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arXiv:2505.02297 · Annalen der Physik

Estimating the Schmidt Numbers of Quantum States via Symmetric Measurements

Hao‐Fan Wang, Shao‐Ming Fei

Published Aug 4, 2025• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/andp.202500259

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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.
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 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.
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.
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.
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.