Distribution of genuine high-dimensional entanglement over 10.2 km of noisy metropolitan atmosphere

Our study investigates the presence of high-dimensional entanglement in a recent demonstration of a noise-resistant quantum key distribution (QKD) protocol presented in [Phys. Rev. X 13, 021001 (2023)]. We determined that it is possible to certify the dimensionality of the distributed entangled states to be at least three. To demonstrate this, we developed an energy-time entanglement discretization technique, as well as an improved witness for entanglement dimensionality. Our results provide insight into the complex relationship between high-dimensional entanglement and the noise resistance of QKD protocols operating in high dimensions.

Figure 1

(a) The setup at the IQOQI laboratory (Alice) with a hyperentangled photon pair source pumped by a ${}^{39}K$ stabilized Laser at 404.532 nm. The hyperentangled state was set by adjusting a combination of half-wave plate (HWP) and quater-wave plates (QWP). The hyperentangled photons produced at 808.9 nm in a ppKTP crystal are separated and guided to Alice and Bob. A 50:50 beam splitter (BS), at Alice's receiver, randomly sends the incoming photons to either the basis measurement or the temporal superposition (TSUP) basis measurement. The Mach-Zehnder-Interferometer (MZI) in the TSUP basis uses Polarizing beam splitters followed by a measurement in the D/A-basis. The photon detection events are recorded by single-photon avalanche diodes (SPAD), time stamped by the time-tagging module and streamed to the server. (b) Receiver at the Bisamberg laboratory (Bob). The received photons are guided to the detection modules after filtering the background noise with filter combinations. Bob's setup is identical with that of Alice, with the difference of having two $4f$ systems in the MZI to compensate for the atmospheric turbulence and Bobs MZI is phase locked to a ${}^{87}\mathrm{Rb}$ stabilized laser at 780.23 nm. Both MZI's together form a postselection-free Franson interferometer where each side is separated by 10.2 km.

Figure 2

Entanglement dimensionality witness. To certify entanglement dimensionality we use different discretisations with $d=4$. For different time bin length (left to right) $\mathrm{\Delta }t\in \{2700,1350,900,675,540\}\mathrm{ps}$, we evaluate the fidelity of the time-energy part of the discretised distributed state to maximally entangled state $|{\varphi }_4^{+}\rangle =1/\sqrt{4}{\sum }_{i=0}^3|ii\rangle$. The integration time of all data points is 200 s and the fidelity is evaluated throughout the duration of the experiment. Fidelity strictly above 0.25 certifies that the underlying state has a Schmidt number of at least 2 and thus is entangled, while fidelity strictly above 0.5 certifies that the underlying state has a Schmidt number value of at least 3 and thus the correlations present in the state cannot be encoded using any qubit-entangled state.

Figure 3

Discretization we used to calculate the Schmidt number certificates. The time delay between interferometer arms is ${\tau }_{\mathrm{MZI}}=2.7\mathrm{ns}$. Each measured time interval of length 10.8 ns is divided into time bins of size $\mathrm{\Delta }t$. In the figure we used values (top to bottom) $\mathrm{\Delta }t\in \{2700,1350,900,675\}\mathrm{ps}$. Subsequently, time bins in the analyzed time interval with starting times exactly 2.7 ns apart are collected into a time frame. These were chosen specifically because we can use our measurement setup to measure projections onto some superpositions of bins with the same color.