Direct measurement of density-matrix elements using a phase-shifting technique

A direct measurement protocol allows reconstructing specific elements of the density matrix a quantum state without using tomography. However, protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts investigated system extract information about specified elements. Here, we present scheme phase-shifting operations do not need pointers. In this method, estimates at most six expectation values projective observables suffice determine any element unknown matrix. concrete circuit implement for multiqubit states is provided, composed just single-qubit gates and two controlled-phase gates. This also extended continuous-variable states. Our method can be used in applications where only partial needs extracted, example, problems such as entanglement witnessing, fidelity estimation systems, coherence estimation.