Two-qubit pure state tomography by five product orthonormal bases**Project supported partially by the National Key Research and Development Program of China (Grant No. 2016YFB1000902), the National Natural Science Foundation of China (Grant No. 61472412), and the Program for Creative Research Group of the National Natural Science Foundation of China (Grant No. 61621003).

Abstract

In this paper, we focus on two-qubit pure state tomography. For an arbitrary unknown two-qubit pure state, separable or entangled, it has been found that the measurement probabilities of 16 projections onto the tensor products of Pauli eigenstates are enough to uniquely determine the state. Moreover, these corresponding product states are arranged into five orthonormal bases. We design five quantum circuits, which are decomposed into the common gates in universal quantum computation, to simulate the five projective measurements onto these bases. At the end of each circuit, we measure each qubit with the projective measurement {|0⟩⟨0|,|1⟩⟨1|}. Then, we consider the open problem whether three orthonormal bases are enough to distinguish all two-qubit pure states. A necessary condition is given. Suppose that there are three orthonormal bases . Denote the unitary transition matrices from to as U1 and U2. All 32 elements of matrices U1 and U2 should not be zero. If not, these three bases cannot distinguish all two-qubit pure states.