Two-qubit pure state tomography by five product orthonormal bases

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 {B 1 , B 2 , B 3 }. Denote the unitary transition matrices from B 1 to {B 2 , B 3 } as U 1 and U 2 . All 32 elements of matrices U 1 and U 2 should not be zero. If not, these three bases cannot distinguish all two-qubit pure states.

Unrestricted generation of pure two-qubit states and entanglement diagnosis by single-qubit tomography

“…The number of measurements depends on the measured state. Our protocol is thus of the adaptive type [15,16], much in the spirit of a very recent theoretical proposal [17]. In contrast to other layouts (see, e.g., [18]), ours does not require several phase shifters to deal with the binary path DOF.…”