Asymptotic Entanglement Hiding under Stabilizer Restrictions

Quantum entanglement is an essential resource for quantum information tasks, while stabilizer operations form the operational backbone of fault-tolerant quantum computation. Because stabilizer measurements are efficiently simulable and define the free sector of the resource theory of magic, the authors investigate how much entanglement remains visible or distillable when an observer is restricted entirely to stabilizer operations. Specifically, the study asks whether stabilizer restrictions prevent an observer from recognizing or extracting entanglement even when the underlying states are completely free of magic.

To quantify visible entanglement under measurement constraints, the authors introduce stabilizer-visible entanglement by restricting the measured relative entropy of entanglement to the class of stabilizer measurements. They evaluate this quantity across three key scenarios: Haar-random pure states, Werner states across odd-prime local dimensions, and an explicit construction of mixed stabilizer states. To establish upper bounds for mixed states and stabilizer operations, the authors employ Gross's discrete Wigner representation together with the Kullback-Leibler divergence, providing a robust technique to bound relative entropy under stabilizer-restricted channels.

The study demonstrates that stabilizer restrictions can drastically hide extensive entanglement. For Haar-random pure states, while unrestricted visible entanglement scales extensively with dimension, the stabilizer-visible entanglement remains bounded by a constant with high probability. Furthermore, for a constructed sequence of mixed stabilizer states on multi-qutrit systems, the unrestricted visible and LOCC-distillable entanglement both grow as $\Omega(N/\log N)$, whereas their stabilizer-visible and stabilizer-distillable counterparts vanish asymptotically as $N \to \infty$. This proves that magic-free mixed states can harbor unbounded entanglement that is entirely invisible and undistillable through stabilizer operations alone.