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

TL;DR

This paper introduces the absolute Schmidt number, characterizes states with invariant Schmidt number under all global unitaries, and develops detection and resource measures for nonabsolute Schmidt number states, with applications to quantum channels.

Contribution

It defines the absolute Schmidt number, characterizes invariant states, and proposes detection and quantification methods, including resource-theoretic measures and channel analysis.

Findings

Characterized states with invariant Schmidt number under all global unitaries.

Developed witness-based and moment-based techniques for detecting nonabsolute Schmidt number states.

Formulated resource measures and identified conditions for channels with the absolute Schmidt number property.

Abstract

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.