Absolute Schmidt number: Characterization, detection, and resource-theoretic quantification

The dimensionality of entanglement, quantified by the Schmidt number, is a valuable resource for a wide range of quantum information processing tasks. In this work, we introduce the notion of the absolute Schmidt number, referring to states whose Schmidt number cannot be increased by any global unitary transformation. We provide a characterization of the set of arbitrary-dimensional states whose Schmidt number is invariant under all global unitaries. Our approach enables us to develop both witness-based and moment-based techniques to detect nonabsolute Schmidt number states which could provide significant operational advantages through Schmidt number enhancement by global unitaries. We next formulate two resource-theoretic measures of nonabsolute Schmidt number states, based respectively on Schmidt number witness and robustness, and demonstrate an operational utility of the latter in a channel discrimination task. Finally, we extend our analysis to quantum channels by introducing a new class of channels that possess the absolute Schmidt number property. We derive a necessary and sufficient condition for identifying when a channel has the absolute Schmidt number property, confining our analysis to the class of covariant channels.