Tomography of the quantum state of photons entangled in high dimensions

Systems entangled in high dimensions have recently been proposed as important tools for various quantum information protocols, such as multibit quantum key distribution and loophole-free tests of nonlocality. It is therefore important to have precise knowledge of the nature of such entangled quantum states. We tomographically reconstruct the quantum state of the two photons produced by parametric downconversion that are entangled in a $d$-dimensional orbital angular momentum basis. We determine exactly the density matrix of the entangled two-qu$d$it state with $d$ ranging from 2 to 8. The recording of higher-dimensional states is limited only by the number of data points required and therefore the length of time needed to complete the measurements. We find all the measured states to have fidelities and linear entropies that satisfy the criteria required for a violation of the appropriate high-dimensional Bell inequality. Our results therefore precisely characterize the nature of the entanglement, thus establishing the suitability of such states for applications in quantum information science.

Figure 1

Experimental setup. L1 $=$ 300 mm, L2 $=$ 750 mm, L3 $=$ 1000 mm, L4 $=$ 3.2 mm. SLM, spatial light modulator; NPBS, nonpolarizing beam splitter; SF, 710 $\pm$ 5-nm spectral filter; SMF, single-mode fiber.

Figure 2

(a) The complete set of measured probabilities for dimension 5. (b) The pure OAM states. (c) A sample of the superposition states. Here, ${\alpha }_i$ denotes the phase difference in the idler arm, from Eq. (7), and ${\alpha }_s$ denotes the same in the signal arm. The OAM states separated by a comma denote ${\ell }_1,{\ell }_2$ as in Eq. (7), while the single states denote $\ell$ as in Eq. (6).

Figure 3

The density matrices for even dimensions 2–8. The axes for dimension 2 are labeled, and the higher dimensions follow the same convention. For example, the labels for the $d=4$ case would read $\langle 2,2|,\langle 2,1|,\langle 2,-1|,...$ and $|-2,-2\rangle ,|-2,-1\rangle ,|-2,1\rangle$, etc., where we use the convention $|{\ell }_s,{\ell }_i\rangle$ to be equivalent to ${|\ell \rangle }_s{|\ell \rangle }_i$.

Figure 4

(a) Linear entropy and (b) fidelity as a function of dimension. The error for both of these measurements is $\pm 0.01$, which is too small to be seen clearly on the graphs. In each case, the squares represent the measured data, while the circles represent the threshold states in Eq. (13). The shaded area represents the set of states that will not violate the appropriate high-dimensional Bell inequality.