Spread and circuit complexity as a measure of particle content and phase space fluctuations

TL;DR

This paper explores how quantum complexity measures relate to particle content and phase space fluctuations in a time-dependent harmonic oscillator, providing insights relevant to quantum field theory in curved spacetime.

Contribution

It establishes a direct link between complexity measures and physical quantities like particle number and phase space variances, with potential universal relations between complexity types.

Findings

Complexity is determined by the mean number of quanta and its rate of change.

Complexity growth correlates with position and momentum variances.

A universal relation between spread and circuit complexity is proposed.

Abstract

In this work, we investigate the relation between different notions of quantum complexity, namely, circuit and spread complexity and physically meaningful quantities such as the particle content of the quantum state and the variances of position and momentum operators. Using a harmonic oscillator with time-dependent mass and frequency as a toy model, we show that both circuit and spread complexity at any instant is determined by the mean number of quanta and its rate of change. Furthermore, both complexity and its growth are directly linked to the variances of the position and momentum operators, providing a clear physical interpretation of complexity in terms of the state's excitation and phase-space fluctuation. Although the analysis is carried out for a single time-dependent oscillator, the results have direct relevance for quantum field theory in curved backgrounds, where individual field modes effectively behave as time-dependent oscillators. This offers new insights into how quantum complexity encodes particle production and phase space fluctuations in non-holographic systems. Finally, we establish a precise and potentially universal relation between spread and circuit complexity for the time evolved state suggesting deeper connections between different complexity measures in the context of field theories on curved backgrounds.