Direct Measurement Methods of Density Matrix of an Entangled Quantum State

In general, the state of a quantum system represented by density operator and its determination is a fundamental problem in quantum mechanics. A method of direct measurement of matrix element of density operator of a single two dimensional quantum system using weak measurement was theoretically proposed[Phys. Rev. Lett. 134, 070402(2012)] and experimentally demonstrated[Phys. Rev. Lett. 117, 120401(2016)] by Lundeen et al. Furthermore, recently the Guo-Guang Can et al.[Phys. Rev. Lett. 123, 150402(2019)] investigated the method of direct measurement of a nonlocal entangled quantum state. However, up to now the methods of directly measure the matrix element of density operator of an entangled quantum state have not been explicitly studied yet. As an extension of previous works, in this study we introduce two theoretical methods such as using postselected weak measurement and sequential measurements of triple products of complementary observables to direct measurement of matrix elements of density operator of two photon entangled quantum system, and discuss the similarity and the feasibility of those methods.