The Verdú-Han [3] and Poor-Verdú[4] bounds (VH and PV bounds for short) are lower bounds on the error probability in classical state discrimination. The PV bound is the tighter of the two bounds. Although the VH bound is known to have at least two proofs, which are the proof based on maximum a posteriori discrimination and the proof based on the Neyman-Pearson's lemma, the proof based on the Neyman-Pearson's lemma for the PV bound has not been reported. This is one of reasons why a quantum version of the PV bound is not obtained although that of the VH bound is [6]. In this paper we show the proof based on the Neyman-Pearson's lemma for the PV bound, extend the PV bound in the quantum case, and discuss the reliability function of classical-quantum channels as an application of the quantum PV bound.
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