Characterizing high-Schmidt-number witnesses in arbitrary-dimensional systems
Phys. Rev. A 114, 012443 – Published 17 July, 2026
DOI: https://doi.org/10.1103/8153-3wh6
Abstract
A profound comprehension of quantum entanglement is crucial for the progression of quantum technologies. The degree of entanglement can be assessed by enumerating the entangled degrees of freedom, leading to the determination of a parameter known as the Schmidt number (SN). In this paper, we develop an efficient analytical tool for characterizing high-SN witnesses for bipartite quantum states in arbitrary dimensions. Our methods not only offer viable mathematical methods for constructing high-dimensional SN witnesses in theory but also simplify the quantification of entanglement and dimensionality by providing upper bounds for witness coefficients that depend solely on the operator Schmidt coefficient (OSC), thereby reducing computational complexity and enabling straightforward application in arbitrary-dimensional systems without the need for numerical optimization or extensive matrices diagonalization. Most notably, we develop high-dimensional SN witnesses within arbitrary-dimensional systems, with our Schmidt witness coefficients relying solely on the OSC. Subsequently, we demonstrate our theoretical advancements and computational superiority by constructing SN witnesses in arbitrary-dimensional bipartite quantum systems with SNs 4 and 5. In addition, we illustrate the feasibility and superiority of our approach through various numerical examples of mixed states across different dimensions, including low-dimensional cases with SNs 3–6 composed of convex combinations of different orders (both specific and randomly generated examples), as well as high-dimensional cases with SNs 10, 15, and 20, again involving both specific and randomly generated mixed states.
