Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2402.09972 · Physical Review A

Enhanced Schmidt-number criteria based on correlation trace norms

Armin Tavakoli, Simon Morelli

Published Dec 11, 2024• 16 citations
arXivHTMLPDF

Abstract

The Schmidt number represents the genuine entanglement dimension of a bipartite quantum state. We derive simple criteria for the Schmidt number of a density matrix in arbitrary local dimensions, given that certain symmetric measurements exist. They are based on the trace norm of correlations obtained from seminal families of quantum measurements, specifically symmetric informationally complete measurements and mutually unbiased bases. Our criteria are strictly stronger than both the well-known fidelity witness criterion and the computable cross-norm or realignment criterion.

DOI
10.1103/physreva.110.062417

Related papers

Estimating the Schmidt Numbers of Quantum States via Symmetric Measurements

Hao‐Fan Wang, Shao‐Ming Fei

Abstract The Schmidt numbers quantify the entanglement degree of quantum states. Quantum states with high Schmidt numbers provide a larger advantage in various quantum information processing tasks compared to quantum states with low Schmidt numbers. A Schmidt number criterion is derived based on the trace norm of the correlation matrix obtained from symmetric measurements. It is showed that the result is more effective than and superior to existing Schmidt number criteria by detailed examples.

arXiv:2505.02297•Annalen der Physik•Aug 4, 2025•5 citations

Families of Schmidt-number witnesses for high dimensional quantum states

Xian Shi

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.

arXiv:2403.00248•Communications in Theoretical Physics•Jul 4, 2024•11 citations

Bounding entanglement dimensionality from the covariance matrix

Shuheng Liu, Matteo Fadel, Qiongyi He +2 more

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

arXiv:2208.04909•Aug 9, 2022•0 citations

Bounding entanglement dimensionality from the covariance matrix

Shuheng Liu, Matteo Fadel, Qiongyi He +2 more

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) \cite{guhnecova} and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

Quantum•Jan 30, 2024•21 citations

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

Katarzyna Siudzińska

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.

quant-pharXiv:2512.18807•Journal of Physics A Mathematical and Theoretical•Dec 23, 2025•0 citations

Schmidt number criterion via mutually unbiased equiangular tight frames

Lin qiao Che, Yi Long, Fan Wu +1 more

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.

Journal of Physics A Mathematical and Theoretical•Oct 20, 2026•0 citations
The Schmidt number represents the genuine entanglement dimension of a bipartite quantum state. We derive simple criteria for the Schmidt number of a density matrix in arbitrary local dimensions, given that certain symmetric measurements exist. They are based on the trace norm of correlations obtained from seminal families of quantum measurements, specifically symmetric informationally complete measurements and mutually unbiased bases. Our criteria are strictly stronger than both the well-known fidelity witness criterion and the computable cross-norm or realignment criterion.
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.
High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.
High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) \cite{guhnecova} and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.
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 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.