The Chernoff lower bound for symmetric quantum hypothesis testing

Abstract

We consider symmetric hypothesis testing in quantum statistics, where the hypotheses are density operators on a finite-dimensional complex Hilbert space, representing states of a finite quantum system. We prove a lower bound on the asymptotic rate exponents of Bayesian error probabilities. The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set. In our framework, the classical result is reproduced if the two hypothetic density operators commute. Recently, it has been shown elsewhere [Phys. Rev. Lett. 98 (2007) 160504] that the lower bound is achievable also in the generic quantum (noncommutative) case. This implies that our result is one part of the definitive quantum Chernoff bound.Comment: Published in at http://dx.doi.org/10.1214/08-AOS593 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantum-process tomography: Resource analysis of different strategies

“…However, the very nature of independence from estimation method means that the Fisher information matrix is not so useful for the purpose of comparing different QPT schemes-our goal in this paper. A more promising and physically-motivated approach, that justifies using the Chernoff bound for the purpose of quantum state/process estimation as we did earlier, has been proposed very recently, and is called the quantum Chernoff bound (QCB) [79,80,81,82]. The physical interpretation of this quantity is as follows: assuming that we have access to all types of measurements-whether local or collective-on all ensembles, the QCB measures the error in distinguishing a state ρ from another state ρ.…”

“…Recently, the QCB has been considered as a natural figure-of-merit in evaluating the performance of different measurement scenarios for qubit tomography [81]. It also has been used for quantum hypothesis testing and distinguishability between density matrices [80,82]. Considering the fact that a generic χ matrix is formally in the category of density matrices, the application of the QCB can in principle be extended to QPT.…”

“…There is a huge literature regarding analysis quantum estimation errors or quantum statistics [52,53,54,55,56,57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82]. Our aim here is to give a very brief discussion of estimation errors in different QPT schemes through a special example.…”

Quantum probes for fractional Gaussian processes

“…We thus need to analyze the quantity P e,n = 1 2 1 − Tr 1 2 (|ρ ⊗n γ 2 − ρ ⊗n γ 1 )| . The evaluation of the trace distance for increasing n may be difficult and for this reason, one usually resort to the quantum Chernoff bound, which gives an upper bound to the probability of error [56,57,58,59,60,61] P e,n ≤ 1 2 Q n where…”

Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit

“…The mathematical solution to the problem is known in terms of necessary and sufficient conditions that the optimal POVM must satisfy [8], although for discriminating between more than two states, the explicit solution of these conditions has been obtained only in some specific cases [9]. Over the years, the scope of quantum detection theory has been broadened beyond the above framework to ones such as unambiguous state discrimination [10], maximum confidence discrimination [11], and to specific scenarios of interest such as multi-copy state discrimination using local operations and classical communication (LOCC) [12][13][14][15][16][17], using a quantum computer with limited entanglement [18], and in the asymptotic limit of a large number of copies [19][20][21][22]. …”