Stabilizer R\'enyi Entropy for Translation-Invariant Matrix Product States

TL;DR

This paper introduces a new method to compute the stabilizer Rnyi entropy (SRE) in translation-invariant matrix product states, revealing a fundamental link between quantum magic and entanglement in many-body systems.

Contribution

It derives exact SRE expressions for certain states, develops a stable numerical algorithm (bond-DMRG), and uncovers the relationship between magic and entanglement in quantum systems.

Findings

High-precision SRE densities for the Ising model ground state

Non-local SRE density is bounded by a universal entanglement function

Two-site mutual SRE vanishes asymptotically in injective MPS

Abstract

Magic, capturing the deviation of a quantum state from the stabilizer formalism, is a key resource underpinning the quantum advantage. The recently introduced stabilizer R\'enyi entropy (SRE) offers a tractable measure of magic, avoiding the complexity of conventional methods. We study SRE in translation-invariant matrix product states (MPS), deriving exact expressions for representative states and introducing a numerically stable algorithm, named bond-DMRG, to compute the SRE density in infinite systems. Applying this method, we obtain high-precision SRE densities for the ground state of the one-dimensional Ising model. We also analyze non-local SRE density, showing it is bounded by a universal function of entanglement entropy, and further prove that two-site mutual SRE vanishes asymptotically in injective MPS. Our work not only introduces a powerful method for extracting the SRE density in quantum many-body systems, but also numerically reveals a fundamental connection between magic and entanglement, thereby paving the way for deeper theoretical investigations into their interplay.