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Quantum hypothesis testing (QHT) provides an effective method to discriminatebetween two quantum states using a two-outcome positive operator-valued measure(POVM). Two types of decision errors in a QHT can occur. In this paper we focuson the asymmetric setting of QHT, where the two types of decision errors aretreated unequally, considering the operational limitations arising from thelack of a reference frame for chirality. This reference frame is associatedwith the group $\bbZ_2$ consisting of the identity transformation and theparity transformation. Thus, we have to discriminate between two qubit statesby performing the $\bbZ_2$-invariant POVMs only. We start from thediscrimination between two pure states. By solving the specific optimizationproblem we completely characterize the asymptotic behavior of the minimalprobability of type-II error which occurs when the null hypothesis is acceptedwhen it is false. Our results reveal that the minimal probability reduces tozero in a finite number of copies, if the $\bbZ_2$-twirlings of such two purestates are different. We further derive the critical number of copies such thatthe minimal probability reduces to zero. Finally, we replace one of the twopure states with a maximally mixed state, and similarly characterize theasymptotic behavior of the minimal probability of type-II error.

Quantum hypothesis testing between qubit states with parity