Effective detection of quantum discord by using convolutional neural networks

Abstract

Quantum discord represents a type of correlation defined as the difference between quantum and classical mutual information of two parties. Due to the optimization involved in the definition of classical mutual information of quantum systems, calculating and distinguishing between discordant and non-discordant states is not a trivial task. Furthermore, complete tomography of a quantum state is the prerequisite for computing its quantum discord, which can be resource-intensive. In this work, by using the relationship between the kernels of convolutional layers in artificial neural networks and the expectation values of operators in quantum measurements, we present a convolutional neural network (CNN) that utilizes 16 kernels to completely distinguish between discordant and non-discordant general two-qubit states. Additionally, we introduce a branching convolutional neural network (BCNN) capable of detecting quantum discord with accuracies of approximately 83%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$83\%$$\end{document} and 98%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$98\%$$\end{document} using only 5 and 8 kernels, respectively. We also investigate the robustness of our network’s performance against the noise. Our results demonstrate that, to achieve the desired accuracy in detecting quantum discord, one can employ appropriate quantum circuits to directly measure the expectation values of the kernels, bypassing the need for complete state tomography. Subsequently, a fully connected network can be implemented to address the presence or absence of quantum discord.