Two-qubit pure state tomography by five product orthonormal bases

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 B₁,B₂,B₃. Denote the unitary transition matrices from B₁ to B₂,B₃ 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.