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:2605.18605

Non-Gaussian Entanglement Hierarchy Based on the Schmidt Number

Jiajie Guo, Shuheng Liu, Matteo Fadel, Qiongyi He

quant-phSubmitted May 18, 2026• Announced May 19, 2026• 0 citations
v1 · May 18, 2026v2 · May 26, 2026
arXivHTMLPDF

Abstract

Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four projective measurements, where an analytical Gaussian-entanglable threshold is derived. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.

Related papers

A note on Schmidt-number witnesses based on symmetric measurements

Xiao-Qian Mu, Hao-Fan Wang, Shao-Ming Fei

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.

quant-pharXiv:2511.12887•Laser Physics Letters•Nov 18, 2025•1 citation

Entanglement Dimensionality of Continuous Variable States From Phase-Space Quasi-Probabilities

Shuheng Liu, Jiajie Guo, Matteo Fadel +3 more

The dimensionality of entanglement, quantified by the Schmidt number, is a core resource for quantum information processing. While its certification is well-established in finite-dimensional systems, many platforms harnessing high Schmidt numbers are natively continuous-variable (CV) systems, which are usually truncated to a finite subspace before canonical discrete-variable techniques are applied. For those instances, a core question is whether directly utilizing a genuinely CV toolbox leads to better and more robust characterisations. We answer this question by introducing Schmidt-number witnesses built directly from correlations and covariances of phase-space observables, accessible through characteristic- or Wigner-function measurements. They hold for arbitrary mixed states, including non-Gaussian ones, and only require correlations along a slice of phase space. We show that this direct estimation is more robust and versatile than first discretizing the system and using canonical discrete variable techniques. Moreover, we apply our method to certify arbitrarily-large Schmidt numbers of a bound-entangled Gaussian state. These results motivate genuine CV methods for characterizing high-dimensional entanglement across CV quantum technologies.

arXiv:2509.02743•Sep 2, 2025•0 citations

Certification of high-dimensional entanglement in continuous-variable systems

Ida Mishra, Simon Morelli, Klára Baksová +2 more

The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.

quant-pharXiv:2610.03113•Oct 5, 2026•0 citations

Detecting strongly non-Gaussian entanglement

Serge Deside, Tobias Haas, Nicolas J. Cerf

Efficiently certifying non-Gaussian entanglement in continuous-variable quantum systems is a central challenge for advancing quantum information processing, photonic quantum computing, and metrology. Here, we put forward continuous-variable counterparts to the recently introduced entanglement criteria based on moments of the partially transposed state, together with simple readout schemes that require only a few replicas of the state, passive linear optics, and particle-number measurements. We develop a multicopy method that enables the detection of strongly non-Gaussian entanglement for various relevant state families overlooked by standard approaches, which include the entire class of NOON states. Further, it is robust against realistic experimental constraints (losses, noise, and finite statistics), which we demonstrate by extensive numerical simulations.

arXiv:2504.15831•Physical Review Research•Apr 2, 2026•0 citations

Gaussian superpositions for bosonic encodings

Federico Centrone, Juan Pablo Paz, Augusto Roncaglia

Non-Gaussian bosonic states are ubiquitous in interacting light--matter systems, many-body platforms, and relativistic quantum field settings, but their quantitative characterization is hindered by the infinite-dimensional Hilbert space and by the poor scalability of Fock-space truncation methods. We introduce an exact finite-manifold encoding for states supported on a finite span of Gaussian branches, enabling the use of standard finite-dimensional quantum-information tools directly on an effective density matrix whose entries are determined by Gaussian overlaps. As demonstrations, we obtain closed-form and numerically stable evaluations of entropies and relative-entropy non-Gaussianity, and derive an analytic expression for the bipartite entanglement negativity of arbitrary multimode two-branch Gaussian superpositions, including a minimal which-branch dephasing model. Our framework provides a practical bridge between experimentally accessible continuous-variable resources (e.g., cat-like and measurement-conditioned states) and discrete-variable information measures, with immediate applications to benchmarking non-Gaussian resources in several quantum technology platforms.

quant-pharXiv:2603.15258•Mar 17, 2026•0 citations

Absolute Schmidt number: characterization, detection and resource-theoretic quantification

Bivas Mallick, Saheli Mukherjee, Nirman Ganguly +1 more

The dimensionality of entanglement, quantified by the Schmidt number, is a valuable resource for a wide range of quantum information processing tasks. In this work, we introduce the notion of the absolute Schmidt number, referring to states whose Schmidt number cannot be increased by any global unitary transformation. We provide a characterization of the set of arbitrary-dimensional states whose Schmidt number is invariant under all global unitaries. Our approach enables us to develop both witness-based and moment-based techniques to detect nonabsolute Schmidt number states which could provide significant operational advantages through Schmidt number enhancement by global unitaries. We next formulate two resource-theoretic measures of nonabsolute Schmidt number states, based respectively on Schmidt number witness and robustness, and demonstrate an operational utility of the latter in a channel discrimination task. Finally, we extend our analysis to quantum channels by introducing a new class of channels that possess the absolute Schmidt number property. We derive a necessary and sufficient condition for identifying when a channel has the absolute Schmidt number property, confining our analysis to the class of covariant channels.

quant-pharXiv:2604.02439•Physical Review A•Apr 6, 2026•0 citations
Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness ENGE_{\rm NG}ENG​ in bipartite bosonic systems, which satisfies ENG=1E_{\rm NG}=1ENG​=1 for all Gaussian-entanglable states, while ENG>1E_{\rm NG}>1ENG​>1 certifies non-Gaussian entanglement. Its ceiling d=⌈ENG⌉d=\lceil E_{\rm NG}\rceild=⌈ENG​⌉ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four projective measurements, where an analytical Gaussian-entanglable threshold is derived. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.
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.
The dimensionality of entanglement, quantified by the Schmidt number, is a core resource for quantum information processing. While its certification is well-established in finite-dimensional systems, many platforms harnessing high Schmidt numbers are natively continuous-variable (CV) systems, which are usually truncated to a finite subspace before canonical discrete-variable techniques are applied. For those instances, a core question is whether directly utilizing a genuinely CV toolbox leads to better and more robust characterisations. We answer this question by introducing Schmidt-number witnesses built directly from correlations and covariances of phase-space observables, accessible through characteristic- or Wigner-function measurements. They hold for arbitrary mixed states, including non-Gaussian ones, and only require correlations along a slice of phase space. We show that this direct estimation is more robust and versatile than first discretizing the system and using canonical discrete variable techniques. Moreover, we apply our method to certify arbitrarily-large Schmidt numbers of a bound-entangled Gaussian state. These results motivate genuine CV methods for characterizing high-dimensional entanglement across CV quantum technologies.
The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.
Efficiently certifying non-Gaussian entanglement in continuous-variable quantum systems is a central challenge for advancing quantum information processing, photonic quantum computing, and metrology. Here, we put forward continuous-variable counterparts to the recently introduced entanglement criteria based on moments of the partially transposed state, together with simple readout schemes that require only a few replicas of the state, passive linear optics, and particle-number measurements. We develop a multicopy method that enables the detection of strongly non-Gaussian entanglement for various relevant state families overlooked by standard approaches, which include the entire class of NOON states. Further, it is robust against realistic experimental constraints (losses, noise, and finite statistics), which we demonstrate by extensive numerical simulations.
Non-Gaussian bosonic states are ubiquitous in interacting light--matter systems, many-body platforms, and relativistic quantum field settings, but their quantitative characterization is hindered by the infinite-dimensional Hilbert space and by the poor scalability of Fock-space truncation methods. We introduce an exact finite-manifold encoding for states supported on a finite span of Gaussian branches, enabling the use of standard finite-dimensional quantum-information tools directly on an effective density matrix whose entries are determined by Gaussian overlaps. As demonstrations, we obtain closed-form and numerically stable evaluations of entropies and relative-entropy non-Gaussianity, and derive an analytic expression for the bipartite entanglement negativity of arbitrary multimode two-branch Gaussian superpositions, including a minimal which-branch dephasing model. Our framework provides a practical bridge between experimentally accessible continuous-variable resources (e.g., cat-like and measurement-conditioned states) and discrete-variable information measures, with immediate applications to benchmarking non-Gaussian resources in several quantum technology platforms.
The dimensionality of entanglement, quantified by the Schmidt number, is a valuable resource for a wide range of quantum information processing tasks. In this work, we introduce the notion of the absolute Schmidt number, referring to states whose Schmidt number cannot be increased by any global unitary transformation. We provide a characterization of the set of arbitrary-dimensional states whose Schmidt number is invariant under all global unitaries. Our approach enables us to develop both witness-based and moment-based techniques to detect nonabsolute Schmidt number states which could provide significant operational advantages through Schmidt number enhancement by global unitaries. We next formulate two resource-theoretic measures of nonabsolute Schmidt number states, based respectively on Schmidt number witness and robustness, and demonstrate an operational utility of the latter in a channel discrimination task. Finally, we extend our analysis to quantum channels by introducing a new class of channels that possess the absolute Schmidt number property. We derive a necessary and sufficient condition for identifying when a channel has the absolute Schmidt number property, confining our analysis to the class of covariant channels.