Experimental measurement of bipartite entanglement using parameterized quantum circuits

Entanglement in quantum systems plays a crucial role in various quantum information tasks. Measuring entanglement has been an important issue in both experiments and theories. In this work, we use parameterized quantum circuits (PQCs) to diagonalize density matrices of quantum states and obtain entanglement by only measuring the diagonal elements. With this method, full quantum state tomography can be bypassed, greatly reducing the number of measurements. A comprehensive characterization of entanglement was performed by using Rényi entropy and partially transposed moments. Mutual information, calculated from entropy is also used to characterize dynamical quantum phase transitions. We experimentally demonstrated the method on a four-qubit nuclear magnetic resonance quantum simulator. Our results agree with the theoretical descriptions. The measurement complexity of our PQC-based method grows linearly with the number of diagonal elements in the density matrix, a square root reduction over the full quantum tomography. The proposed method can have great potential in quantum systems with a large number of particles.