Time-bin HD-COW QKD Protocol

• Time-bin HD-COW QKD is a protocol that uses multi-bin frames to encode quantum information, effectively upgrading 2D systems to higher dimensions.
• It achieves significant secure-key rate improvements, with experiments showing up to 3–3.5× gains at metropolitan distances using only software updates.
• The protocol simplifies implementation by leveraging existing optical hardware and scaling feasibly to dimensions beyond 8 with minimal modifications.

Time-bin high-dimensional coherent one-way quantum key distribution (HD-COW QKD) is a protocol that enables arbitrary-dimensional QKD using only the hardware of a standard two-dimensional (2D) time-bin system. By encoding quantum information in frames of multiple time-bins, the protocol enhances secure key rates beyond conventional 2D QKD, without necessitating complex multiport interferometry or additional detectors. This architecture achieves significant performance gains via software and firmware updates alone, holding implications for the upgrade of already deployed time-bin QKD networks (Sulimany et al., 2021).

1. Protocol Structure and Encoding

In the HD-COW QKD protocol, time is divided into frames of $d$ contiguous time-bins, each indexed by $j = 0,1,\dots,d-1$. For every frame, Alice prepares a single weak coherent pulse with amplitude $\alpha$ (mean photon number $\mu \ll 1$) in exactly one of these $d$ bins, with all other bins in the vacuum state. The resulting computational (time) basis states are:

$$|\psi_k\rangle = |0_0, 0_1, \dots, 1_k, \dots, 0_{d-1}\rangle, \quad k \in \{0,\dots,d-1\}$$

where $|1_k\rangle$ denotes a one-photon pulse in bin $k$. To check coherence across the full frame, Alice randomly (with probability $p_\text{mon}$) replaces standard basis states by superpositions in the Fourier (X) basis:

$$|\phi_\ell\rangle = \frac{1}{\sqrt{d}} \sum_{j=0}^{d-1} e^{2\pi i \ell j / d}|\psi_j\rangle, \quad \ell=0,\dots,d-1$$

These states serve as "monitoring" pulses by interfering equally across all $d$ bins.

Transmission is realized with a continuous-wave laser at a telecom wavelength ($\lambda\approx 1550$ nm), temporally modulated into $d$ bins per frame by a high-speed intensity modulator (e.g., LiNbO$_3$) driven by an FPGA. On each frame, Alice probabilistically chooses to send a data or a monitoring state.

2. Receiver Operations and Post-Processing

Upon reception, Bob splits the incoming signal between a data line and a monitoring line:

• Data Line: A single-photon detector (SPD) time-tags photon arrivals, revealing which bin $k$ was chosen when a click occurs.
• Monitoring Line: Either a $(d-1)$-delay cascaded interferometer or a reconfigurable multiport mesh coherently overlaps time bins, projecting onto the Fourier basis. Detection at port $\ell$ corresponds to the outcome $\langle\phi_\ell|$.

After detection, Bob publicly announces which frames resulted in clicks on each line. Alice discloses which frames were data or monitoring, and empty or ambiguous frames are discarded. Parameter estimation is conducted by extracting the time-bin error rate $Q$ from the data line and the monitoring visibility $V$ from interference fringe statistics. Subsequent classical post-processing (sifting, error correction, privacy amplification) follows standard QKD paradigms.

3. Security Analysis and Key Rate Formula

Security is based on reduction to symmetric collective attacks via the quantum de Finetti theorem, applied to large ($n\gg1$) blocks of frames. The error metrics are:

• $Q$: time-bin error rate (probability of Bob detecting the incorrect bin);
• $V$: observed visibility in monitoring basis.

The binary entropy function is defined:

$h_2(x) = -x\log_2 x - (1-x)\log_2(1-x)$

An asymptotic secure-key rate per frame against coherent attacks is:

$r \geq q - h_2(Q) - h_2\left(\frac{1+V}{2}\right)$

where $q = \log_2 d$ is the information content per $d$-ary symbol. Incorporating the data/monitoring ratio, detection probability $p_\text{click}$, and error correction leakage ($\text{leak}_{EC} \approx h_2(Q)$),

$$R_\text{secure} = p_\text{data}\left[p_\text{click}\cdot(q-h_2(Q)-h_2((1+V)/2)) - \text{leak}_{EC}\right]$$

For $d=2$, this reproduces the 2D COW result; for $d>2$, both $q$ and the monitoring term generalize to $d$-level alphabets.

4. Experimental Realization over 40 km Fiber

The protocol has been experimentally validated using a 40 km standard SMF-28 fiber link ($\sim$0.2 dB/km loss, total optical attenuation $\sim$8 dB). Key components and parameters include:

• Laser: CW external-cavity diode, linewidth $\lesssim 100$ kHz, $1550$ nm.
• Intensity Modulator: $10$ GHz bandwidth, $\geq 30$ dB extinction.
• Frame Rate: $250$ MHz/bin, $d=2,4,8$ tested ($2$ GHz modulation).
• Detectors: InGaAs/InP SPDs, $\eta \sim 20\%$, dark count $\sim$10^{-6}$/gate, jitter$\sim$$100$ ps.

Performance for $d=2,4,8$:

• QBER $Q\approx2-3\%$ (rises slightly with $d$),
• Monitoring visibility $V\geq98\%$ for all $d$,
• Raw sifted rates: $1.2$, $2.3$, $3.8$ Mbit/s respectively,
• Secure key rates after post-processing at $40$ km:
  • $d=2$: $\sim150$ kbit/s,
  • $d=4$: $\sim300$ kbit/s,
  • $d=8$: $\sim500$ kbit/s.

Table: Secure Key Rates at $L=40$ km

$L$ (km)	$R_\text{2D}$ (kbit/s)	$R_\text{4D}$ (kbit/s)	$R_\text{8D}$ (kbit/s)
40	150	300	500

Direct implementation required no hardware changes at Bob; only the FPGA pattern generator was updated to support larger $d$.

5. Comparative Performance and Benefits

Secure-key rate $R_\text{secure}(d, L)$ performance was benchmarked as a function of distance $L$ and dimension $d$. For $d=4,8$, $R_\text{secure}$ consistently exceeds the 2D benchmark up to $L\sim80$ km, achieving a factor-of-two improvement for $d=4$ and up to $\sim$3–3.5× for $d=8$ at metropolitan distances. Improvement arises solely from software-level reparameterization; no additional optical or detection channels are required. Further benefits include reduced per-bit overhead in classical post-processing, as higher-dimensional encoding yields more secret bits per successfully transmitted frame. The use of higher $d$ also partially mitigates channel loss since information per photon increases.

6. Scalability, Alternative Encodings, and Network Implications

The HD-COW architecture readily scales to $d>8$; with current components, $d\sim16-32$ is achievable, limited primarily by detector timing jitter. The same min-entropy–based security arguments transpose to other high-dimensional encodings, such as frequency-bin or OAM spatial-mode qudits, provided the physical implementation supports projective measurements in both a “data” basis and a “monitoring” basis spanning the $d$-dimensional Hilbert space. In principle, time-multiplexed high-dimensional frames enable multi-user QKD by allocating subsets of bins per user without additional center-station hardware.

7. Context and Outlook

The HD-COW protocol generalizes the conventional two-pulse coherent one-way QKD scheme to frames of $d$ time-bins, raising per-frame capacity from $1$ bit to $\log_2 d$ bits. By replacing the standard COW two-dimensional basis and monitoring with their $d$-dimensional analogs and leveraging optical hardware already ubiquitous in deployed QKD systems, the protocol enables immediate practical performance enhancement via software. This suggests considerable potential for near-term increases in urban and metropolitan-scale QKD throughput with minimal physical infrastructure intervention (Sulimany et al., 2021).