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arXiv:2602.19307

Learning partial transpose signatures in qubit ququart states from a few measurements

Christian Candeago, Paolo Da Rold, Michele Grossi, Pawel Horodecki, Antonio Mandarino

quant-phSubmitted Feb 22, 2026• Announced Feb 24, 2026• 0 citations
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Abstract

Higher-dimensional quantum systems are attracting interest for improving quantum protocol performance by increasing memory space. Characterizing quantum resources of such systems is fundamental but experimentally costly. We tackle the first non-trivial example: a qubit-ququart system, focusing on partial-transpose spectral classification. Entanglement distillation extracts maximally entangled states from noisy resources, but determining distillability typically requires full state tomography, experimentally prohibitive for high-dimensional systems. We explore a machine learning framework to classify distillable bipartite quantum states using fewer measurements than complete tomography. Our approach employs the PPT criterion, categorizing states by negative eigenvalues in the partial transpose. We use various ML algorithms, including Support Vector Machines, Random Forest, and Artificial Neural Networks, with features from fixed measurements and learnable observables. Results show learnable observables consistently outperform Collective Measurement Witnesses methods. While all models distinguish between non-distillable (PPT) and distillable (NPT) states, differentiating NPT subclasses remains challenging, underscoring the intricate Hilbert space geometry. This work provides an experimentally friendly tool for distillability verification in high-dimensional quantum systems without full state reconstruction

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We show that a bipartite quantum state of rank four is distillable if its partial transpose has at least one negative eigenvalue; that is, the state is a non-positive-partial transpose (NPT). For this purpose we prove that if the partial transpose of a two-qutrit NPT state has at least two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT state of rank five which is not 1-distillable and show that it is not $n$-distillable for any given $n$ when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.

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Higher-dimensional quantum systems are attracting interest for improving quantum protocol performance by increasing memory space. Characterizing quantum resources of such systems is fundamental but experimentally costly. We tackle the first non-trivial example: a qubit-ququart system, focusing on partial-transpose spectral classification. Entanglement distillation extracts maximally entangled states from noisy resources, but determining distillability typically requires full state tomography, experimentally prohibitive for high-dimensional systems. We explore a machine learning framework to classify distillable bipartite quantum states using fewer measurements than complete tomography. Our approach employs the PPT criterion, categorizing states by negative eigenvalues in the partial transpose. We use various ML algorithms, including Support Vector Machines, Random Forest, and Artificial Neural Networks, with features from fixed measurements and learnable observables. Results show learnable observables consistently outperform Collective Measurement Witnesses methods. While all models distinguish between non-distillable (PPT) and distillable (NPT) states, differentiating NPT subclasses remains challenging, underscoring the intricate Hilbert space geometry. This work provides an experimentally friendly tool for distillability verification in high-dimensional quantum systems without full state reconstruction
Negativities in quasiprobability distributions, a foundational concept originating in quantum optics, serve as a fundamental signature of quantum nonclassicality, with entanglement quasiprobabilities offering a necessary and sufficient criterion for entanglement. However, practical reconstruction of entanglement quasiprobabilities conventionally requires full quantum state tomography, severely limiting scalability. Here, we propose a deep-learning framework that reconstructs entanglement quasiprobabilities directly from incomplete local projective measurements, bypassing full state reconstruction. Using a residual neural network, partial measurement outcomes are mapped to high-fidelity entanglement quasiprobabilities. Numerical benchmarks up to three qubits show more than a 30×30\times30× reduction in reconstruction error compared with state-of-the-art tomographic methods. Experimental validation on photonic entangled states demonstrates reconstruction and entanglement detection with substantially reduced measurement resources. Our results establish machine-learning-assisted reconstruction of entanglement quasiprobabilities as a scalable and practical tool for entanglement characterization in quantum optical systems.
Abstract We present a systematic comparative analysis of machine learning and traditional approaches for quantum entanglement characterization in bi-partite ququart systems. Quantifying entanglement traditionally requires full quantum state tomography, necessitating Nmub2D2NcopN_{mub}^2D^2N_{cop}Nmub2​D2Ncop​ measurements, where NmubN_{mub}Nmub​ is the number of mutually unbiased bases, DDD is the subsystem dimensionality, and NcopN_{cop}Ncop​ is the number of identical copies. For ququart systems with D=4D=4D=4, this translates to hundreds of distinct measurement settings, each requiring multiple copies (NcopN_{cop}Ncop​ typically in thousands)—resulting in hundreds of thousands of total measurements in practice. Our research demonstrates that neural networks achieve comparable or superior accuracy with orders of magnitude fewer measurement settings. We evaluate three architectures—Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), and Transformer—against Maximum Likelihood Estimation (MLE) and Bayesian methods across measurement counts ranging from 10 to 400. Neural approaches achieve up to 1650× faster computation times compared to traditional methods while maintaining competitive accuracy. At 100 measurements, the Transformer achieves Mean Squared Error (MSE) of 1.49×10−11.49 \times 10^{-1}1.49×10−1, while MLE yields 2.04×10−12.04 \times 10^{-1}2.04×10−1—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately 1/N1/\sqrt{N}1/N​ with increased measurements. While this observed scaling might be influenced by our specific architectures, it provides important practical guidance for experimental design. The Transformer architecture demonstrates exceptional sample efficiency, achieving with 100 measurements what traditional methods require 250-400 measurements to accomplish—a significant advantage for resource-constrained quantum experiments. This research provides a viable pathway for real-time entanglement characterization in higher-dimensional quantum systems where traditional methods become computationally prohibitive.
Abstract This work presents a machine learning approach based on support vector machines (SVMs) for quantum entanglement detection. Particularly, we focus on bipartite systems of dimensions 3×33\times 33×3 3 × 3 , 4×44\times 44×4 4 × 4 , and 5×55\times 55×5 5 × 5 , where the positive partial transpose criterion (PPT) provides only partial characterization. Using SVMs with quantum-inspired kernels, we develop a classification scheme that distinguishes between separable and entangled states, including PPT-detectable entangled states, and entangled states that evade PPT detection. Our method achieves increasing accuracy with system dimension, reaching 80%80\%80% 80 % , 90%90\%90% 90 % , and nearly 100%100\%100% 100 % for 3×33\times 33×3 3 × 3 , 4×44\times 44×4 4 × 4 , and 5×55\times 55×5 5 × 5 systems, respectively. Our results show that principal component analysis significantly enhances performance for small training sets. The study reveals important practical considerations regarding purity biases in the generation of data for this problem and examines the challenges of implementing these techniques on near-term quantum hardware. Our results demonstrate that machine learning can be an effective alternative to entanglement detection of higher-dimensional systems where conventional entanglement detection methods struggle. Our approach provides improved data generation protocols and can be readily implemented in hybrid classical-quantum architectures, overcoming current limitations.
This paper explores the use of machine learning techniques for detecting quantum entanglement in two-qubit systems. A dataset of randomly generated quantum states was constructed and labeled using the Positive Partial Transpose (PPT) criterion. Several supervised learning models, including Random Forest and Multi-Layer Perceptron neural networks, were trained to classify quantum states as entangled or separable. The results show that machine learning models can effectively learn patterns from quantum state data, while also highlighting the impact of dataset imbalance on classification performance. This study demonstrates the potential of machine learning as a complementary tool for quantum entanglement detection and provides insights into the challenges and opportunities of integrating artificial intelligence with quantum physics.
The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing. Despite the many methods---such as the positive partial transpose criterion and the kkk-symmetric extendibility criterion---to tackle this problem, none of them enables a general, practical solution due to the problem's NP-hard complexity. Explicitly, separable states form a high-dimensional convex set of vastly complicated structures. In this work, we build a different separability-entanglement classifier underpinned by machine learning techniques. We use standard tools from machine learning to learn the entanglement feature of arbitrary given quantum states. We perform substantial numerical tests on two-qubit and two-qutrit systems, and the results indicate that our method can outperform the existing methods in generic cases in terms of both speed and accuracy. This opens up avenues to explore quantum entanglement via the machine learning approach.
We show that a bipartite quantum state of rank four is distillable if its partial transpose has at least one negative eigenvalue; that is, the state is a non-positive-partial transpose (NPT). For this purpose we prove that if the partial transpose of a two-qutrit NPT state has at least two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT state of rank five which is not 1-distillable and show that it is not nnn-distillable for any given nnn when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.