\section{\label{sec:introduction} Introduction}

Secure and hacking-proof communications is a vital requirement in today’s world. Traditional public key cryptography relies on lengthy and hard to decipher mathematical functions to encrypt and decrypt data. However, with the advancements of quantum computers, secured communication is increasingly vulnerable to hacking attempts. Quantum Key Distribution (QKD)~\cite{diamanti2016practical,xu2020secure,vajner2022quantum} the best-known application of quantum cryptography, offers an information-theoretic secure communication system, largely on account of the quantum non-cloning theorem~\cite{scarani2009security}. It enables two users to generate the exact same key without sharing any part of it publicly, providing a solution to secure key exchange. Since its inception in mid-1980s, and the first successful proof-of-concept test~\cite{Bennett_2014}, QKD has evolved to include various approaches such as entanglement-based QKD~\cite{basso2021quantum, basso2022daylight, yin2020entanglement}, measurement-device-independent QKD~\cite{lo2012measurement}, quantum teleportation-based QKD~\cite{bashar2009review}, and satellite-based QKD~\cite{liao2017satellite}. So far the majority of QKD systems rely on either nonlinear down-converted sources or attenuated lasers~\cite{ma2008quantum,kupko2020tools}. The advantage of these latter sources is their potentially high repetition rate. However, the issue remains that these sources are probabilistic, and remain unsecured even with proper counter measures in place, such as the decoy protocol~\cite{lo2005decoy}. One alternative approach is using a deterministic, triggered single photon source (SPS) that emits a single photon per excitation cycle. Over the last few decades, significant effort has been put forward to develop such sources, with prime challenges being their purity (i.e., minimisation of multiphoton events) and extraction of light (i.e., collection efficiencies)~\cite{senellart2017high, aharonovich2016solid}. While semiconductor quantum dots are a great choice for a bright and pure source~\cite{basso2021quantum, heindel2012quantum, takemoto2015quantum}, their operation is limited to cryogenic temperatures. For wide deployment and practical implementation of QKD in real-world settings, compact, room temperature, sources are required~\cite{samaner2022free, leifgen2014evaluation, piparo2017measurement, gao2023atomically}.

Among the various solid-state materials, single-photon sources in hexagonal boron nitride (hBN) are considered a prime candidate for QKD owing to the material's favourable physical and optical properties~\cite{aharonovich2022quantum}. In particular, properties such as a high single-photon purity, and high-brightness operating in ambient conditions give a competitive advantage over other sources~\cite{zeng2022integrated}. 

After having demonstrated the in-principle usability of hBN SPSs for QKD in Ref. \onlinecite{zeng2022integrated}, in this work, we include a full implementation of a free-space, discrete-level QKD system using an integrated SPS in hBN. We implement the BB84 protocol, demonstrating the sending, receiving and encryption/decryption process of an image from one device to another. We perform all security protocols, including privacy amplification, to demonstrate the most reliable QKD realised with SPSs to date.



\section{Setup Configuration}

\begin{figure*}[ht]
\includegraphics[width=0.8\textwidth]{figure01.pdf}
\caption{(a) Schematic diagram of the single-photon source. DM, dichroic mirror. (b) Schematic diagram of the QKD setup. EOM, electro-optic modulator; LP, linear polariser; APD, avalanche photodiode; PBS, polarising beam splitter; HWP, half-wave plate. (c) The optical components of the transmitter (Alice) and the receiver (Bob).}
\label{figure_1}
\end{figure*}

The optical apparatus used to perform the measurements as a whole can be split into two main systems; the source to generate the single photons, and the QKD apparatus to perform the key distribution measurements. The hBN single photon sources are the primary source of photons for the QKD system. The sources are integrated with a solid immersion lens and packaged into a compact, portable device as described previously~\cite{zeng2022integrated} and shown schematically in \cref{figure_1}a. For the QKD experiments, the sources are excited using a 515 nm pulsed laser (PicoQuant PDL-800D), and the collection is coupled via single mode optical fibre to the QKD sender (Alice).

The QKD-side setup is shown in \cref{figure_1}b. The management and distribution of keys are performed by elements which modulate and/or measure the polarisation of the photons. The elements lie in a transmitter and receiver, termed Alice and Bob, respectively, that are separated by tens of centimetres in free space. The signal from the source is sent to the Alice, firstly. Alice consists of a half-wave plate (FBR-AH1, Thorlabs), linear polariser (FBRP, Thorlabs), and an electro-optic modulator (EOM) (EO-AM-NR-C1, Thorlabs). The half-wave plate and linear polariser are used to align and filter vertically polarised photons, respectively, minimising losses as they enter the EOM – a requirement for its function. The EOM is driven by a digital-to-analogue converter (DAC) (EVAL-AD5754REBZ, Analog Devices), multiplexer (MUX36D04EVM-PDK, Texas Instruments), and a high voltage amplifier (HVA 200, Thorlabs), supplying the four voltages (two voltages in Bob’s case) to the EOMs to induce the desired polarisation states.  Bob consists of an EOM, polarising beam splitter (PBS052, Thorlabs), and two avalanche photodiodes (APD, SPCMAQRH-12-FC, Excelitas). 


\section{Experimental Procedure and Results}

\begin{figure*}[htb]
\includegraphics[width=0.8\textwidth]{figure02.pdf}
\caption{Optical characteristics of the hBN SPS as measured through the SIL. (a) Spectrum of the SPS used for QKD, showing a sharp ZPL at 645 nm. (b) Pulsed second-order autocorrelation function of the QKD SPS at 10 MHz, showing a $g^{(2)}(0) = 0.08$. (c)  Power-dependent saturation measurements showing count rate as a function of pulsed excitation power at 40 MHz repetition rate. (d) Pulsed second-order autocorrelation measurements of three characteristic SIL-integrated hBN SPSs, showing typical single-photon purity. All measurements were performed at 10 MHz repetition rate.}
\label{figure_2}
\end{figure*}

We start by choosing the most appropriate hBN SPSs – in particular, those with a high single-photon purity. \cref{figure_2}a shows the spectrum of the hBN SPS used in all QKD runs reported here.
This SPS has a characteristic sharp zero-photon line (ZPL) at 645 nm. The emission was then bandpass filtered (Semrock, $650\pm13$~nm) to select only photons from around the ZPL for the QKD experiment. The bandpass filtering was crucial in overcoming the wavelength dependence of the EOM and ensuring that the single-photon purity remains as low as possible, with \cref{figure_2}b showing the second-order autocorrelation function $g^{(2)}(0) = 0.08$. Additionally, the count rate was measured as a function of pulsed excitation power at a pulse repetition rate of 40 MHz, as shown in \cref{figure_2}c. The SPS was found to saturate at an average pulsed-excitation power of 217 $\mu$W with a count rate of 5.08×10$^5$ cts/s as out-coupled from the single-mode fibre. From this, the calculation of the source-side total setup efficiency can be quantified with a mean photon number per pulse of $\mu = 0.012$. This can be substantially improved by optimising the collection further, employing different designs~\cite{chen2011_99, sortino2021bright}. Importantly, our modular design enables rapid characterisation and swapping of other sources. Examples of three other suitable sources with low $g^{(2)}(0)$ values are shown in \cref{figure_2}d. 

Having characterised and established the source of single photons, the testbed for the QKD system was then initiated. \cref{figure_3} shows the operation sequence of the QKD process. The process starts with Alice generating a sequence of random integer numbers, each of which is 0, 1, 2 or 3, while Bob generates a sequence of random numbers between 0 and 1. Alice’s numbers are used to encode the measurement bases and bit values onto the photons, while Bob’s numbers are used to select the measurement bases for the measurements. 

More specifically, Alice’s numbers map to one of four polarisation states imparted to the photons, horizontal (H,$\leftrightarrow$), vertical (V,$\updownarrow$), right-handed circular (R,$\circlearrowright$), or left-handed circular (L,$\circlearrowleft$). For each laser pulse cycle, at a rate of 500~kHz, Alice’s randomly generated number switches a multiplexer to select a specific one of four channels from the digital-to-analogue converter (DAC) to apply a predetermined voltage to the EOM and, as such, encodes the desired polarisation state onto the photon. 

Bob uses a corresponding setup to randomly select a basis, H/V or R/L, and measure the photon to obtain one of four outcomes. Bob’s EOM either leaves the incoming photon in its polarisation base or interchanges linear and circular polarisation. The PBS (polarising beam splitter) then separates linearly polarised photons with certainty to be detected at one of two detectors, whereas circularly polarised photons are unpredictably registered at either one of the two detectors. Bob’s APDs are gated to the laser pulse, and only the detection event of a single photon by one of the APDs per time bin is considered a valid bit and stored to the computer, while the rest is discarded.

\begin{figure*}[htb]
\includegraphics[width=0.8\textwidth]{figure03.pdf}
\caption{QKD operation sequence.}
\label{figure_3}
\end{figure*}


\begin{table*}[htp]
\begin{center}
\begin{tabular}{l|R{1.5cm}|R{1.5cm}|R{1.5cm}|R{1.5cm}|R{1.5cm}|R{1.5cm}}
\textbf{Experiment no.} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6}\\
\hline
\hline
Clock rate (kHz) & 500,000 & 500,000 & 500,000 & 500,000 & 500,000 & 500,000\\
Photon detection rate at Bob (cts/s) & 1527 & 1486 & 1375 & 1722 & 1666 & 1001\\
\hline
Raw key length (bits) & 1,000,000 & 1,000,000 & 100,000 & 100,000 & 10,000 & 10,000\\
Raw key rate (bits/s) & 88 & 96 & 80 & 100 & 96 & 56\\
Bounded raw key rate (bits/s) & 396 & 432 & 360 & 450 & 432 & 252\\
QBER & 0.07 & 0.06 & 0.08 & 0.05 & 0.03 & 0.08\\
\end{tabular}
\end{center}
\caption{Selected experimental runs with their respective parameters.}
\label{table_1}
\end{table*}%



After revealing their bases used, Alice and Bob “sift” their events, discarding those which have mismatched bases. By defining H and R to correspond to a bit value of “0”, and V and L to correspond to a bit value of “1”, respectively, Alice and Bob retain a partially-correlated string of random bits. This string must be processed further to obtain a \textit{secret key}.

The required number of bits in the partially-correlated string must be large enough to provide a non-zero secret key rate at the pre-assigned $\varepsilon$-security (total failure probability), as discussed in the appendix. After accumulating the required number of bits, the post-processing procedure is initiated, a first component of which is parameter estimation. After this, a next phase is commenced, which consists of reconciliation and privacy amplification, after which the final secret key is generated. 


In the QKD experiments we transmitted blocks of 1 million random bits at a clock rate of 500~kHz (limited by the voltage amplifier and the EOM), resulting in a 2~s transmission phase during which Bob received photons at a detection rate of $\approx1500$~cts/s. After this transmission phase, our system required 7~s for the data processing phase (dominated by the data transfer between the FPGA board and the PC), resulting in a total of 9~s per block of 1 million random bits. During the experiments no. 1 and no. 2 (in \cref{table_1}) we repeated the transmission of these blocks until 4 million valid bits were accumulated (see also flow chart in \cref{figure_3}). 50\% of the accumulated bits are sacrificed during the sifting process, while another 50\% of the remaining bits are used for the quantum bit error rate (QBER) estimation. The partially-correlated keys that exist at this phase are referred to as the `\textit{raw keys}'.  Error correction is then instigated to turn the partially-correlated pair of keys into identical keys. Following this, the now correlated keys are input to the privacy amplification process of which, approximately $1/16^{\rm th}$  end up as the final secret keys (see appendix for details on error correction and privacy amplification).

We repeated the QKD measurement numerous times, accumulating raw keys of lengths up to 10$^6$ bits (after sifting and QBER estimation). Representative results for several independent runs of different lengths are shown in \cref{table_1}, with QBER values ranging between 3\% and 8\%. 

Our experimental set up was primarily aimed at the development of the most reliable key in terms of $\varepsilon$-security, not the key rate. Hardware limitations imposed certain constraints on us, related to the amount of quantum information that could be transferred before classical protocols were implemented (e.g., TCP socket connections, memory-read functions), which led to time delays of 7~s per million bit-transfer attempts. Of course, such processing delays can never be set to zero in any implementation, but if we simply set all such delays to zero, we derive what we refer to as the \textit{bounded raw key} rate in \cref{table_1}. Beyond the time delays, the three most important factors influencing the raw key rate are (i) the input pump rate; (ii) capture rates from the photon source; and (iii) losses in our optical components. Achievable improvements in these three factors alone would readily lead us into the $\sim 10$~kHz raw key rate range.

For runs no. 1 and no. 2 (in \cref{table_1}) we also performed privacy amplification at $\varepsilon$-security levels of 10$^{-10}$ and below, and achieved secret key lengths of 33176~bits \& 68516~bits at secret key rates of 4~(bits/s) \& 6 (bits/s) or bounded secret key rates of 18 (bits/s) \& 26 (bits/s), respectively. For the other runs in \cref{table_1}, finite key effects  result in zero secret key rates at a $\varepsilon$-security level of 10$^{-10}$.



\begin{figure*}[htb]
\includegraphics[width=0.8\textwidth]{figure04.png}
\caption{(a), Original image, (b) image encrypted with Alice’s secure key, and (c) the decrypted image after decoding it using Bob’s secure key.}
\label{figure_4}
\end{figure*}


Finally, to demonstrate the QKD utility, we transfer an image in a secured way from Alice to Bob. The image of a toy car consisting of 48 kbits is shown in \cref{figure_4}a. We use the key from the QKD experiment no. 2 (see \cref{table_1}) as a one-time keypad to encrypt the image (\cref{figure_4}b). Alice then transmits the image to Bob classically, and Bob decrypts the image using the secure key at his end, as shown in \cref{figure_4}c. 




\section{Discussion and Outlook}

We report a functional, free-space, room-temperature QKD prototype with high-purity hBN SPSs. We implement a complete BB84 protocol, including privacy amplification and QBER corrections at a security level (total failure probability) of 10$^{-10}$. As discussed further in the appendix, although our rates are lower relative to others seen in the literature~\cite{leifgen2014evaluation, gao2022quantum, rau2014free}, straightforward comparison of the many published results should be done with caution as many different assumptions and security settings can be in place. In addition, our clock rate operates at 500~kHz, compared to other experiments that trigger at a few MHz rates, or faster. 

Furthermore, we have included all possible assumptions in our derivation of security in the finite key limit, including the possibility of vacuum contributions affecting the probability of multiple photon events~\cite{grunwald2019effective, chavez2020estimating}. We believe our experimental secret key rates currently represent the most reliable in terms of security for the type of photon source used. Increasing the rate at a given security level, can be achieved via various improvements in the system – including an increased photon collection rate from the SPS, increased speed in the electronics, and increased computational power for the classical reconciliation. For instance, one key limitation originates from the high voltage amplifiers needed to drive the EOMs, which can be improved using alternative hardware. Using different field-programmable gate array (FPGA) hardware and data transfer protocols, can also further increase the rates.

All in all, our source has one of the best purities at room temperature, with an excellent brightness, and operation under ambient conditions that can enable a straightforward integration with existing QKD networks. Our work paves the way for scalable implementation of QKD systems and holds great promise for using triggered room temperature SPSs based on hBN. 

\section*{Acknowledgements}
The authors thank Timm Kupko, Tobias Heindel, Oliver Benson, Esteban Gómez-López, Çağlar Samaner, Serkan Ateş, and Francois Ladouceur for useful discussions. We also thank Rich Mildren and Adam Bennett for the initial assistance of the source integration. This work is supported by the Australian Research Council (CE200100010), and the Office of Naval Research Global (N62909-22-1-2028). We thank the NSW Defence Innovation Network, the NSW State Government, and the Next Generation Technologies Fund for financial support of this project funded by the DIN Strategic Investment Initiative.


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