Abstract
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 are 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.
Highlights
As fundamental tools for quantum sensing, various different strategies have been developed for quantum hypothesis testing in various scenarios
In a minimum error discrimination (MED) [23,24,25,26] strategy, the overall error probability is minimized, while in an unambiguous discrimination (UD) [27,28,29,30,31] strategy, an inconclusive result is allowed to vanish any possible errors. Both strategies have wide applications: MED is important for applications like target detection and digital-memory reading; and UD can be utilized in applications related to quantum key distribution protocols [32] and optimal cloning [33]
To make UD practically relevant, given that the experimentation of sensing protocols is never perfect, relaxations of exact UD have been considered in many different approaches, including allowing a fixed inconclusive probability [34,35,36,37,38,39,40], maximum confidence [41,42], error-margin tuning [43], and general cost-function approaches [40,44,45]
Summary
Quantum sensing [1,2,3,4] has enabled quantum advantages in various applications, such as positioning and timing [5,6], target detection [7,8,9,10,11], digital-memory reading [12], distributed sensing [13,14,15,16,17,18], entangled-assisted spectroscopy [19], and most prominently the Laser Interferometer Gravitationalwave Observatory (LIGO) [20,21,22]. In a minimum error discrimination (MED) [23,24,25,26] strategy, the overall error probability is minimized, while in an unambiguous discrimination (UD) [27,28,29,30,31] strategy, an inconclusive result is allowed to vanish any possible errors Both strategies have wide applications: MED is important for applications like target detection and digital-memory reading; and UD can be utilized in applications related to quantum key distribution protocols [32] and optimal cloning [33]. Extensions to parameter-estimation scenarios have led to quantum metrology protocols assisted by abstention [46,47,48] Another complication beyond the different strategies is that the hypotheses being discriminated often involve physical processes modeled as quantum channels. Different from previous approaches, we enable the fine tuning of all conclusive conditional error probabilities Such a relaxation allows the proof of a general continuity inequality for states. When the Choi states are pure, this achievable lower bound can be directly calculated; while for mixed Choi states, we further obtain an efficiently calculable but nontight lower bound
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