Pseudo complexity of purification for free scalar field theories

Abstract

We compute the pseudo complexity of purification corresponding to the reduced transition matrices for free scalar field theories with an arbitrary dynamical exponent. We plot the behaviour of complexity with various parameters of the theory under study and compare it with the complexity of purification of the reduced density matrices of the two states $|\psi_1\rangle$ and $|\psi_2\rangle$ that constitute the transition matrix. We first find the transition matrix by reducing to a small number ($1$ and $2$) of degrees of freedom in lattice from a lattice system with many lattice points and then purify it by doubling the degrees of freedom ($2$ and $4$ respectively) for this reduced system. This is a primary step towards the natural extension to the idea of the complexity of purification for reduced density matrices relevant for the studies related to post-selection.