Direct measurement of density-matrix elements with a phase-shifting technique on a quantum photonic chip

A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. Until now, most direct measurement protocols have relied on ancillary pointers, which add complexity and are not always easy to implement experimentally. In this paper, we experimentally validate a direct measurement protocol requiring no ancillary pointers that calculates the value of a specific element of a quantum state???s density matrix using only six projection measurements. We prepare path-encoded arbitrary four-dimensional quantum states on a silicon-based quantum photonic chip and measure the density-matrix elements of the four-dimensional quantum states using this protocol, with experimental results deviating from ideal values by an average of only 0.010 ?? 0.004. This approach has the potential to be applied to 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.