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arXiv:0812.4061 · Physical Review A

Physical qubits from charged particles: Infrared divergences in quantum information

Juan León, Eduardo Martín-Martínez

Published May 12, 2009• 2 citations
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Abstract

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.

DOI
10.1103/physreva.79.052309

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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.
The classical limit and the infrared divergence problem for a non-relativistic charged quantum particle interacting with the quantized electromagnetic field are analized. Overall three momentum conservation is taken into account. A unitary transformation associated to the coherent state corresponding to a particle surrounded by a cloud of soft photons is performed upon the hamiltonian and the particle-field states. The transformed state represnting a moving dressed quantum particle and its energy are given by the Brillouin-Wigner perturbation theory. It is shown formally that the quantum energy approaches the classical electromagnetic field at the Plank constant h goes to zero. Moreover,all Feynman diagrams contributing to this quantum energy are infrared finite, without needing add diagrams of the same order in the electric charge to obtain the infrared finiteness. Those properties justify the usefulness of the unitary transformation. The Compton effect in the forward direction is studied using dressed charged particle states after the unitary transformation has been performed. The quantum cross section approaches the classical limit (Thomson´s formula) as h ͢͢͢͢͢ 0, and the Feynman diagrams are free of infrared divergences.
We discuss information-theoretic properties of low-energy photons and gravitons in the S matrix. Given an incoming n-particle momentum eigenstate, we demonstrate that unobserved soft photons decohere nearly all outgoing momentum superpositions of charged particles, while the universality of gravity implies that soft gravitons decohere nearly all outgoing momentum superpositions of all the hard particles. Using this decoherence, we compute the entanglement entropy of the soft bosons and show that it is infrared-finite when the leading divergences are resummed in the manner of Bloch and Nordsieck.
In this thesis we discuss applications of quantum information theoretic concepts to quantum gravity and the low-energy regime of quantum field theories. The first part of this thesis is concerned with how quantum information spreads in four-dimensional scattering experiments for theories coupled to quantum electrodynamics or perturbative quantum gravity. In these cases, every scattering process is accompanied by the emission of an infinite number of soft photons or gravitons, which cause infrared divergences in the calculation of scattering probabilities. There are two methods to deal with IR divergences: the inclusive and dressed formalisms. We demonstrate that in the late-time limit, independent of the method, the hard outgoing particles are entangled with soft particles in such a way that the reduced density matrix of the hard particles is essentially completely decohered. Furthermore, we show that the inclusive formalism is ill-suited to describe scattering of wavepackets, requiring the use of the dressed formalism. We construct the Hilbert space for QED in the dressed formalism as a representation of the canonical commutation relations of the photon creation/annihilation algebra, and argue that it splits into superselection sectors which correspond to eigenspaces of the generators of large gauge transformations. In the second part of this thesis, we turn to applications of quantum information theoretic concepts in the AdS/CFT correspondence. In pure AdS, we find an explicit formula for the Ryu-Takayanagi (RT) surface for special subregions in the dual conformal field theory, whose entangling surface lie on a light cone. The explicit form of the RT surface is used to give a holographic proof of Markovicity of the CFT vacuum on a light cone. Relative entropy of a state on such special subregions is dual to a novel measure of energy associated with a timelike vector flow between the causal and entanglement wedge. Positivity and monotonicity of relative entropy imply positivity and monotonicity of this energy, which yields a consistency conditions for solutions to quantum gravity.
In quantum field theory, bare particles are dressed by a cloud of virtual particles to form physical particles. The virtual particles affect properties such as the mass and charge of the physical particles, and it is only these modified properties that can be measured in experiments, not the properties of the bare particles. The influence of virtual particles is prominent in the ultrastrong-coupling regime of cavity quantum electrodynamics (QED), which has recently been realised in several condensed-matter systems. In some of these systems, the effective interaction between atom-like transitions and the cavity photons can be switched on or off by external control pulses. This offers unprecedented possibilities for exploring quantum vacuum fluctuations and the relation between physical and bare particles. We consider a single three-level quantum system coupled to an optical resonator. Here we show that, by applying external electromagnetic pulses of suitable amplitude and frequency, each virtual photon dressing a physical excitation in cavity-QED systems can be converted into a physical observable photon, and back again. In this way, the hidden relationship between the bare and the physical excitations can be unravelled and becomes experimentally testable. The conversion between virtual and physical photons can be clearly pictured using Feynman diagrams with cut loops.