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

TL;DR

The paper proposes a measurement-efficient framework to classify distillability of $2\times4$ quantum states by predicting the PPT vs NPT signature and the number $\xi$ of negative eigenvalues in the partial transpose $\rho^{\Gamma}$. It compares fixed collective measurement witness features with learnable observables across ANN, SVM, and RF, finding that learnable observables consistently improve performance and saturate around $k\approx64$ observables. A key finding is the intrinsic difficulty of separating NPT$_1$ from NPT$_2$ due to the complex geometry of the $2\times4$ Hilbert space, as evidenced by t-SNE analyses and cross-model results. The work frames its ML approach as a practical screening tool for distillability that avoids full tomography, with potential applicability to quantum repeaters, QKD, and high-dimensional resource management, and it highlights future directions toward quantum-enhanced learning and scalable generalizations to $2\times N$ systems.

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

Figures (8)

• Figure 1: Schematic representation of the machine learning pipeline for quantum state classification. (a) Bipartite quantum state $\rho$ shared between subsystems $\mathcal{H}_A$ and $\mathcal{H}_B$. (b) Measurements performed on the quantum state, yielding observable expectation values as input for learning algorithms. (c) Machine learning classifiers: Support Vector Machine (SVM), Artificial Neural Network (ANN), and Random Forest (RF) trained on the measurement data to distinguish between different entanglement classes. (d) Classification output: states are categorized as either non-distillable (PPT, including separable and bound entangled states) or distillable (NPT, offering resources for entanglement distillation).
• Figure 2: Summary of the results obtained with the different models as a function of $k$. All plots are generated using $10^6$ quantum states, split into training and test sets, and each data point represents the average over 10 independent training runs. (a) Mean Macro-F1 score, defined in Eq. \ref{['eq:Macro-F1']}, as a function of $k$. Models employing learnable observables achieve comparable performance and consistently outperform the model based on CWM. Insets show the corresponding standard deviations. (b) Class-wise F1 scores, with colors and markers matching those in panel (a). The model performance is lowest for the NPT$_1$ and NPT$_2$ classes, whereas NPT$_0$ states are correctly classified in the majority of cases.
• Figure 3: Plots representing the t-SNE of the expectation values of 64 learned observables for different classes, each made up of $10^3$ different states. Each subplot highlights a specific class to improve readability, though all points are part of the same t-SNE embedding. In the top row the embedding includes NPT$_0$, NPT$_1$ and NPT$_2$ states. These classes appear largely intertwined, although the NPT$_0$ shows some clustering. On the bottom row we changed the NPT$_0$ states with separable product states. In this case, while NPT$_1$ and NPT$_2$ remain heavily mixed, the product states forms a well separated cluster.
• Figure 4: Summary of the mixed state method for generate quantum states. The number of sampled states per class depends on the value of $n$. For each $n$ 5000 samples have been generated. (a) Histogram for the mixed state method. (b) Purity behavior for the different classes.
• Figure 5: Analysis of the transition from NPT$_2$ to NPT$_1$ for $\alpha \to 1$ in Eq. \ref{['eq:mixture_of_two']}. (a) Probability of sampling an NPT$_1$ or NPT$_2$ state as a function of $\alpha \to 1$ highlighting the point $\tilde{\alpha}$ where the probability is $50\%$ showing the transition from NPT$_2$ to NPT$_1$. (b) Mean value with standard deviation of the $2^{nd}$ negative eigenvalue as a function of the parameter $\alpha$ in Eq. \ref{['eq:mixture_of_two']}. To obtain the mean value, we sample 1000 density matrices for 200 values of $\alpha$ between $0.5$ and $1$ and we take the lower of the two negative eigenvalues after partial transposition. Inset is a zoom for $\alpha \in [0.990,1]$