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First, by means of explicit examples, we show that, for R\u00e9nyi index <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>0<\/mml:mn><mml:mo>&amp;#x2264;<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>&amp;#x2264;<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>, 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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><mml:mo>&amp;#x2265;<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>). Next, we show that, for any R\u00e9nyi 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 R\u00e9nyi 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.<\/jats:p>","DOI":"10.22331\/q-2023-08-28-1092","type":"journal-article","created":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T14:22:42Z","timestamp":1693232562000},"page":"1092","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":98,"title":["Stabilizer entropies and nonstabilizerness monotones"],"prefix":"10.22331","volume":"7","author":[{"given":"Tobias","family":"Haug","sequence":"first","affiliation":[{"name":"QOLS, Blackett Laboratory, Imperial College London SW7 2AZ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lorenzo","family":"Piroli","sequence":"additional","affiliation":[{"name":"Philippe Meyer Institute, Physics Department, \u00c9cole Normale Sup\u00e9rieure (ENS), Universit\u00e9 PSL, 24 rue Lhomond, F-75231 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2023,8,28]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Scott Aaronson and Daniel Gottesman. 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