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