Effective quantum channel for minimum error discrimination

Yiran Wang, Yongming Li

Abstract In this paper, we mainly study the indistinguishability of classically simulable measurements (measurements that can be effectively simulated on classical computers) in the tasks of quantum channel discrimination. First, we investigate the limitations of classically simulable measurements under unambiguous and perfect discrimination strategies. For unambiguous discrimination, we establish a conditional limitation result for channel-induced output-state discrimination scenarios, while for perfect discrimination we derive a sufficient condition under which classically simulable measurements fail to perfectly distinguish quantum channels. When considering the minimum-error discrimination strategy, we formulate the corresponding bilinear optimization problem and discuss its formal Lagrange dual formulation. Subsequently, we give the lower bound of the minimum-error probability obtained by using classically simulable measurements in the minimum-error discrimination task under the adaptive protocol. Furthermore, we give two measures of the nonstabilizerness (magic) of measurements. We prove that one measure we proposed can be used as an exact quantifier for the advantage in some tasks of quantum state discrimination. Finally, we investigate the relationship between classically simulable measurements and the data hiding ratio.

Elloá B. Guedes, Francisco de Assis, Rex A. C. Medeiros

Chih‐Lung Chou

Quntao Zhuang

Quantum hypothesis testing is an important tool for quantum information processing.Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained.In both scenarios, the testing between quantum states is relatively well understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channels is less understood.Although the ultimate limit of minimum error discrimination between channels has been explored recently, the corresponding limit of unambiguous discrimination is unknown.In this paper, we formulate an approximate unambiguous discrimination scenario, and derive the ultimate limits of the performance for both states and channels.In particular, in the channel case, our lower bound of the inconclusive probability holds for arbitrary adaptive sensing protocols.For the special class of "teleportation-covariant" channels, the lower bound is achievable with maximum entangled inputs and no adaptive strategy is necessary.

Massimiliano Federico Sacchi

We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.

László Bacsárdi, László Gyöngyösi, Sándor Imre