Understanding and utilizing the inner bonds of process tensors

TL;DR

This paper reveals that the inner bonds of process tensor matrix product operators (PT-MPOs) have a concrete physical meaning, representing environment subspaces influencing open quantum system dynamics, and demonstrates their practical extraction and application.

Contribution

It establishes a physical interpretation of PT-MPO inner bonds and introduces methods to extract environment observables from these bonds, enhancing understanding and utility.

Findings

Inner bonds correspond to environment subspaces affecting system dynamics

Methods to extract environment observables from PT-MPO bonds are demonstrated

Different PT-MPO algorithms compress environments to similar subspaces

Abstract

Process tensor matrix product operators (PT-MPOs) enable numerically exact simulations for an unprecedentedly broad range of open quantum systems. By representing environment influences in MPO form, they can be efficiently compressed using established algorithms. The dimensions of inner bonds of the compressed PT-MPO may be viewed as an indicator of the complexity of the environment. Here, we show that the inner bonds themselves, not only their dimensions, have a concrete physical meaning: They represent the subspace of the full environment Liouville space which hosts environment excitations that may influence the subsequent open quantum systems dynamics the most. This connection can be expressed in terms of lossy linear transformations, whose pseudoinverses facilitate the extraction of environment observables. We demonstrate this by extracting the environment spin of a central spin problem, the current through a quantum system coupled to two leads, the number of photons emitted from quantum emitters into a structured environment, and the distribution of the total absorbed energy in a driven non-Markovian quantum system into system, environment, and interaction energy terms. Numerical tests further indicate that different PT-MPO algorithms compress environments to similar subspaces. Thus, the physical interpretation of inner bonds of PT-MPOs both provides a conceptional understanding and it enables new practical applications.