Pseudo complexity of purification for free scalar field theories

TL;DR

This paper calculates the pseudo complexity of purification for free scalar field theories with arbitrary dynamical exponents, analyzing how it varies with different parameters and comparing it to the complexity of the original reduced density matrices.

Contribution

It introduces a method to compute the pseudo complexity of purification for reduced transition matrices in free scalar field theories with arbitrary dynamical exponents.

Findings

Complexity varies with theory parameters.

Comparison shows differences between transition matrix and reduced density matrices.

Method applicable to systems with small degrees of freedom.

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.