Entanglement and discord classification via deep learning

TL;DR

This work addresses the challenge of identifying entanglement and quantum discord in bipartite $d\times d$ states with $d\in\{2,...,7\}$, including hard cases like bound entanglement. It introduces a convolutional autoencoder trained unsupervised on separable states, using reconstruction error as a discriminant to separate entangled from separable states and to reveal local-unitary invariance in the learned representations. A bound-entangled-state generation scheme is built atop the classifier, validated via the symmetric-extension criterion, while a parallel CAE setup achieves rapid, high-accuracy discord detection on CC/CQ/QC states, often in a single epoch. The approach yields strong performance across dimensions, provides a scalable tool for entanglement verification, and supplies a practical method to generate and certify bound entangled samples for $d=3$ to $7$, with potential impact on quantum resource tasks and benchmarking.

Abstract

In this work, we propose a deep learning-based approach for quantum entanglement and discord classification using convolutional autoencoders. We train models to distinguish entangled from separable bipartite states for $d \times d$ systems with local dimension $d$ ranging from two to seven, which enables identification of bound and free entanglement. Through extensive numerical simulations across various quantum state families, we demonstrate that our model achieves high classification accuracy. Furthermore, we leverage the learned representations to generate samples of bound entangled states, the rarest form of entanglement and notoriously difficult to construct analytically. We separately train the same convolutional autoencoders architecture for detecting the presence of quantum discord and show that the model also exhibits high accuracy while requiring significantly less training time.

Figures (8)

• Figure 1: Schematic of the CAE architecture. The input density matrix $\rho$ undergoes dimensionality reduction through the encoder $E$ to a latent representation. The decoder $D$ then maps this representation back to the original Hilbert space dimensions to produce the reconstructed state $\hat{\rho} = D_\theta(E(\rho))$, where $\theta$ represents the weights of the entire network.
• Figure 2: Reconstruction errors $\left\lVert \hat{\rho}_i - \rho_i \right\rVert_1$ of bipartite $d\times d$ quantum states. Orange squares correspond to NPT states whereas blue circles correspond to separable states generated according to Equation (\ref{['eq:bipartite_sep']}).The horizontal black line indicates the decision threshold $\epsilon_d$; reconstruction errors below this line are classified as separable, while those above are identified as non-separable.
• Figure 3: Reconstruction errors $\left\lVert \hat{\rho}_i - \tilde{\rho}_i \right\rVert_1$ of the validation set after applying local unitaries $\tilde{\rho}_i =U_A \otimes U_B\left( \rho_i \right) U^{\dagger}_A \otimes U^{\dagger}_B$. Orange squares correspond to NPT states whereas blue circles correspond to separable states generated according to Equation (\ref{['eq:bipartite_sep']}).
• Figure 4: Validation accuracy for entanglement classification as a function of dimension $d$. The model demonstrates robust performance ($>98\%$) for most states, with the exception of NPT and NPT-rotated (after applying local unitaries) in the $d=2$ bipartite system.
• Figure 5: Reconstruction errors $\left\lVert \hat{\rho}_i - \rho_i \right\rVert_1$ for samples of the Horodecki-like family of bound entangled states before and after local unitaries are applied.