Stabilizer Entropy of Subspaces

TL;DR

This paper investigates how embedding quantum states into larger systems affects their nonstabilizerness, revealing that certain embeddings can reduce or even negate the resource cost associated with magic, with implications for quantum error correction and simulation.

Contribution

It provides analytical formulas and numerical methods to evaluate the stabilizer entropy gap in subspace embeddings, identifying conditions for zero or negative gaps and optimizing subspace choices.

Findings

Stabilizer entropy gap is usually positive but can be zero or negative.

Certain stabilizer codes enable negative magic gaps, reducing resource costs.

Optimized subspaces can enhance efficiency in quantum simulations.

Abstract

We consider the costs and benefits of embedding the states of one quantum system within those of another. Such embeddings are ubiquitous, e.g., in error correcting codes and in symmetry-constrained systems. In particular we investigate the impact of embeddings in terms of the resource theory of nonstabilizerness (also known as magic) quantified via the stabilizer entropy (SE). We analytically and numerically study the stabilizer entropy gap or magic gap: the average gap between the SE of a quantum state realized within a subspace of a larger system and the SE of the quantum state considered on its own. We find that while the stabilizer entropy gap is typically positive, requiring the injection of magic, both zero and negative magic gaps are achievable. This suggests that certain choices of embedding subspace provide strong resource advantages over others. We provide formulas for the average nonstabilizerness of a subspace given its corresponding projector and sufficient conditions for realizing zero or negative gaps: in particular, certain classes of stabilizer codes provide paradigmatic examples of the latter. Through numerical optimization, we find subspaces which achieve both minimal and maximal average SE for a variety of dimensions, and compute the magic gap for specific error-correcting codes and symmetry-induced subspaces. Our results suggest that a judicious choice of embedding can lead to greater efficiency in both classical and quantum simulations.