Asymptotic Error Rates in Quantum Hypothesis Testing
By Koenraad M. R. Audenaert, Michael Nussbaum, Arleta Szkola, Frank Verstraete. Communications in Mathematical Physics, 2007
266 citations recorded by the source corpus.
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing.The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support.The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.
Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/b470e7945c26b1faf41173792c7df0d3bc56314e
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Citations in the corpus
- Efficient quantum tomography (2015, unknown): “ult, if dH2(ρ,σ) = ǫ, then 1 −ǫ/2 ≤ F(ρ,σ). Proof. The upper bound A(ρ,σ) ≤ F(ρ,σ) is immediate, as tr(M) ≤ kMk1 for any matrix M. As for the lower bound, it follows from Equations (28) and (32) from [ANSV08]. As a result, fidelity and affinity are essentially equivalent in the “1 − ǫ” regime, and furt”
- A Generalization of Quantum Stein’s Lemma (2009, unknown): “[7, 8, 13].”
- THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING (2006, background): “Another discussion of the geometric properties of the quantum Chernoff bound and a derivation of the related quantum Hoeffding bound can be found in [2].”