Discrimination with error margin between two states: Case of general occurrence probabilities

Abstract

We investigate a state discrimination problem which interpolates minimum-error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with general occurrence probabilities. The optimal measurements are classified into three types. One of the three types of measurement is optimal depending on parameters (occurrence probabilities and error margin). We determine the three domains in the parameter space and the optimal discrimination success probability in each domain in a fully analytic form. It is also shown that when the states to be discriminated are multipartite, the optimal success probability can be attained by local operations and classical communication. For discrimination of two mixed states, an upper bound of the optimal success probability is obtained.

Programmable discrimination with an error margin

“…The strong condition is obviously more restrictive, as it sets a margin on both types of errors separately. However, as we will see, the two conditions are directly related: the strong one just corresponds to the weak one with a tighter error margin [14]. Note that both error margin schemes have the unambiguous (when r = 0) and the minimum-error schemes (when r is large enough) as extremal cases.…”

“…This result was derived in [13] and its generalization to arbitrary prior probabilities in [14] (also in [15], by fixing an inconclusive rate Q instead of an error margin). Note that the POVM E is fully determined by the angle φ, which in turn is fully determined by the margin r through Eq.…”

“…Hence by introducing an error margin we can unify minimum-error and unambiguous discrimination. Both become extremal points of the unified discrimination with error margin scheme [13][14][15]. Interpolating between these two extemal cases may have practical interest in some situations.…”

Upper and lower bounds on optimal success probability of quantum state discrimination with and without inconclusive results

“…In contrast, the square root measurement (SRM, also called the pretty good measurement), is well known as a suboptimal measurement of the success probability criterion; the success probability of the SRM is a good lower bound on the optimal one. In the case of optimal inconclusive measurements, an upper bound on the optimal success probability for binary quantum states has been derived by Sugimoto et al [28].…”

“…Instead of computing an exact optimal success probabilities, several previous studies have given its upper and/or lower bounds [20][21][22][23][24][25][26][27][28]. These methods are especially useful for large scale problems of which it is hard to compute an exact value within feasible time; for example, in Ref.…”