Three-parameter symmetric quantum joint measurement on two qubits

A standard two-qubit joint measurement is the well-known Bell state measurement, in which each reduced state (traced out one qubit) is the completely mixed state. Recently, a novel quantum joint measurement known as the elegant joint measurement (EJM) has been proposed, where the reduced states of the EJM basis have tetrahedral symmetry. In this paper, we first propose a five-parameter entangled state and reveal its inherent symmetry. Based on this, we define a generalized EJM parametrized by z, phi, and 9, and provide the quantum circuits for preparing and detecting these basis states. There are three main results: (i) The previous singleparameter EJM can be directly derived by specifying the parameters z and phi; (ii) the initial unit vectors related to the four vertices of the regular tetrahedron are not limited to the original choice and not all the unit vectors in cylindrical coordinates are suitable for forming the EJM basis; and (iii) the reduced states of the present EJM basis can always form two mirror-image tetrahedra, robustly preserving its elegant properties. We aim to determine what kind of states the EJM basis belongs to and to provide a method for constructing the generalized three-parameter EJM, which may contribute to the multisetting measurement and the potential applications for quantum information processing.