Improving direct state measurements by using rebits in real enlarged Hilbert spaces

We propose a protocol to improve the accuracy of direct complex state measurements (DSM) by using rebits in real Hilbert spaces. We show that to improve the accuracy, the initial complex state should be decomposed into the real and imaginary parts and stored in an extended state (rebit) which can be tracked individually by two bases of an extra qubit. For pure states, the numerical calculations show that the trace distances between the true state and the reconstructed state obtained from the rebit method are more precise than those ones obtained from the usual DSM and quantum state tomography (SQT) because the number of projective measurements is reduced. For mixed states, the rebit method gives the same accuracy in comparison to the usual DSM, while it is less precise than QST. Its precision is also significantly improved when using nearly-pure states. Our proposal holds promises as a reliable tool for quantum computation, testing of quantum circuits by using only real amplitudes.