Zipping many-body quantum states: a scalable approach to diagonal entropy

TL;DR

This paper introduces a scalable method using Lempel-Ziv compression to estimate diagonal entropy in many-body quantum states, enabling efficient analysis of phase transitions and symmetry breaking with polynomial resources.

Contribution

It demonstrates that Lempel-Ziv compression can accurately estimate diagonal entropy density in quantum many-body systems, offering a scalable alternative to exponential tomographic methods.

Findings

Compression accurately recovers entropy density in various models.

Diagonal entropy density exhibits specific scaling near critical points.

Method requires only polynomially many images for reliable estimates.

Abstract

The outcomes of projective measurements on a quantum many-body system in a chosen basis are inherently probabilistic. The Shannon entropy of this probability distribution (the "diagonal entropy") often reveals universal features, such as the existence of a quantum phase transition. A brute-force tomographic approach to estimating this entropy scales exponentially with the system size. Here, we explore using the Lempel-Ziv lossless image compression algorithm as an efficient, scalable alternative, readily implementable in a quantum gas microscope or programmable quantum devices. We test this approach on several examples: one-dimensional quantum Ising model, and two-dimensional states that display conventional symmetry breaking due to quantum fluctuations, or strong-to-weak symmetry-breaking due to local decoherence. We also employ the diagonal mixed state to put constraints on the phase boundaries of our models. In all examples, the compression method accurately recovers the entropy density while requiring at most polynomially many images. We also analyze the singular part of the diagonal entropy density using renormalization group on a replicated action. In the 1+1-D quantum Ising model, we find that it scales as $|t| \log|t|$, where $t$ is the deviation from the critical point, while in a 2+1-D state with amplitudes proportional to the Boltzmann weight of the 2D Ising model, it follows a $t^2 \log|t|$ scaling.