Quantification of multidimensional entanglement stored in a crystal

The use of multidimensional entanglement opens new perspectives for quantum information processing. However, an important challenge in practice is to certify and characterize multidimensional entanglement from measurement data that are typically limited. Here, we report the certification and quantification of two-photon multidimensional energy-time entanglement between many temporal modes, after one photon has been stored in a crystal. We develop a method for entanglement quantification which makes use of only sparse data obtained with limited resources. This allows us to efficiently certify an entanglement of formation of 1.18 ebits after performing quantum storage. The theoretical methods we develop can be readily extended to a wide range of experimental platforms, while our experimental results demonstrate the suitability of energy-time multidimensional entanglement for a quantum repeater architecture.

Figure 1

Experimental setup. A pair of photons (signal and idler) is generated in a ppKTP waveguide via SPDC of a 532-nm pump photon. Both photons are spectrally filtered using optical cavities. Since the resulting coherence time of the photon pair is much smaller than the coherence time of the pump laser, this leads to the generation of two-photon energy-time entanglement. The signal photon is sent to a quantum memory (QM) based on a ${\mathrm{Nd}}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystal and stored for ${\tau }_M=50$ ns. The pump laser intensity is modulated using an acousto-optic modulator to generate a square pulse with duration ${\tau }_p$ smaller than the storage time of the QM. Finally, the photon pair is analyzed via unbalanced interferometers, with controllable phases ${\varphi }_s$ and ${\varphi }_i$ and identical delays $\mathrm{\Delta }=5.5$ ns, and single-photon detectors ($D_s$ and $D_i$). Hence, the entangled state generated and measured in our experiment can be compactly described by an entangled state of $d$ temporal modes of the form $|{\mathrm{\Phi }}_d\rangle$. The experimental parameters allow for up to $d=9$ modes.

Figure 2

Illustration of the method. Given a submatrix where only the diagonal and first off-diagonal are known (a), the method allows us to complete the matrix (b), giving lower bounds (7) on all unknown elements based on positivity constraints. Finally, this construction leads to a lower bound on the entanglement of formation via relation (2).

Figure 3

Results for one experimental run. (a) The measured intensities in the time-of-arrival basis (diagonal elements $r_{j,j}$) and visibilities (first off-diagonal $r_{j,j+1}$) for ten temporal modes, separated by $\mathrm{\Delta }=5.5$ ns. (b) Lower bounds for the entanglement of formation (number of ebits) as a function of the number $d$ of temporal modes taken into account when reconstructing the density matrix. Here, the optimal value is $\sim 1.25\left(11\right)$ ebits. The data show good agreement with our model considering the measured visibility of $\mathcal{V}$ of 97%. The case $\mathcal{V}=1$, corresponding to the maximally entangled state (1), gives ${log}_2\left(d\right)$ ebits.