Dispersive Optics HD-QKD

• Dispersive Optics HD-QKD is a quantum key distribution method that leverages temporal entanglement and dispersive measurements to encode multi-bit key symbols.
• It utilizes mutually unbiased time-of-arrival and frequency-conjugate bases, achieved through optical dispersion, ensuring high photon-information efficiency and robust security.
• Experimental results show high secret-key rates over long distances with loss tolerance up to 51 dB, demonstrating its scalability and compatibility with existing fiber networks.

Dispersive Optics High-Dimensional Quantum Key Distribution (HD-QKD) refers to quantum key distribution protocols that employ temporal entanglement and interferometric or dispersive optical components to realize high-dimensional quantum cryptographic alphabets. In these protocols, information is encoded in photon arrival times or their conjugate spectral modes, with all security-critical measurements effected via optical dispersion. The approach enables the realization of multi-bit per photon secure communications and is compatible with standard fiber-based telecommunications infrastructure.

1. Temporal Entanglement and Basis Construction

Dispersive-optics HD-QKD utilizes the joint time-energy correlations of SPDC-generated photon pairs. In the low-flux regime, the biphoton wavefunction is

$$\Psi_{AB} = \int dt_A dt_B\, \psi(t_A, t_B)\, e^{-\frac{i\omega_p}{2}(t_A+t_B)} |t_A\rangle_A |t_B\rangle_B$$

with

$$\psi(t_A, t_B) \propto \exp\left[ -\frac{(t_A-t_B)^2}{4\sigma_{\mathrm{cor}}^2} - \frac{(t_A + t_B)^2}{16\sigma_{\mathrm{coh}}^2}\right]$$

where $\sigma_{\mathrm{coh}}$ is the pump coherence time and $\sigma_{\mathrm{cor}}$ the biphoton correlation time. The dimensionality ("Schmidt number") of the temporal Hilbert space is $$d \simeq \sigma_{\mathrm{coh}}/\sigma_{\mathrm{cor}} \gg 1$$, yielding a $d$-ary key symbol per photon.

Measurements are performed in two mutually unbiased bases:

• Time-of-arrival (T basis): Photons are binned into time intervals, corresponding to the projectors $T_j = \int_{t_j} |t_j\rangle\langle t_j| dt_j$.
• Dispersed (frequency-conjugate, D basis): Each photon passes through a dispersive medium implementing $U = \exp[i(\beta_2 L)\omega^2/2]$, such that the output timing corresponds to the conjugate basis $D_j = (\beta_2L)^{-1} U T_j U^\dagger$. This realizes an effective frequency-bin measurement.

Mutual unbiasedness holds for these bases on the positive-frequency subspace, with $|\langle t_k|f_\ell\rangle| = 1/\sqrt{d}$.

2. Implementation: Sources, Dispersion, and Detection

Dispersive-optics HD-QKD is implemented using entangled-photon sources (continuous-wave SPDC or SFWM). After heralding the signal photon, only frames with single-pair events are accepted, suppressing multi-pair background to $<1\%$.

Quantum basis choice is effected by an optical switch or beamsplitter, routing photons to either a time-of-arrival detector or dispersive module plus detector. Dispersion is typically realized using fiber Bragg gratings, long fiber links, or on-chip photonic crystal waveguides tailored for group velocity dispersion (GVD).

Detection is optimized for single-photon sensitivity and sub-$20$ ps jitter (SNSPDs). For the D basis, careful balancing of anomalous and normal GVD ensures restoration of sharp timing correlations. Channel loss and detector imperfections degrade mutual information but do not increase Eve's information, as per the covariance matrix security proofs (Mower et al., 2012).

3. Security Framework: Gaussian States and Covariance Matrices

The security of dispersive-optics HD-QKD is based on covariance-matrix analysis of the measured joint quantum state, enabling collective attack proof. For each accepted frame, Alice and Bob obtain one key symbol if both register exactly one detection in the same basis.

The per-frame secure key capacity is

$\Delta I = \beta\,I(A;B) - \chi(A;E)$

where $\beta$ is the reconciliation efficiency, $I(A;B)$ is Alice–Bob mutual information, and $\chi(A;E)$ is Eve's Holevo bound. For Gaussian measurements,

$$I(A;B) = \frac{1}{2} \log_2 \frac{1}{1-\rho_t^2} + \frac{1}{2}\log_2 \frac{1}{1-\rho_f^2}$$

where $\rho_t, \rho_f$ are temporal and spectral correlation coefficients. The Holevo bound is computed from the symplectic eigenvalues $d_{\pm}$ of the full covariance matrix.

Decoy-state protocols are used to bound the single-pair fraction in the presence of multi-pair emissions. Only one or two decoy states suffice for near-optimal key rates at distances $>200$ km (Bunandar et al., 2014).

4. Performance Metrics and Experimental Results

State-of-the-art experiments demonstrate high secret-key rates and noise tolerance:

• Per-photon information: Up to $4$ bits per photon for $d=64$; experimentally $>1.2$ bits/photon at $d=4 \text{ or } 8$.
• Absolute rates: Secure key rates exceed $100$ Mb/s for short urban links; positive key rates persist beyond $200$ km fiber (Mower et al., 2012, Liu et al., 2022).
• Loss tolerance: Secret key persists at loss levels up to $51$ dB, enabled by high-dimensional encoding (Liu et al., 2022).
• Noise tolerance: QBER thresholds increase with $d$: for $d=4$, up to $18.9\%$ (coherent attacks), and higher $d$ raises the tolerable error (Ding et al., 2016).

Theoretical analysis (finite-key regime) confirms high photon-information efficiency (PIE) over metropolitan-area distances, even against the strongest eavesdropping attacks (Niu et al., 2016).

5. Composability and Finite-Key Security

Recent advances employ entropic uncertainty relations tailored for time and conjugate-time bins, yielding rigorous finite-key bounds. The min-entropy for Alice's raw key, conditioned on Eve's side information, is bounded by observed statistics in both bases: $$H_{\min}^\epsilon(X_A|E) + H_{\max}^\epsilon(Y_A|Y_B) \geq -n\,\log_2[c(\delta, \beta_D)]$$ with $c(\delta, \beta_D)$ reflecting the overlap between coarse-grained T and D bin POVMs (Niu et al., 2016). These relations provide the foundation for composable security proofs, ensuring strict key-independence from Eve even in fluctuating atmospheric channels (Kanitschar et al., 6 May 2025).

Variable-length key extraction and acceptance tests (accept/reject per block or per observation) significantly improve overall throughput and resilience to statistical channel variations, with order-of-magnitude gains over naive fixed-length post-processing (Kanitschar et al., 6 May 2025).

6. Integration, Telecommunication Compatibility, and Scalability

Dispersive-optics HD-QKD is inherently compatible with standard fiber networks and DWDM systems (Mower et al., 2012, Liu et al., 2022). Multi-wavelength capability allows parallelization, and integration with silicon photonic circuits provides a route toward compact, mass-manufacturable transmitters and receivers.

Emerging implementations leverage:

• Fan-in/fan-out couplers for multicore fiber transmission and reception (Ding et al., 2016).
• On-chip GVD control and photonic integration.
• Resource-efficient detection (single detector per measurement basis) using Talbot effects (Ogrodnik et al., 2024).
• Error reconciliation optimized for large-alphabet QKD, including nonbinary LDPC and HD-Cascade algorithms to nearly saturate the Slepian-Wolf limit (Mueller et al., 2023).

7. Comparison and Advantages over Alternative High-Dimensional Protocols

Dispersive optics HD-QKD combines the information-theoretic advantages of high-dimensional encoding (multi-bit per photon, elevated QBER threshold) with telecom hardware compatibility and relative immunity to polarization or OAM channel fluctuations. By contrast, spatial-mode/OAM and multicore approaches face greater challenges from modal crosstalk, turbulence, and alignment, requiring adaptive optics and active mode correction (Zhang et al., 12 Dec 2025).

Decoy-state dispersive protocols achieve comparable or superior rates and distances to OAM-based schemes, with easier integration into deployed fiber. Heralded-qudit sources and efficient reconciliation further enhance the protocol's practical throughput and security margin.

---

References:

• High-dimensional quantum key distribution using dispersive optics (Mower et al., 2012)
• High-dimensional quantum key distribution using energy-time entanglement over 242 km partially deployed fiber (Liu et al., 2022)
• Practical high-dimensional quantum key distribution with decoy states (Bunandar et al., 2014)
• Finite-key analysis for time-energy high-dimensional quantum key distribution (Niu et al., 2016)
• High-Dimensional Quantum Key Distribution based on Multicore Fiber using Silicon Photonic Integrated Circuits (Ding et al., 2016)
• Efficient Information Reconciliation for High-Dimensional Quantum Key Distribution (Mueller et al., 2023)
• Composable Finite-Size Security of High-Dimensional Quantum Key Distribution Protocols (Kanitschar et al., 6 May 2025)
• High-dimensional quantum key distribution with resource-efficient detection (Ogrodnik et al., 2024)