Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing
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In this article, a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing is derived, and a projection measurement characterized by the irreducible representation theory of the special linear group SL is proposed.Abstract:
We derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing. We discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0. As an asymptotically optimal measurement, we propose a projection measurement characterized by the irreducible representation theory of the special linear group SL(). Especially, in the spin-1/2 system, it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction. As a by-product, we obtain another proof of quantum Stein's lemma. In addition, an asymptotically optimal measurement is constructed in the quantum Gaussian case, and it is physically meaningful.read more
Citations
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Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges
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References
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The classical groups : their invariants and representations
TL;DR: Weyl as discussed by the authors discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations using basic concepts from algebra, and examines the various properties of the groups.
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Roe Goodman,Nolan R. Wallach +1 more
TL;DR: In this paper, the basic structure of classical groups is described as linear algebraic groups and representations of these groups are described. But the representation of these representations is not defined. And the representation is not restricted to linear groups, but also to algebraic algebras.
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太舜 韓,Hiroki Koga +1 more
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The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability
Fumio Hiai,Dénes Petz +1 more
TL;DR: Umegaki's relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andD ϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint.
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