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arXiv:2512.18807 · Journal of Physics A Mathematical and Theoretical

Families of $k$-positive maps and Schmidt number witnesses from generalized equiangular measurements

Katarzyna Siudzińska

quant-phmath-phSubmitted Dec 21, 2025• Announced Dec 23, 2025• 0 citations• Revised Jul 20, 2026 (v4)
v1 · Dec 21, 2025v2 · Apr 30, 2026v3 · May 10, 2026v4 · Jul 20, 2026
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Abstract

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 $k$-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of $k$-positive linear maps and the corresponding Schmidt number witnesses. We present examples of witnesses that detect two-parameter entangled states with Schmidt number $k+1$ in any dimension. Our approach allows for a more efficient entanglement quantification than other known Schmidt number witnesses constructed from symmetric measurement operators.

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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 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.
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 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.
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.
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.
In quantum information theory, measurements are represented by positive, operator-valued measures. In this paper, we use the operators corresponding to generalized equiangular measurements to construct positive maps. Their positivity follows from the properties of index of coincidence for few equiangular tight frames. These maps give rise to entanglement witnesses, which include as special cases many important classes considered in the literature. Additionally, we introduce separability criteria based on the correlation matrix and analyze them for various types of measurements.