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arXiv:1806.01089 · Physics Letters B

Entanglement entropy, entropy production and time evolution in high energy QCD

Alex Kovner, Michael Lublinsky, Mirko Serino

Published Oct 25, 2018• 43 citations
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Abstract

Working in the framework of the Color Glass Condensate effective theory of high energy QCD , we revisit the momentum space entanglement entropy of the soft gluons produced in high energy dilute-dense collisions. We extend the work of [1] by considering entropy produced in a single event. This entropy arises due to decoherence of eigenstates with different energies during the time evolution after the collisions with the target. We define it rigorously as the entanglement entropy of the produced system with the experimental apparatus . We compute the time dependent single event entropy in the limit of weak projectile field. Further we compute the entropy for the ensemble of events defined by the McLerran–Venugopalan model for the projectile wave function. Interestingly the entropy of the ensemble has a much weaker time dependence than the entropy in any single event. We attribute this feature to the so called monogamy of entanglement.

DOI
10.1016/j.physletb.2018.10.043

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Working in the framework of the Color Glass Condensate effective theory of high energy QCD , we revisit the momentum space entanglement entropy of the soft gluons produced in high energy dilute-dense collisions. We extend the work of [1] by considering entropy produced in a single event. This entropy arises due to decoherence of eigenstates with different energies during the time evolution after the collisions with the target. We define it rigorously as the entanglement entropy of the produced system with the experimental apparatus . We compute the time dependent single event entropy in the limit of weak projectile field. Further we compute the entropy for the ensemble of events defined by the McLerran–Venugopalan model for the projectile wave function. Interestingly the entropy of the ensemble has a much weaker time dependence than the entropy in any single event. We attribute this feature to the so called monogamy of entanglement.
We consider the following conundrum which attracted attention recently: on one hand the proton as a quantum object is in a pure state and is described by a completely coherent wave function with zero entropy. On the other hand in high energy experiments (DIS) when probed by a small external probe, it behaves like an incoherent ensemble of (quasi-free) partons. In this talk, we define the “entropy of ignorance,” which quantifies the entropy associated with the ability to perform only a partial set of measurements on a quantum system. For a parton model, the entropy of ignorance is equal to a Boltzmann entropy of a classical system of partons. We analyze a calculable model used for describing low x gluons in the Color Glass Condensate approach, which has similarities with the parton model of QCD. In this model we calculate the entropy of ignorance in the particle number basis as well as the entanglement entropy of the observable degrees of freedom. We find that the two are similar at high momenta, but differ by a factor of order unity at low momenta. This holds for the Renyi as well as von Neumann entropies. We conclude that entanglement does not seem to play an important role in the context of the parton model.
We define the ``entropy of ignorance'' which quantifies the entropy associated with ability to perform only a partial set of measurement on a quantum system. For the parton model the entropy of ignorance is equal to the Boltzmann entropy of a classical system of partons. We analyze a calculable model used for describing low x gluons in Color Glass Condensate approach, which has similarities with the parton model of QCD. In this model we calculate the entropy of ignorance in the particle number basis as well as the entanglement entropy of the observable degrees of freedom. We find that the two are similar at high momenta, but differ by a factor of order unity at low momenta. This holds for the Renyi as well as von Neumann entropies. We conclude that the entanglement does not seem to play an important role in the context of the parton model.
Abstract We compute the entropy per particle, S / N , produced by the decoherence of a coherent state interacting with an environment, using an analytical open quantum system approach. The coherent state considered is characterized by occupation numbers borrowed from the Glasma fields produced in the early stages of high-energy nuclear collisions. The environment is modeled as the vacuum, and decoherence arises from the interaction of the state with vacuum fluctuations. We describe the system–environment interaction via a phase-damping model, which represents continuous measurements on the system without altering its energy or particle number. Starting from the occupation numbers typical of the Glasma in high-energy proton–nucleus and nucleus–nucleus collisions, we find that the final S / N after decoherence is lower than that of a two-dimensional thermal bath of ultra-relativistic gluons, except for proton–nucleus collisions at small values of gμg\mugμ g μ . Our results indicate that quantum decoherence alone does not generate sufficient entropy to transform the initial coherent state into a thermalized gluon bath.
Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in Lipatov's spin chain and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal{S}(t)=1/3\mathrm{ln}(t/\ensuremath{\tau})$ with $\ensuremath{\tau}=1/m$ for $1/m\ensuremath{\le}t\ensuremath{\le}(mx{)}^{\ensuremath{-}1}$, where mmm is the proton mass and xxx is the Bjorken xxx. The central charge ccc of Lipatov's spin chain is determined here to be c=1c=1c=1; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small xxx as $xG(x)\ensuremath{\sim}1/{x}^{1/3}$.
We propose a new prescription for evaluating a von Neumann entropy in the initial stage of high-energy heavy-ion collisions utilizing the time evolution of classical Yang-Mills (CYM) field: The von Neumann entropy is computed for the quantum coherent states constructed so as to give the classical gluon fields as the expectation values. The entropy is to be liberated when the complete decoherence is achieved. As a demonstration, the time evolution of the CYM dynamics is solved with an initial condition which mimics the Glasma state, though in a non-expanding geometry; the Glasma state is characterized by the longitudinal color-electric and -magnetic fields with gluon fields' fluctuations around it. We find that the initial longitudinal fluctuations of the fields play essential roles for the entropy production in two ways: First, the field fluctuations at t=0t=0t=0 themselves act as a source of the von Neumann entropy prepared before the time evolution. Second, the initial fluctuations triggers field instabilities, and hence the larger the strength of them, the more the entropy production at later time.
For quantum many-body systems with short-range correlations (SRCs), the intimate relationship between their magnitude, the behavior of the single-particle occupation probabilities at momenta larger than the Fermi momentum, and the entanglement entropy is a new qualitative aspect not studied and exploited yet. A large body of recent condensed matter studies indicates that the time evolution of the entanglement entropy describes the nonequilibrium dynamics of isolated and strongly interacting many-body systems, in a manner similar to the Boltzmann entropy, which is strictly defined for dilute and weakly interacting many-body systems. Both theoretical and experimental studies in nuclei and cold atomic gases have shown that the fermion momentum distribution has a generic behavior n(k)=C/k4n(k)=C/{k}^{4}n(k)=C/k4 at momenta larger than the Fermi momentum, due to the presence of SRCs, with approximately 20% of the particles having momenta larger than the Fermi momentum. The presence of the long momentum tails in the presence of SRCs changes the textbook relation between the single-particle kinetic energy and occupation probabilities, ${n}_{\text{mf}}(k)=1/{1+exp\ensuremath{\beta}[\ensuremath{\epsilon}(k)\ensuremath{-}\ensuremath{\mu}]}$ for momenta very different form the Fermi momentum, particularly for dynamics processes. SRCs induced high-momentum tails of the single-particle occupation probabilities increase the entanglement entropy of fermionic systems, which in its turn affects the dynamics of many nuclear reactions, such as heavy-ion collisions and nuclear fission.