Krylov complexity and gluon cascades in the high energy limit

We point out an interesting connection between the mathematical framework of the Krylov basis, which is used to quantify quantum complexity, and the entanglement entropy in high energy QCD. In particular, we observe that the cascade equation of the dipole model is equivalent to the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>L</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mo>,</a:mo><a:mi>R</a:mi><a:mo stretchy="false">)</a:mo></a:math> Schrödinger equation in the Krylov basis. Consequently, the Krylov complexity corresponds to the average distribution of partons and the Krylov entropy is the counterpart of the entanglement entropy computations of [D. E. Kharzeev and E. M. Levin, .]. Our work not only brings new tools for exploring quantum information and complexity in QCD, but also gives hope for experimental tests of some of the recent, physical probes of quantum complexity. Published by the American Physical Society 2024