Characterizing High Schmidt Number Witnesses in Arbitrary Dimensions System

TL;DR

The paper tackles the challenge of certifying high Schmidt number entanglement in bipartite quantum states across arbitrary dimensions. It develops an analytical framework based on the Operator Schmidt decomposition and nonnegative-matrix spectral-radius theory to bound and construct Schmidt-number witnesses, expressing witness coefficients solely in terms of Operator Schmidt Coefficients. The authors establish a hierarchy of computable bounds ($\theta_{k+1},\eta_{k+1},\zeta_{k+1},P_{k+1}$) that bound the optimal witness coefficient $\lambda_{k+1}$ and demonstrate their approach by constructing witnesses for Schmidt numbers four and five, including explicit examples and numerical evidence. This yields a scalable, mathematically grounded method for entanglement-dimensionality quantification with potential practical impact on high-dimensional quantum information tasks.

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. In this paper, we develop an efficient analytical tool for characterizing high Schmidt number witnesses for bipartite quantum states in arbitrary dimensions. Our methods not only offer viable mathematical methods for constructing high-dimensional Schmidt number witnesses in theory but also simplify the quantification of entanglement and dimensionality. Most notably, we develop high-dimensional Schmidt number witnesses within arbitrary-dimensional systems, with our Schmidt witness coefficients relying solely on the operator Schmidt coefficient. Subsequently, we demonstrate our theoretical advancements and computational superiority by constructing Schmidt number witnesses in arbitrary dimensional bipartite quantum systems with Schmidt numbers four and five.

Figures (2)

• Figure 1: The optimal witnesses coefficient $\lambda_{k+1}$ and upper bounds $\theta_{k+1},\zeta_{k+1}, \eta_{k+1}, P_{k+1}$ of the Schmidt number witnesses associated with a mixed state generated randomly in MATLAB are determined. The four images (a), (b), (c), and (d) depict 50 randomly generated mixed states for Schmidt number $k=2, 3, 4,$ and $5,$ respectively. Each mixed state is produced through 10,000 convex combinations of pure states.
• Figure 2: The optimal witnesses coefficients $\lambda_{11}$ and upper bounds $\theta_{11}$, $\zeta_{11}$, $\eta_{11}$, and $P_{11}$ for mixed bipartite states with the Schmidt numbers $10$ are obtained through random generation using MATLAB. As can be seen from the Figure 2, the coefficients $\lambda_{11}$ and the upper bounds $\theta_{11}$, $\zeta_{11}$, $\eta_{11}$, and $P_{11}$ of 50 randomly generated mixed states. Descriptions in Figures 2(a), Figures 2(b), Figures 2(c), and Figures 2(d) correspond that each mixed state is created through convex combinations of 5, 100, 10000, and 100000 pure states, respectively.

Theorems & Definitions (19)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• Proposition 1
• Proposition 2
• Remark 1