Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination

Abstract

We consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states σ_1,...,σ_r. By splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities. These upper bounds allow us to deduce various bounds on the asymptotic error rate, for which it has been hypothesised that it is given by the multi-hypothesis quantum Chernoff bound (or Chernoff divergence) C(σ_1,...,σ_r), as recently introduced by Nussbaum and Szkoła in analogy with Salikhov's classical multi-hypothesis Chernoff bound. This quantity is defined as the minimum of the pairwise binary Chernoff divergences min_{j<k}C(σ_j,σ_k). It was known already that the optimal asymptotic rate must lie between C/3 and C, and that for certain classes of sets of states the bound is actually achieved. It was known to be achieved, in particular, when the state pair that is closest together in Chernoff divergence is more than 6 times closer than the next closest pair. Our results improve on this in two ways. Firstly, we show that the optimal asymptotic rate must lie between C/2 and C. Secondly, we show that the Chernoff bound is already achieved when the closest state pair is more than 2 times closer than the next closest pair. We also show that the Chernoff bound is achieved when at least $r-2$ of the states are pure, improving on a previous result by Nussbaum and Szkoła. Finally, we indicate a number of potential pathways along which a proof (or disproof) may eventually be found that the multi-hypothesis quantum Chernoff bound is always achieved.

50 pages. v3: Slightly restructured, main results unchanged, connection to Barnum and Knill's result (arXiv:quant-ph/0004088) clarified. Accepted for JMP

References in corpus (12)

• The Quantum Chernoff Bound
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Minimum-error discrimination between mixed quantum states
• Generalized relative entropies and the capacity of classical-quantum channels
• Multiparty data hiding of quantum information
• Error exponents in hypothesis testing for correlated states on a spin chain
• Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo-Curlander bounds
• Large deviations and Chernoff bound for certain correlated states on a spin chain
• Hypothesis testing for Gaussian states on bosonic lattices
• Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
• Asymptotic distinguishability measures for shift-invariant quasi-free states of fermionic lattice systems
• Attainment of the multiple quantum Chernoff bound for certain ensembles of mixed states

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