Direct Measurement of the Quantum Density Matrix without Ancillary Pointers.

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 ancillary pointers, which interacts with the investigated system to extract information about the specified elements. Here we present a new direct measurement protocol based on phase-shifting technique which do not need ancillary pointers. In this protocol, at most six different projective measurements suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement the phase-shifting measurement protocol for multi-qubit states is provided, where the circuit is composed of just single-qubit gates and two multi-qubit controlled-phase gates. This protocol is also extended to the continuous-variable cases for directly measuring the Wigner function. Furthermore, we show that the protocol has the advantage of reducing measurement and device complexity in the task of measuring the complete density matrix compared to quantum state tomography in some quantum experiments. Our method provides an efficient way to characterize arbitrary quantum systems, which may be used to predict various properties of a quantum system and find applications in quantum information processing.