Draft:Generalized quantum stein's lemma
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Generalized quantum Stein's lemma is a result for quantum hypothesis testing, which extends the quantum version of Stein's lemma to correlated quantum states[1][2]. The result has implications in Quantum thermodynamics, because it establishes an analogous second law for Entanglement distillation[3][4]. Specifically, it shows that the asymptotic rate of conversion between any two quantum states is given by the asymptotic relative entropy of entanglement, providing a bridge between quantum hypothesis testing and quantum resource theories.
History
[edit]The result was first described in[5], in which
which claimed to prove the result, though this turned out to be false. This lead to a gap in the theory of quantum thermodynamics.[6][7]
TODO: Mention the fact that there were two proofs of the result, and then both proofs turned out to be wrong. Then mention how there was a race to finish the result, leading to two different proofs of the result (that should be in the section after it).
Mathematical formulation
[edit]Consider the hypothesis testing problem, given where and are ...
The Stein exponent for a quantum hypothesis testing problem is ...
Generalized quantum Stein's lemma tells us that the Stein exponent is given the regularized quantum relative entropy, ...
Significance
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Further reading
[edit]
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Category:Quantum information theory Category:Statistical mechanics
- ↑ Lami, Ludovico (June 2025). "A Solution of the Generalized Quantum Stein's Lemma". IEEE Transactions on Information Theory. 71 (6): 4454–4484. arXiv:2408.06410. doi:10.1109/TIT.2025.3543610. Retrieved 27 August 2026.
- ↑ Hayashi, Masahito; Yamasaki, Hayata (December 2025). "The generalized quantum Stein's lemma and the second law of quantum resource theories". Nature Physics. 21 (12): 1988–1993. arXiv:2408.02722. doi:10.1038/s41567-025-03047-9. Retrieved 27 August 2026.
- ↑ Christandl, Matthias (December 2025). "A cornerstone of entanglement theory restored". Nature Physics. 21 (12): 1881–1882. doi:10.1038/s41567-025-03084-4. Retrieved 27 August 2026.
- ↑ Brandão, Fernando G. S. L.; Plenio, Martin B. (November 2008). "Entanglement theory and the second law of thermodynamics". Nature Physics. 4 (11): 873–877. arXiv:0810.2319. doi:10.1038/nphys1100. Retrieved 27 August 2026.
- ↑ Brandão, Fernando G. S. L.; Plenio, Martin B. (May 2010). "A Generalization of Quantum Stein's Lemma". Communications in Mathematical Physics. 295 (3): 791–828. arXiv:0904.0281. doi:10.1007/s00220-010-1005-z.
- ↑ Chitambar, Eric; Gour, Gilad (4 April 2019). "Quantum resource theories". Reviews of Modern Physics. 91 (2). arXiv:1806.06107. doi:10.1103/RevModPhys.91.025001. Retrieved 27 August 2026.
- ↑ Berta, Mario; Brandão, Fernando G. S. L.; Gour, Gilad; Lami, Ludovico; Plenio, Martin B.; Regula, Bartosz; Tomamichel, Marco (February 2024). "The tangled state of quantum hypothesis testing" (PDF). Nature Physics. 20 (2): 172–175. doi:10.1038/s41567-023-02289-9. Retrieved 27 August 2026.