Complexity of quantum states in the stabilizer formalism

TL;DR

This paper introduces an information-theoretic measure of quantum state complexity within the stabilizer formalism, linking it to nonstabilizerness and classical-quantum distinctions.

Contribution

It proposes a novel complexity quantifier based on Jordan and Lie products, connecting state complexity to nonstabilizerness through characteristic functions.

Findings

The quantifier is shown to have fundamental properties.

State complexity relates to nonstabilizerness via the $L^4$-norm.

Establishes a link between classicality and quantumness in states.

Abstract

We initiate an investigation into a notion of state complexity for discrete-variable quantum systems. Specifically, we propose an information-theoretic quantifier for the complexity of quantum states within the stabilizer formalism of quantum computation. This is achieved by leveraging the symmetric Jordan product (associated with classicality) and the skew-symmetric Lie product (linked to quantumness) between the square root of the quantum state and the Heisenberg-Weyl displacement operators. We establish the fundamental properties of this quantifier and demonstrate that state complexity is closely related to the nonstabilizerness of quantum states via the $L^4$-norm of their characteristic functions.