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arXiv:2403.00248 · Communications in Theoretical Physics

Families of Schmidt-number witnesses for high dimensional quantum states

Xian Shi

Published Jul 4, 2024• 11 citations
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Abstract

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.

DOI
10.1088/1572-9494/ad48fb

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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.
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.
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.
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 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 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.