Ultimate precision of direct tomography of wave functions

In contrast to the standard quantum state tomography, the direct tomography seeks a direct access to the complex values of the wave function at particular positions. Originally put forward as a special case of weak measurement, it has been extended to arbitrary measurement setup. We generalize the idea of “quantum metrology,” where a real-valued phase is estimated, to the estimation of complex-valued phase. We show that it enables to identify the optimal measurements and investigate the fundamental precision limit of the direct tomography. We propose a few experimentally feasible examples of direct tomography schemes and, based on the complex phase estimation formalism, demonstrate that direct tomography can reach the Heisenberg limit.