Pseudo complexity of purification for free scalar field theories

TL;DR

The paper defines a pseudo complexity of purification for reduced transition matrices $\tau_A^{1|2}$ derived from two pure states $|\psi_1\rangle$, $|\psi_2\rangle$ in free scalar/Lifshitz field theories in $(1+1)$D. It builds Gaussian purifications in an enlarged Hilbert space and minimizes a Nielsen-type complexity functional to obtain $\mathcal{C}_P$, then compares with the complexities of purifications of the individual reduced density matrices. Numerical results reveal nontrivial, state-dependent behavior of $\mathcal{C}_P$ with theory parameters $(z,m)$ and subsystem size, including subadditivity $\Delta C(\tau_A^{1|2})\le 0$ and saturation at large reference frequency $\omega$, with saturation approaching the half-sum of the two purities. The work provides a practical framework for analyzing the complexity of operators arising from post-selection and suggests extensions to mode-by-mode purifications and open-system contexts, as well as possible holographic considerations.

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 $|ψ_1\rangle$ and $|ψ_2\rangle$ 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.