basis amplitude for real single soft photon emission, which leads to a logarithmic infrared divergence when integrated over the photon momentum. We argue that the infrared divergences in the entanglement entropy (per particle flux per unit time) in this latter case persist to all orders in perturbation theory in the infinite volume limit.
I formulate unresolved soft-photon sector of QED as open quantum system. Resolved charged particles and hard photons form the system, unresolved soft photons form the environment, and basic object is the reduced density matrix. A resolved outcome fff of multiple hard particles and photons has probability Pf(i)=∑n∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff∣i⟩P_{f}(i) =\sum_n|\langle f,n|S|i,0\rangle|^2=\langle i|F_f|i\ranglePf​(i)=∑n​∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff​∣i⟩, with Kraus operators Kn=soft⟨n∣S∣0⟩softK_n= {}_{\rm soft} \langle n|S|0 \rangle_{\rm soft}Kn​=soft​⟨n∣S∣0⟩soft​ and effect Ff=∑nKn†ΠfKnF_f=\sum_nK_n^\daggerΠ_fK_nFf​=∑n​Kn†​Πf​Kn​. The SK formulation places unresolved virtual and real terms in one doubled-contour expansion. At one loop they carry the same on-shell eikonal kernel with opposite signs. This elegantly organizes the QED probability: for the same observable, perturbative order, diagrams, and phase space, the OQS gives the same infrared-finite terms as the full-QED. The soft-photon evolution is a unitary coherent-state displacement driven by the scattering current. The equal-history identity of influence functional exactly normalizes this soft evolution; together with the soft-photon theorem it removes the IR divergent leading-soft factor from inclusive probability. I also derive explicit leading-soft QED realization of scale-parametrized Lindblad evolution on a fixed hard-branch space. Tracing an infinitesimal soft-photon shell produces diagonal jump operators whose entries are fixed by the corresponding eikonal emission amplitudes. The finite-shell map is a completely positive unital Schur channel and, in the sharp scale-invariant leading-soft regime, a dephasing semigroup of a completely-positive-divisible scale flow. The resulting logarithmic visibility slope and monotonic purity loss are off-diagonal predictions of the reduced-state description. The same controlled-displacement dilation gives the Sudakov probability, Poisson soft-photon multiplicities, and the bremsstrahlung number spectrum.