Soft photon radiation and entanglement

Abstract

We study the entanglement between soft and hard particles produced in generic scattering processes in QED. The reduced density matrix for the hard particles, obtained via tracing over the entire spectrum of soft photons, is shown to have a large eigenvalue, which governs the behavior of the Renyi entropies and of the non-analytic part of the entanglement entropy at low orders in perturbation theory. The leading perturbative entanglement entropy is logarithmically IR divergent. The coefficient of the IR divergence exhibits certain universality properties, irrespectively of the dressing of the asymptotic charged particles and the detailed properties of the initial state. In a certain kinematical limit, the coefficient is proportional to the cusp anomalous dimension in QED. For Fock basis computations associated with two-electron scattering, we derive an exact expression for the large eigenvalue of the density matrix in terms of hard scattering amplitudes, which is valid at any finite order in perturbation theory. As a result, the IR logarithmic divergences appearing in the expressions for the Renyi and entanglement entropies persist at any finite order of the perturbative expansion. To all orders, however, the IR logarithmic divergences exponentiate, rendering the large eigenvalue of the density matrix IR finite. The all-orders Renyi entropies (per unit time, per particle flux), which are shown to be proportional to the total inclusive cross-section in the initial state, are also free of IR divergences. The entanglement entropy, on the other hand, retains non-analytic, logarithmic behavior with respect to the size of the box (which provides the IR cutoff) even to all orders in perturbation theory.

Scattering, IR dynamics and entanglement

“…As it was shown in [10,11], the leading entanglement entropy is independent of the Faddeev-Kulish dressing function.…”

“…The variation matrix admits a perturbative expansion in terms of the coupling constant at fixed cutoff scale , which is set by the size of the box. Unitarity ensures that Tr = 0, as expected [10,11].…”

“…Soft particles with energy below some IR energy scale escape detection, and we would like to quantify the information carried by these states. In particular, we will study the behavior of the entanglement entropy as a function of and , the IR cutoff of the theory [4][5][6][7][8][9][10][11]. We focus on scattering processes in perturbative QED, but the results are expected to generilize to perturbative gravity.…”

“…The non-diagonal elements contain IR divergences at any finite order in perturbation theory. To all orders, however, these divergences exponentiate leading to the (faster) vanishing of the non-diagonal elements as → 0, and to decoherence [4][5][6][7][8][9][10][11]. For example, the coefficient of the | | = | | term in the reduced density matrix is the Bloch -Nordsieck rate associated with the inclusive rate → + any number of photons with ≤ :…”

“…These hard and soft particles are highly correlated. Therefore, it is important to understand the entanglement between them, and to quantify the information carried by the soft particles [4][5][6][7][8][9][10][11].…”