Determining the mixed high-dimensional Bell state of a photon pair through the measurement of a single photon

Recently, a theoretical method and corresponding experiment have shown that the information about entanglement in the state of the qubit photon pair can be retrieved by measuring only one of the photons, even though the state is mixed. In this paper, we investigate this process with the high-dimensional photon pair and more information about the state. We present a method that enables us to determine the state of the photon pair fully. Our method measures only one of the photons while dropping the other one. Importantly, no adjustment of the transformation is required in our method. Moreover, our method is robust to photon loss so that the state can be perfectly determined as long as the photon is not all lost.