Preparation and characterization of arbitrary states of four-dimensional qudits based on biphotons

We report interferometric schemes to prepare arbitrary states of four-dimensional qudits (ququarts) based on biphoton states of ultrafast-pumped frequency-nondegenerate spontaneous parametric down conversion. Preparation and tomographic characterization of a few examples of general single-ququart states, a pure state, a mixed state (fully diagonal), and a mixed state (partially coherent), are experimentally demonstrated.

Figure 1

(Color online) Schematic of the experiment. Ququart preparation: The collinear frequency-nondegenerate biphoton polarization state is transformed by the WP to prepare an arbitrary four-dimensional quantum state, a biphoton ququart. The theoretically expected quantum state of the ququart, ${\rho }_{\mathit{\text{theory}}}$, can be calculated by using the information on the experimental settings. Ququart tomography: The biphoton ququart is tomographically characterized to obtain the experimentally reconstructed density matrix ${\rho }_{\mathit{exp}}$. See the text for details.

Figure 2

(Color online) Real part of the experimentally reconstructed $\left({\rho }_{\mathit{exp}}\right)$ and the theoretically expected $\left({\rho }_{\mathit{\text{theory}}}\right)$ density matrices for a pure state ququart. The calculated fidelity is $F=0.938\pm 0.019$.

Figure 3

(Color online) Real parts of two mixed (fully diagonal) state ququarts with different entropy values. These states are generated using the double-crystal scheme shown in Fig. 1 with different pump polarization. (a) For the state shown in Eq. (9). $S\left({\rho }_{\mathit{exp}}\right)=0.405\pm 0.022$. (b) For the state shown in Eq. (12). $S\left({\rho }_{\mathit{exp}}\right)=0.510\pm 0.016$.

Figure 4

(Color online) A ququart with a mixed (partially coherent) state between a pure state and a mixed (fully diagonal) state. The calculated fidelity is $F=0.878\pm 0.029$. We obtain $\mathrm{Tr}\left[{\rho }_{\mathit{exp}}^2\right]=0.483$ and $S\left({\rho }_{\mathit{exp}}\right)=0.550\pm 0.003$.

Figure 5

(Color online) Scheme for preparing a general biphoton ququart. Pump HWP is used to control the relative magnitude of the SPDC amplitudes generated at the BBO. Four BBO (two type-I and two type-II) crystals are needed for this scheme.

Figure 6

(Color online) Proposed setup for arbitrary ququart preparation. This scheme reduces the four-path interferometer shown in Fig. 5 to a Mach-Zehnder interferometer, making the experimental implementation much easier. QP2, QP3, and PZT adjust the phase terms in Eq. (17), ${\varphi }_{03}$, ${\varphi }_{12}$, and ${\varphi }_{01}$, respectively.

Figure 7

(Color online) Proposed experimental setup for preparation of an arbitrary pure ququart state. DBS represents the dichroic beam splitter.