Low resource entanglement classification from neural network interpretability

TL;DR

The paper tackles entanglement identification from incomplete Pauli measurements by proposing a unified, interpretable ML framework that handles 2- and 3-qubit states across pure and mixed regimes. It trains dense and 1D CNN architectures on Pauli-tomography inputs, benchmarks performance, and uses SHAP-derived measurement-importance profiles to guide measurement-reduction schemes. A two-tier Shapley-approximation approach and cross-model averaging yield robust global rankings of measurements, revealing redundancy in pure-state tomography and structured importance blocks in mixed states. Comparisons with analytic entanglement criteria and witnesses validate the approach and expose the limitations of Shapley interpretability in high-redundancy settings. Overall, the work provides practical guidelines for minimal measurement resources and highlights when Shapley-based explanations produce reliable physical insight for quantum entanglement classification.

Abstract

Entanglement is a central resource in quantum information and quantum technologies, yet its characterization remains challenging due to both theoretical complexity and measurement requirements. Machine learning has emerged as a promising alternative, enabling entanglement characterization from incomplete measurement data, however model interpretability remains a challenge. In this work, we introduce a unified and interpretable framework for SLOCC entanglement classification of two- and three-qubit states, encompassing both pure and mixed states. We train dense and convolutional neural networks on Pauli-measurement outcomes, provide design guidelines for each architecture, and systematically compare their performance across types of states. To interpret the models, we compute Shapley values to quantify the contribution of each measurement, analyze measurement-importance patterns across different systems, and use these insights to guide a measurement-reduction scheme. Accuracy-versus-measurement curves and comparisons with analytical entanglement criteria demonstrate the minimal resources required for reliable classification and highlight both the capabilities and limitations of Shapley-based interpretability when using machine learning models for entanglement detection and classification.

Figures (27)

• Figure 1: In this graph, nodes represent the different coalitions $S$ (subsets of players that may contribute to the game) with associated values $f(S)$. Arrows represent the process of adding one player $i$ to $S$. Labels on the arrows show the (marginal) contribution of each addition. That is the difference between the value of the coalition with and without such player $f(S\cup\{i\})-f(S)$. Any full path $\emptyset \to \{1,2,3\}$ corresponds to one ordering of players. To compute the Shapley value of a given player, list all the orders that can be constructed $\emptyset \to \{1,2,3\}$. For each path, extract the marginal contribution to each player to the game (this is shown in the table under marginal contributions on the figure). Then, compute the average of all the $3! = 6$ marginal contributions. This corresponds to the coefficient $\frac{|S|! (|F| - |S| - 1)!}{|F|!}$ in Eq. \ref{['eq.:shap']}.
• Figure 2: Schematic representation of the method presented. (a) We first prepare the dataset, which is divided into subsets corresponding to all SLOCC entanglement class. (b) For each state of the dataset, we compute its complete Pauli basis decomposition by simulating a full tomography measurement scenario. (c) The measuremets are used as data for different machine learning models, whose task is to perform SLOCC entanglement classification on the states of the dataset. (d) After the machine learning models are trained, we apply an interpretability algorithm (approximate Shapley values), to obtain the importance of each measurement setting for the entanglement classification. (e) Finally, we bechmark the importance distributions on number of measurements vs. accuracy numerical experiments. As a result, we find minimum sets of measurement required for entanglement classification as a function of the accuracy requirements.
• Figure 3: Matrix plots showing the re-scaled Shapley values for individual states. The re-scaling is done through the function $g = \sqrt{1 - (|\chi| - 1)^2}$, where $|\chi|$ denotes Shapley values normalized to the interval $[0, 1]$. Orange plots show the measurement settings ordered by lexicographic order $j$, while blue plots show measurement settings ordered by average Shapley value importance $\tilde{\jmath}$. Panels (a) and (b) are for mixed two-qubit states, while panels (c) and (d) are for pure three-qubit states.
• Figure 4: Normalized distribution of Shapley values for different models sorted in increasing order of importance, $\tilde{\jmath}$. Panels (a) and (b) show the distributions for two different two-qubit models. We have highlighted with different colors the Shapley values corresponding to measurement settings $j = 8$ (red), and $j=15$ (purple). Due to variability between models orders, they appear in different positions when ordered by importance ($\tilde{\jmath}$) deppending on the model. Measurement setting $j = 8$ and $j = 15$ are assigned importance orders $\tilde{\jmath} = 3, 14$ and $\tilde{\jmath} = 13, 1$ on models models (a) and (b), respectively. Panel (c) corresponds to a three-qubit model.
• Figure 5: Test accuracies of neural networks trained with a progressively reduced number of measurements. (a) For two-qubit states, results are shown for measurement removal in both increasing and decreasing order of importance for a single model (model-dependent), and for the aggregated order across all models (model-independent). (b) For three-qubit states, results are shown for the aggregated increasing/decreasing orders and a random order for comparison.