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Laser Physics Letters

Schmidt number witnesses from purity-controlled k -positive maps

Wen Huang, Bo Li

Published Jun 1, 2026• 0 citations
Publisher

Abstract

Abstract Quantum entanglement is a fundamental resource in quantum information tasks. Schmidt number is a key quantity for characterizing the entanglement dimension of bipartite states. Constructing entanglement detection methods with physical interpretability and experimental feasibility is a current research focus. In this work, we utilize the geometric symmetry of the generalized equiangular measurement conical 2-design to derive a family of k -positive maps. Using the Choi–Jamiołkowski isomorphism, we construct a general family of Schmidt number witnesses. For isotropic states, we derive a structure-independent analytical threshold. Our approach offers a mathematically concise criterion while retaining clear experimental feasibility.

DOI
10.1088/1612-202x/ae7704

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Abstract Quantum entanglement is a fundamental resource in quantum information tasks. Schmidt number is a key quantity for characterizing the entanglement dimension of bipartite states. Constructing entanglement detection methods with physical interpretability and experimental feasibility is a current research focus. In this work, we utilize the geometric symmetry of the generalized equiangular measurement conical 2-design to derive a family of k -positive maps. Using the Choi–Jamiołkowski isomorphism, we construct a general family of Schmidt number witnesses. For isotropic states, we derive a structure-independent analytical threshold. Our approach offers a mathematically concise criterion while retaining clear experimental feasibility.
Families of kkk-positive maps and Schmidt number witnesses from generalized equiangular measurements
Quantum entanglement is an important resource in many modern technologies, like quantum computation or quantum communication and information processing. Therefore, most interest is given to detect and quantify entangled states. Entanglement degree of bipartite mixed quantum states can be measured using the Schmidt number. Witnesses of the Schmidt number are closely related to kkk-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of kkk-positive linear maps and the corresponding Schmidt number witnesses. We present examples of witnesses that detect two-parameter entangled states with Schmidt number k+1k+1k+1 in any dimension. Our approach allows for a more efficient entanglement quantification than other known Schmidt number witnesses constructed from symmetric measurement operators.
Abstract Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.
The Schmidt number is an important kind of characterization of quantum entanglement. Quantum states with higher Schmidt numbers demonstrate significant advantages in various quantum information processing tasks. By deriving a class of k-positive linear maps based on symmetric measurements, we present new Schmidt-number witnesses of class (k + 1). By detailed example, we show that our Schmidt number witnesses identify better the Schmidt number of quantum states in high-dimensional systems. Furthermore, we note that the Fedorov ratio, which coincides with the Schmidt number for pure Gaussian states and provides a close approximation in non-Gaussian cases such as spontaneous parametric down-conversion, serves as an experimentally accessible tool for validating the proposed (k +1)-class Schmidt-number witnesses.
Abstract With the rapid evolution of quantum information science, quantum entanglement remains a fundamental resource for quantum tasks. Nevertheless, the reliable and efficient detection of entanglement, especially the quantitative determination of the Schmidt number in high-dimensional quantum systems, continues to pose a significant and long-standing challenge. In this work, we introduce a novel Schmidt number criterion for bipartite quantum systems via mutually unbiased equiangular tight frames. Notably, this criterion unifies and substantially generalizes previous approaches that relied on mutually unbiased bases and equiangular measurements (Tavakoli and Morelli 2024 Phys. Rev. A 110 062417). Based on this framework, we also derive a lower bound on concurrence. Moreover, detailed examples demonstrate that our result outperforms several existing approaches. Consequently, this contribution advances both the foundational theory of entanglement quantification and the development of scalable, experimentally feasible tools for quantum information processing.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.