Spectral geometry of nonlocal stabilizer entropy

arXiv:2609.13106v1 Announce Type: new Abstract: Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer R\'enyi entropy (SRE), the resulting nonlocal SRE has been conjectured to be attained by the computational-basis state, whose Schmidt vectors are computational-basis states ordered by decreasing Schmidt coefficient. In this work, we prove this conjecture for two families of states at every system size: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. Without restriction on the spectrum, we show that the nonlocal SRE equals the SRE of the computational-basis state up to a bounded constant, which fixes the leading term of any divergent scaling of the nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy, so that it remains finite for area-law states and grows at most doubly logarithmically with the system size for critical states. We illustrate these results with the transverse-field Ising chain, in which we determine the nonlocal SRE both in the gapped phases and at the critical point. Our results establish a quantitative spectral geometry of the nonlocal SRE and enclose its value in many-body states between computable bounds.

Abstract:Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the resulting nonlocal SRE has been conjectured to be attained by the computational-basis state, whose Schmidt vectors are computational-basis states ordered by decreasing Schmidt coefficient. In this work, we prove this conjecture for two families of states at every system size: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. Without restriction on the spectrum, we show that the nonlocal SRE equals the SRE of the computational-basis state up to a bounded constant, which fixes the leading term of any divergent scaling of the nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy, so that it remains finite for area-law states and grows at most doubly logarithmically with the system size for critical states. We illustrate these results with the transverse-field Ising chain, in which we determine the nonlocal SRE both in the gapped phases and at the critical point. Our results establish a quantitative spectral geometry of the nonlocal SRE and enclose its value in many-body states between computable bounds.

18+24 pages, comments welcome!

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech)

arXiv:2609.13106 [quant-ph]

(or arXiv:2609.13106v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.13106 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Spectral geometry of nonlocal stabilizer entropy

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