Quantifying Photonic High-Dimensional Entanglement

High-dimensional entanglement offers promising perspectives in quantum information science. In practice, however, the main challenge is to devise efficient methods to characterize high-dimensional entanglement, based on the available experimental data which is usually rather limited. Here we report the characterization and certification of high-dimensional entanglement in photon pairs, encoded in temporal modes. Building upon recently developed theoretical methods, we certify an entanglement of formation of 2.09(7) ebits in a time-bin implementation, and 4.1(1) ebits in an energy-time implementation. These results are based on very limited sets of local measurements, which illustrates the practical relevance of these methods.

Figure 1

Schematic view of the experiment for creating and quantifying high-dimensional entanglement using limited measurement data. The time-bin entangled two-photon state is produced by the spontaneous parametric down-conversion process using a ${\chi }^{(2)}$ nonlinear crystal. The crystal is pumped by a mode-locked laser with a high repetition rate producing a photon pair in well-defined temporal modes $|j,j⟩$. A Franson type setup [6] consisting of two interferometers is used to analyze the resulting state and reveal its entanglement.

Figure 2

Results for the time-bin experiment. (a) Here, $(j,j)$ corresponds to the intensity probabilities [$|c_j{|}^2$ in Eq. (1)], while $(j,j+1)$ and $(j,j+2)$ represent the visibilities of the two-photon interference between two neighboring and next-neighboring temporal modes. We consider eight temporal modes. (b) Lower bounds on the entanglement of formation (in terms of ebits) as a function of the number of temporal modes $n$ considered. The data allow us to certify an entanglement of formation of 2.09(7) ebits, taking $n=7$. The black dots correspond to the mean number of ebits and the one $\sigma$ statistical error is represented by the filled red curve.

Figure 3

Results of the energy-time experiment. (a) Visibility of the two-photon interference representing coherence between temporal modes $|j,j⟩$ and $|j+n,j+n⟩$ where $n$ is a bin number. The bin $n=0$ corresponds to the first order interference with visibility of 98(1)%. Up to bin number 11 (blue), the visibility is rather low due to the jitter of the detection system. For the remaining bins (green), the average visibility is 98.5(8)%. In order to lower bound the entanglement of formation, we use bins 12 to 29. (b) Lower bounds on the entanglement of formation (in terms of ebits) as a function of the number of temporal modes $n$ for the CW experiment. Energy-time entanglement containing up to 4.1(1) ebits is certified for $n=42$. The filled red curve represents the one $\sigma$ statistical error.