Stabilizer Rényi entropy of the Affleck-Kennedy-Lieb-Tasaki ground state

Abstract

Nonstabilizerness, or quantum magic, is an important quantum computing resource that characterizes the complexity of state preparation on quantum hardware. In this work, we use a matrix product state (MPS) framework and a replica-based approach to quantify the nonstabilizerness of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model ground state as measured by the stabilizer Rényi entropy (SRE). This AKLT state is a spin-1 chain so we utilize the Heisenberg–Weyl operator framework to extend the concept of Pauli qubit strings to qutrits. Through the use of operator-twisted transfer matrices, we develop a replicated transfer operator with a dominant eigenvalue that establishes the SRE in the limit of long chains. Our findings indicate that the SRE is extensive and grows linearly with the system size, with an estimated value of 0.68 per site, reflecting a finite nonstabilizerness and verifying that the AKLT state possesses a nonzero quantum magic per spin even in the thermodynamic limit. These findings provide insight into the relationship between entanglement and nonstabilizerness as separate yet coexisting quantum resources, demonstrating that extensive quantum magic can emerge in systems that follow an area law for entanglement.