Stabilizer entropies and nonstabilizerness monotones

Tobias Haug1 and Lorenzo Piroli2

1QOLS, Blackett Laboratory, Imperial College London SW7 2AZ, UK
2Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS), Université PSL, 24 rue Lhomond, F-75231 Paris, France

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Abstract

We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for Rényi index $0\leq n\leq2$, the SEs are not monotones with respect to stabilizer protocols which include computational-basis measurements, not even when restricting to pure states (while the question remains open for $n\geq2$). Next, we show that, for any Rényi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of many-body states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them.
Finally, we discuss exact or efficient tensor-network numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the Rényi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.

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Cited by

[1] Davide Rattacaso, Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, "Stabilizer entropy dynamics after a quantum quench", Physical Review A 108 4, 042407 (2023).

[2] Xhek Turkeshi, Marco Schirò, and Piotr Sierant, "Measuring nonstabilizerness via multifractal flatness", Physical Review A 108 4, 042408 (2023).

[3] Guglielmo Lami and Mario Collura, "Nonstabilizerness via Perfect Pauli Sampling of Matrix Product States", Physical Review Letters 131 18, 180401 (2023).

[4] Poetri Sonya Tarabunga, Emanuele Tirrito, Titas Chanda, and Marcello Dalmonte, "Many-Body Magic Via Pauli-Markov Chains—From Criticality to Gauge Theories", PRX Quantum 4 4, 040317 (2023).

[5] Emanuele Tirrito, Poetri Sonya Tarabunga, Gugliemo Lami, Titas Chanda, Lorenzo Leone, Salvatore F. E. Oliviero, Marcello Dalmonte, Mario Collura, and Alioscia Hamma, "Quantifying non-stabilizerness through entanglement spectrum flatness", arXiv:2304.01175, (2023).

[6] Andi Gu, Lorenzo Leone, Soumik Ghosh, Jens Eisert, Susanne Yelin, and Yihui Quek, "A little magic means a lot", arXiv:2308.16228, (2023).

[7] Junjie Chen, Yuxuan Yan, and You Zhou, "Magic of quantum hypergraph states", arXiv:2308.01886, (2023).

[8] Oliver Hahn, Alessandro Ferraro, Lina Hultquist, Giulia Ferrini, and Laura García-Álvarez, "Erratum: Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding [Phys. Rev. Lett. 128, 210502 (2022)]", Physical Review Letters 131 4, 049901 (2023).

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