High-dimensional quantum communication with scalable photonic entanglement in time and frequency

Abstract

High-dimensional photonic entanglement holds significant promise for advancing quantum communication, computation, and metrology. For example, large-alphabet quantum communication protocols are known to benefit from enhanced noise resilience and information capacity via multi-bit time-bin encoding. Yet, characterizing high-dimensional entangled states is challenging, as full state tomography becomes prohibitively costly and often requires unrealizable measurements. Here, we demonstrate a scan-free method to characterize high-dimensional entanglement in the time-frequency domain. Our reconstruction achieves a record $5.70\pm0.07$ ebits and a fidelity of $65.4\pm0.4\%$ with the maximally entangled state of local dimension $1021$, certifying the presence of $668$-dimensional entanglement. We further prove the attainability of a secure key rate of $15.6$ kB/s in a composable finite-size, entanglement-based protocol, and show that in continuous operation, the setup can quickly approach asymptotic key rates. Using commercial telecom components and state-of-the-art low-jitter single-photon detectors, our scalable architecture offers a practical path towards high-rate, noise-resilient quantum communication testbeds.

Figures (6)

• Figure 1: Two-shots measurements for high-dimensional qudit entanglement, high-rate QKD, and 31-dimensional time-frequency resolved joint temporal intensities (JTIs). a, In spontaneous parametric down-conversion (SPDC) photons, the detection of one photon fixes the arrival time of the other photon, yielding strong temporal correlations in the JTI. We denote temporal measurement basis of Alice and Bob as $T_A-T_B$. By using time-to-frequency convertor, we can perform the frequency-resolved measurements in basis $F_A-F_B$. For $T_A-T_B$ the local timing jitter errors are marked as light blue slots, while bin-width $\tau$ and number of bins $N$ define the time frame length $N\tau$, optimized for the JTI. Orange slots indicate there are no registered coincidence photons. For $F_A-F_B$, we utilize non-local dispersion cancelation to recover the narrow temporal correlation and to convert temporal information of SPDC into frequency-domain. b, The experimental setup involves separating signal and idler photons, with Alice and Bob each using 50:50 fiber beam splitters and superconducting nanowire single-photon detectors (SNSPDs) for both $T_A-T_B$ and $F_A-F_B$ measurements. c, and d, Exemplary 31-dimensional JTIs in $T_A-T_B$ and $F_A-F_B$ bases. The full width at half maximum (FWHM) of temporal correlation peak are 32.9 ps and 125.5 ps, respectively. For d, we optimize the $F_A-F_B$ measurements by adjusting the pump wavelength. The slight asymmetry of temporal correlation peak comes from the limitation of time-to-frequency convertor. Parameters ($\tau$, $N$) are chosen to optimize the JTI: $\tau=200$ ps, $N=1024$ for $T_A$–$T_B$; $\tau=800$ ps, $N=1024$ for $F_A$–$F_B$. The duration of coincidence counting for the data in c and d is 3 seconds, and no accidental subtraction is used.
• Figure 2: Experimental time-frequency bases up to 509$\times$509 dimensions. a, and b, An experimental measured 3$\times$3, 13$\times$13, 61$\times$61, 127$\times$127, 331$\times$331, and 509$\times$509 Hilbert space dimensional JTI for temporal and frequency-resolved measurement basis. We can observe that our JTIs from both bases are indeed dimensionally independent with respect to the measurements, owing to the large-alphabet arrival-time encoding, and the sufficient detected coincidence counts in our experimental setup. For all the measurements presented here, the JTI of $T_A-T_B$ basis has higher diagonal coincidence counts than that of $F_A-F_B$ basis, which is mainly due to the losses in of the time-to-frequency converter (which is in total of $\approx$ 6 dB). From the same reason, we observe that the $F_A-F_B$ basis is noisier than the $T_A-T_B$ basis. For all experimental data in a and b, the duration of measured coincidence counting is 3 seconds, and the raw data is presented here.
• Figure 3: High-dimensional entanglement certification from a maximum of 1021-dimensional JTI measured in the temporal and frequency bases. a and b, Coincidence counts for measuring in the $T_A\!-\!T_B$ and $F_A\!-\!F_B$ bases in a two-shot setting, with consistent bin width $\tau$ and number of bins $N$ as shown in Figures \ref{['fig:Figure1']} and \ref{['fig:Figure2']}. The timing for coincidence counting is 3 seconds, and the data are reported without accidental subtractions. The strong correlations in both measurement bases signify robustness of large-alphabet arrival-time encoding. c and d, The certified Schmidt number $k$ for each measured local dimension $d$, and their associated lower bound of the fidelity $\tilde{F}(\rho,\Phi)$ with respect to the maximally entangled state are shown. The maximum certified Schmidt number is 668 at a fidelity of $65.4\pm0.4\%$ in $d=1021$. e and f, The distillable entanglement $E_D$ for various local dimensions $d$ are shown together with the certified Schmidt numbers and the theoretical upper bound of $E_D$, $\text{log}_2(d)$.
• Figure 4: a Illustration of the security argument: Based on the recorded coincidence-click matrices, we derive two quantities. First, based on the FF-clicks, we derive the expectation of an entanglement witness, which allows us to bound Eve's guessing probability on the final key, which, in turn, allows us to derive a statistical estimator. That is a high probability lower-bound on the private entropy of the key given Eve's side information. Second, from the disclosed TT-clicks, we derive the error-correction leakage. Subtracting the error-correction leakage from the statistical estimator yields (up to second-order correction terms) a reliable high-probability lower bound on the secure key length. b Secure key rate in bits per second versus system dimension in four different scenarios: asymptotic (black pluses), i.i.d. collective attacks with block size $N=10^{8}$ (orange crosses), i.i.d. collective attacks with block size $N=10^{11}$ (peach circles), and coherent attacks with $M=10^{11}$ (blue triangles). Based on our data, we obtain optimal system dimensions of $d_{\mathrm{opt}}^{\mathrm{asym}}(\infty)=232$ for the asymptotic scenario, $d_{\mathrm{opt}}^{\mathrm{coll}}(10^{8})=96$ for i.i.d. collective attacks with $N=10^8$, $d_{\mathrm{opt}}^{\mathrm{coll}}(10^{11})=196$ for i.i.d. collective attacks with $N=10^{11}$, and $d_{\mathrm{opt}}^{\mathrm{coh}}(10^{11})=9$ for coherent attacks with $N=10^{11}$. c Secure key rate in bits per second versus block size $N$ for three different dimensions. In all three cases, the curves converge to the asymptotic rates (horizontal lines) already for practically viable block sizes. d Examination of the secure key rate in bits per second versus splitting ratio between time and frequency measurements for two different block sizes and fixed dimension $d=61$. For $N=10^{8}$, we find an optimal splitting ratio of $29\%$, while for $N=10^{11}$, the optimal splitting ratio is found to be $6\%$. This highlights that optimal splitting is far below the default $50\%$ and the key rates can be improved significantly by optimising this parameter.
• Figure S1: Comparison of the required number of local projective measurement settings in different local dimensions $d$ for different techniques. In this work, we highlight the constant number of measurement settings, since we only need a single setting for each of the $T_A-T_B$ and $F_A-F_B$ bases, independent of the dimensions $d$. Hence, our work represents many orders-of-magnitude improvement over traditional FST and prior literature Bavaresco_2018Erker_2017Valencia_2020Bertlmann_2008. We note that there are only a few fundamental limitations of our scheme: the number of measurable coincidence counts from the photon-pair source, loss in the time-to-frequency converter and other fiber components, as well as the timing jitter and detection efficiency of accessible single-photon detectors.