Abstract Quantum discord characterizes non-classical correlations between two or more subsystems that persist even in the absence of entanglement. Unlike quantum entanglement, this measure captures more general quantum correlations, yet its estimation remains challenging due to the inherent difficulty in determining the optimal measurement orientations. In this work, we present a machine learning approach to estimate quantum discord for two-qubit systems with the help of artificial neural networks. Two distinct input strategies are investigated: one involving complete quantum state information derived from the density matrix elements and the other involving partial information derived from the expected values of measurements. Numerical simulations demonstrate that both artificial neural network architectures can predict the quantum discord of new quantum states with high precision. Our findings provide a new way to estimate quantum correlations, demonstrating the effectiveness of machine learning in quantum information tasks.
Abstract Quantum discord characterizes non-classical correlations between two or more subsystems that persist even in the absence of entanglement. Unlike quantum entanglement, this measure captures more general quantum correlations, yet its estimation remains challenging due to the inherent difficulty in determining the optimal measurement orientations. In this work, we present a machine learning approach to estimate quantum discord for two-qubit systems with the help of artificial neural networks. Two distinct input strategies are investigated: one involving complete quantum state information derived from the density matrix elements and the other involving partial information derived from the expected values of measurements. Numerical simulations demonstrate that both artificial neural network architectures can predict the quantum discord of new quantum states with high precision. Our findings provide a new way to estimate quantum correlations, demonstrating the effectiveness of machine learning in quantum information tasks.
Quantum entanglement acts as a crucial part in quantum computation and quantum information, hence quantifying unknown entanglement is an important task. Due to the fact that the amount of entanglement cannot be achieved directly by measuring any physical observables, it remains an open problem to quantify entanglement experimentally. In this work, we provide an effective way to quantify entanglement for the unknown quantum states via artificial neural networks. By choosing the expectation values of measurements as input features and the values of entanglement measures as labels, we train artificial neural network models to predict the entanglement for new quantum states accurately. Our method does not require the full information about unknown quantum states, which highlights the effectiveness and versatility of machine learning in exploring quantum entanglement.
The paper suggest employing machine learning for resource-efficient classification of quantum correlations in entanglement distribution networks. Specifically, artificial neural networks (ANN) are utilized to classify quantum correlations based on collective measurements conducted in the geometry of entanglement swapping. ANNs are trained to categorize two-qubit quantum states into five mutually exclusive classes depending on the strength of quantum correlations exhibited by the states. The precision and recall of the ANN models are analyzed as functions of the quantum resources consumed, i.e. the number of collective measurements performed.
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 nondiscordant. We find that the near-zero value of the recognition loss reproduces the shape of the nondiscordant separable states with high accuracy, especially considering the nontrivial 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.
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well-known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord for two-qubit states. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.
Quantum entanglement is a key resource in quantum computing and quantum information processing tasks. However, its quantification remains a major challenge since it cannot be directly extracted from physical observables. To address this issue, we study a few machine-learning based models to estimate the amount of entanglement in two-qubit as well as three-qubit systems. We use measurement outcomes as the input features and entanglement measures as the training labels. Our models predict entanglement without requiring the full state information. This demonstrates the potential of machine learning as an effcient and powerful tool for characterizing quantum entanglement