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International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions (Hard Probes)

Entanglement, partial set of measurements, and diagonality of the density matrix in the parton model

Vladimir V. Skokov

Published Sep 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.22323/1.387.0180

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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.
We calculate the entanglement entropy of a model proton wave function in coordinate space by integrating out degrees of freedom outside a small circular region A‾\overline{A}A of radius LLL, where LLL is much smaller than the size of the proton. The wave function provides a nonperturbative distribution of three valence quarks. In addition, we include the perturbative emission of a single gluon and calculate the entanglement entropy of gluons in A‾\overline{A}A. For both quarks and gluons, we obtain the same simple result: ${S}_{E}=\ensuremath{-}\ensuremath{\int}\frac{dx}{\mathrm{\ensuremath{\Delta}}x}{N}_{{L}^{2}}(x)\mathrm{log}[{N}_{{a}^{2}}(x)]$, where aaa is the UV cutoff in coordinate space and $\mathrm{\ensuremath{\Delta}}x$ is the longitudinal resolution scale. Here NS(x){N}_{S}(x)NS​(x) is the number of partons (of the appropriate species) with longitudinal momentum fraction xxx inside an area SSS. It is related to the standard parton distribution function by ${N}_{S}(x)=\frac{S}{{A}_{p}}\mathrm{\ensuremath{\Delta}}xF(x)$, where Ap{A}_{p}Ap​ denotes the transverse area of the proton.
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.
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Relying on previous results, that link entanglement entropy and parton distribution functions in deep inelastic scattering, and focusing on the small Bjorken scaling region we present here indications that gluon shadowing might indeed be explained as due to a depletion of the entanglement entropy, between observed and unobserved degrees of freedom, per nucleon within a nucleus, with respect to the free nucleon. We apply to gluon shadowing the general Page approach to the calculation of the entanglement entropy in bipartite systems, giving physical motivations of the results.
The aim of this work is to compute the entanglement entropy of real and virtual particles by rewriting the generating functional of ${\ensuremath{\phi}}^{4}$ theory as a mean value between states and observables defined through the correlation functions. Then the von Neumann definition of entropy can be applied to these quantum states and in particular, for the partial traces taken over the internal or external degrees of freedom. This procedure can be done for each order in the perturbation expansion showing that the entanglement entropy for real and virtual particles behaves as ln(m0)\mathrm{ln}({m}_{0})ln(m0​). In particular, entanglement entropy is computed at first order for the correlation function of two external points showing that mutual information is identical to the external entropy and that conditional entropies are negative for all the domain of m0{m}_{0}m0​. In turn, from the definition of the quantum states, it is possible to obtain general relations between total traces between different quantum states of a ${\ensuremath{\phi}}^{r}$ theory. Finally, discussion about the possibility of taking partial traces over external degrees of freedom is considered, which implies the introduction of some observables that measure space-time points where an interaction occurs.