Entanglement and Stabilizer entropies of random bipartite pure quantum states

Daniele Iannotti; Gianluca Esposito; Lorenzo Campos Venuti; Alioscia Hamma

doi:10.22331/q-2025-07-21-1797

Entanglement and Stabilizer entropies of random bipartite pure quantum states

Daniele Iannotti^1,2, Gianluca Esposito^1,2, Lorenzo Campos Venuti^3,4, and Alioscia Hamma^1,2,3

¹Scuola Superiore Meridionale, Largo S. Marcellino 10, 80138 Napoli, Italy
²Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli
³Università degli Studi di Napoli Federico II , Dipartimento di Fisica Ettore Pancini
⁴Department of Physics and Astronomy, University of Southern California, Los Angeles, USA

Published:	2025-07-21, volume 9, page 1797
Editor:	Himadri Shekhar Dhar
Eprint:	arXiv:2501.19261v4
Doi:	https://doi.org/10.22331/q-2025-07-21-1797
Citation:	Quantum 9, 1797 (2025).

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Abstract

The interplay between non-stabilizerness and entanglement in random states is a very rich arena of study for the understanding of quantum advantage and complexity. In this work, we tackle the problem of such interplay in random pure quantum states. We show that while there is a strong dependence between entanglement and magic, they are, surprisingly, perfectly uncorrelated. We compute the expectation value of non-stabilizerness given the Schmidt spectrum (and thus entanglement). At a first approximation, entanglement determines the average magic on the Schmidt orbit. However, there is a finer structure in the average magic distinguishing different orbits where the flatness of entanglement spectrum is involved.

Featured image: Illustration of Schmidt orbits (colored pink and violet lines on the sphere) categorized by their entanglement in a coarse-grained view (left) and the fine structure detailed by the averaged magic valuated over each orbit (right).

Popular summary

Quantum computing promises capabilities far beyond classical machines, but not all quantum states are equally powerful. Some, like stabilizer states, can be highly entangled yet are still efficiently simulable on a classical computer. Others possess a different kind of quantum resource known as magic—also called non-stabilizerness—which is essential for achieving universal quantum computation.
This paper explores the intricate relationship between entanglement and magic in random bipartite quantum states. Intuitively, one might expect that more entanglement implies more magic, suggesting a dependence between the two.
Using analytical tools and numerical simulations, we demonstrate a striking and nontrivial result: entanglement and magic are statistically (linearly) uncorrelated, even though they are not independent. To reach states with higher combined resource value, one must optimize along directions of constant entanglement or constant magic—much like choosing a direct path or a steady incline when climbing a mountain to avoid misleading slopes. In this sense, the linear measures of entanglement and magic are "intrinsically orthogonal."
To probe this further, we compute the average magic of quantum states that share the same Schmidt spectrum (i.e., the same entanglement structure). This leads us to define and study Schmidt orbits, families of states with fixed entanglement but varying local unitary transformations. We find that, at leading order, the average magic within such orbits depends only on the amount of entanglement. However, a deeper structure emerges: finer properties of the entanglement spectrum—especially its anti-flatness—affect the amount of magic.
Our results show that even though entanglement is necessary for magic, it is not sufficient. Magic depends on the distribution of entanglement—not just its total amount.

► BibTeX data

@article{Iannotti2025entanglement,
  doi = {10.22331/q-2025-07-21-1797},
  url = {https://doi.org/10.22331/q-2025-07-21-1797},
  title = {Entanglement and {S}tabilizer entropies of random bipartite pure quantum states},
  author = {Iannotti, Daniele and Esposito, Gianluca and Campos Venuti, Lorenzo and Hamma, Alioscia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1797},
  month = jul,
  year = {2025}
}

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 04:50:24) and SAO/NASA ADS (last updated successfully 2026-08-12 04:50:28). The list may be incomplete as not all publishers provide suitable and complete citation data.

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