Data-driven criteria for quantum correlations

TL;DR

A machine learning model to detect correlations in a three-qubit system using a neural network trained in an unsupervised manner on randomly generated states finds that the proposed detector performs much better at distinguishing a weaker form of quantum correlations, namely, the quantum discord, than entanglement.

Abstract

We build a machine learning model to detect correlations in a three-qubit system using a neural network trained in an unsupervised manner on randomly generated states. The network is forced to recognize separable states, and correlated states are detected as anomalies. Quite surprisingly, we find that the proposed detector performs much better at distinguishing a weaker form of quantum correlations, namely, the quantum discord, than entanglement. In fact, it has a tendency to grossly overestimate the set of entangled states even at the optimal threshold for entanglement detection, while it underestimates the set of discordant states to a much lesser extent. In order to illustrate the nature of states classified as quantum-correlated, we construct a diagram containing various types of states -- entangled, as well as separable, both discordant and non-discordant. We find that the near-zero value of the recognition loss reproduces the shape of the non-discordant separable states with high accuracy, especially considering the non-trivial shape of this set on the diagram. The network architecture is designed carefully: it preserves separability, and its output is equivariant with respect to qubit permutations. We show that the choice of architecture is important to get the highest detection accuracy, much better than for a baseline model that just utilizes a partial trace operation.

Figures (7)

• Figure 1: The 3-qubit neural network separator model. The input density matrix is convolved separately with three convolution layers, each with $4\times4$ kernels that work in parallel. The stride and dilation parameters defining these convolution layers are arranged in such a way that they operate on each qubit subspace separately (details can be found in SM sm). After the convolution layers, shape-preserving fully-connected layers are applied independently for each qubit matrix.
• Figure 2: (a) Separator performance for detecting discordant (green) and entangled (red) states, tested on $S_\mathrm{mixed}$. The results present: (left) precision vs. recall curves, and (right) balanced accuracy, $BA$ depending on the threshold $\tau$ value. For comparison, partial trace-based baseline model performance is also presented by blue and orange curves, respectively. (b) Same as (a) but for separator model with removed FC layers -- cf. Fig. \ref{['fig:separator']}. (c) Same as in (a) but for the separator trained on the non-product states only.
• Figure 3: Reconstruction loss $\mathcal{L}$ for the already trained separator model tested on the family of 3-qubit states with their separability/discordance conditions known and parameterized on 2D map (see inset for the map division into the classes).
• Figure S1: The separator autoencoder composed of trainable encoder neural network (gray triangle) and analytical decoder that simply calculates the Kronecker product.
• Figure S2: Same as in Fig. \ref{['fig:similarity_plot_0']} but for separator trained only on some part of the original training set: pure separable states (Pure), product states (Prod), non-discordant states (ZD), separable states (Sep) and non-product separable states (NPS). Results for separator without four fully-connected layers (containing only convolutional ones) are marked as "No FC4".