© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Annalen der Physik

Efficient Self‐Guided Estimation of Schmidt Number via Local Measurements

Tian‐Ze Yang, Jun Tang, Wen‐Hao Zhang, Liu Ye, Dong Wang

Published Jan 1, 2026• 0 citations

Abstract

ABSTRACT Recent advances in experimental quantum platforms have enabled the development of devices capable of generating high‐dimensional quantum states, facilitating their use in quantum information processing tasks. High‐dimensional entangled states offer significant advantages in areas such as quantum key distribution, quantum communication, and quantum metrology. While the Schmidt rank (SR) characterizes entanglement for pure bipartite states in high‐dimensional spaces, its extension to mixed states is the Schmidt number (SN). However, experimentally determining the SN of unknown mixed states remains a significant challenge. Here, we propose an experimentally feasible method to determine the SN of unknown states, constructing a self‐guided SN witness scheme. Leveraging techniques from quantum state verification (QSV), this method allows SN estimation using only local measurements, thereby avoiding the need for direct projections onto high‐dimensional maximally entangled states. The self‐guided aspect is implemented through a Stochastic Gradient Algorithm (SGA) that systematically explores the space of local unitary operations to maximize a fidelity related to Schmidt number witnesses. We numerically demonstrate that for four‐dimensional entangled states, our scheme accurately estimates the SN, typically converging within approximately 200 iterations. Our results also show that the method is robust to realistic imperfections, highlighting its potential for practical applications in quantum information processing.

DOI: 10.1002/andp.202500201

Related papers

Construction of efficient Schmidt-number witnesses for high-dimensional quantum states

Nikolai Wyderka, Giovanni Chesi, Hermann Kampermann +2 more

Recent progress in quantum optics has led to setups that are able to prepare high-dimensional quantum states for quantum information processing tasks. As such, it is of importance to benchmark the states generated by these setups in terms of their quantum mechanical properties, such as their Schmidt numbers, i.e., the number of entangled degrees of freedom. In this paper, we develop an iterative algorithm that finds Schmidt number witnesses tailored to the measurements available in specific experimental setups. We then apply the algorithm to find a witness that requires the measurement of a number of density matrix elements that scales linearly with the local dimension of the system. As a concrete example, we apply our construction method to an implementation with photonic temporal modes.

arXiv:2210.05272•Physical Review A•Feb 27, 2023•22 citations

Characterizing High Schmidt Number Witnesses in Arbitrary Dimensions System

Liang Xiong, Nung-Sing Sze

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.

arXiv:2504.11213•Apr 15, 2025•0 citations

Detecting high-dimensional entanglement by randomized product projections

Jin‐Min Liang, Shuheng Liu, Shao-Ming Fei +1 more

Abstract The characterization of high-dimensional entanglement plays a crucial role in the field of quantum information science. Conventional entanglement criteria measuring coherent superpositions of multiple basis states face experimental bottlenecks on most physical platforms due to limited multi-channel control. Here, we introduce a practically efficient detection strategy based on randomized product projections. We show that the first-order moments of such projections can be used to estimate entanglement fidelity, thereby enabling practical and efficient certification of the Schmidt number (SN) in high-dimensional bipartite systems. By constructing optimal observables, it is sufficient to merely measure a single basis state, substantially reducing experimental overhead. Moreover, we present an algorithm to obtain a lower bound of the SN with a high confidence level from a limited number of experimental data. Our results open up resource-efficient experimental avenues to detect high-dimensional entanglement and test its implementations in modern information technologies.

arXiv:2501.01088•Quantum Science and Technology•Jan 21, 2026•1 citation

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

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

Measurement-device-independent Schmidt number certification of all entangled states

Saheli Mukherjee, Bivas Mallick, Arun Kumar Das +2 more

Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information-processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilization. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: It cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with a Schmidt number exceeding $r$, a semiquantum nonlocal game exists that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.

arXiv:2502.13296•Physical Review A•Nov 21, 2025•1 citation

ABSTRACT Recent advances in experimental quantum platforms have enabled the development of devices capable of generating high‐dimensional quantum states, facilitating their use in quantum information processing tasks. High‐dimensional entangled states offer significant advantages in areas such as quantum key distribution, quantum communication, and quantum metrology. While the Schmidt rank (SR) characterizes entanglement for pure bipartite states in high‐dimensional spaces, its extension to mixed states is the Schmidt number (SN). However, experimentally determining the SN of unknown mixed states remains a significant challenge. Here, we propose an experimentally feasible method to determine the SN of unknown states, constructing a self‐guided SN witness scheme. Leveraging techniques from quantum state verification (QSV), this method allows SN estimation using only local measurements, thereby avoiding the need for direct projections onto high‐dimensional maximally entangled states. The self‐guided aspect is implemented through a Stochastic Gradient Algorithm (SGA) that systematically explores the space of local unitary operations to maximize a fidelity related to Schmidt number witnesses. We numerically demonstrate that for four‐dimensional entangled states, our scheme accurately estimates the SN, typically converging within approximately 200 iterations. Our results also show that the method is robust to realistic imperfections, highlighting its potential for practical applications in quantum information processing.

Recent progress in quantum optics has led to setups that are able to prepare high-dimensional quantum states for quantum information processing tasks. As such, it is of importance to benchmark the states generated by these setups in terms of their quantum mechanical properties, such as their Schmidt numbers, i.e., the number of entangled degrees of freedom. In this paper, we develop an iterative algorithm that finds Schmidt number witnesses tailored to the measurements available in specific experimental setups. We then apply the algorithm to find a witness that requires the measurement of a number of density matrix elements that scales linearly with the local dimension of the system. As a concrete example, we apply our construction method to an implementation with photonic temporal modes.

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 The characterization of high-dimensional entanglement plays a crucial role in the field of quantum information science. Conventional entanglement criteria measuring coherent superpositions of multiple basis states face experimental bottlenecks on most physical platforms due to limited multi-channel control. Here, we introduce a practically efficient detection strategy based on randomized product projections. We show that the first-order moments of such projections can be used to estimate entanglement fidelity, thereby enabling practical and efficient certification of the Schmidt number (SN) in high-dimensional bipartite systems. By constructing optimal observables, it is sufficient to merely measure a single basis state, substantially reducing experimental overhead. Moreover, we present an algorithm to obtain a lower bound of the SN with a high confidence level from a limited number of experimental data. Our results open up resource-efficient experimental avenues to detect high-dimensional entanglement and test its implementations in modern information technologies.

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.

Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information-processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilization. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: It cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with a Schmidt number exceeding

r

, a semiquantum nonlocal game exists that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.