High-Dimensional Quantum Key Distribution

• HD-QKD is a quantum cryptographic protocol that encodes key material in a Hilbert space of dimension greater than two, allowing log2(d) bits per photon with enhanced eavesdropping sensitivity.
• Experimental implementations leverage diverse encoding methods such as time–energy, spatial modes, and multicore fiber to achieve robust, scalable, and noise-tolerant secure communications.
• Advanced security proofs combine finite-key composable analysis and decoy-state techniques to ensure practical, high-rate key generation even in fluctuating channel conditions.

High-dimensional Quantum Key Distribution (HD-QKD) is a class of quantum cryptographic protocols that encodes secret key material in a Hilbert space of dimension $d>2$. By employing multilevel quantum systems—qudits rather than qubits—HD-QKD achieves increased information capacity per detected photon and enhanced robustness to errors. Protocols capitalize on diverse encoding degrees of freedom: time–energy, spatial modes, orbital angular momentum (OAM), multicore fiber, and others. Rigorous finite-key, composable security has been demonstrated for several major modalities, establishing HD-QKD as an advanced route toward scalable, high-rate, and noise-tolerant quantum communications.

1. Protocol Principles and Hilbert-Space Structure

HD-QKD generalizes and extends the qubit-based BB84 protocol to $d$-dimensional state spaces. The two primary design strategies are:

• Mutually unbiased bases (MUBs): For key distillation, two (or more) bases are selected so that the probability $|\langle \psi_i | \phi_j \rangle|^2 = 1/d$ for all $i,j$. This maximizes eavesdropping sensitivity.
• High information capacity: Each detected photon can ideally convey $\log_2 d$ secret bits, compared to 1 bit for standard qubit protocols.

Example Encodings

• Time–energy encoding: Qudits are represented as photons localized in one of $d$ time bins; complementary measurements exploit group-velocity dispersion for conjugate (“frequency-like”) bases (Niu et al., 2016).
• OAM and spatial modes: Photons are prepared in superpositions or eigenstates of OAM; multidimensional state preparation is accomplished via spatial light modulators, multi-plane light converters, or integrated photonics (Zhang et al., 12 Dec 2025, Lib et al., 2024).
• Multicore fiber: Core index in a multicore fiber serves as the computational basis; MUBs are constructed using on-chip interferometers (Ding et al., 2016).

The raw Hilbert space thus scales exponentially with the number of available degrees of freedom.

2. Protocol Realizations and State Preparation

Diverse experimental architectures have been developed to instantiate HD-QKD:

• Time–energy entanglement via SPDC: A pulsed or continuous-wave spontaneous parametric downconversion (SPDC) source with a coherence time $\sigma_{\text{coh}} \gg \sigma_{\text{cor}}$ (correlation time), yielding a biphoton Schmidt rank $d \sim \sigma_{\text{coh}}/\sigma_{\text{cor}}$ (Niu et al., 2016, Mower et al., 2012).
• Mode sorting and spatial encoding: Programmable multi-plane light converters (MPLC) can apply arbitrary $d\times d$ unitaries for efficient and robust encoding and detection without cascaded interferometers (Lib et al., 2024).
• Qubit-like (Fourier-qubit) superpositions: Rather than full $d$-term superpositions, efficient protocols use only two-level superpositions with $d$ possible phases (Fourier-qubits), simplifying state preparation and analysis (Scarfe et al., 4 Apr 2025).
• Hybrid spin–orbit and 3D polarization modes: Exploiting hybrid degrees of freedom (polarization, OAM, radial profile, longitudinal polarization) can offer self-healing and improved robustness to channel impediments (Nape et al., 2018, Otte et al., 2023).

Table 1 summarizes selected implementations:

Encoding	State Preparation Tool	Example $d$	Ref.
Time–energy	SPDC, time-bin selector	64	(Mower et al., 2012)
OAM (spatial)	SLM, MPLC, q-plate	4–25+	(Zhang et al., 12 Dec 2025, Lib et al., 2024, Ding et al., 2016)
Fourier-qubit	2-term interferometers, SLM	4–8	(Scarfe et al., 4 Apr 2025)
3D polarization	Integrated inverse-designed chip	4	(Otte et al., 2023)

3. Security Proofs: Asymptotic and Finite-Key, Composability

HD-QKD protocols have evolved from asymptotic, collective-attack security proofs to full finite-key, universal-composable proofs against general attacks.

• Entropic uncertainty relations: Key lengths are bounded via entropic uncertainty relations for coarse-grained time and conjugate time (or analogous) measurements. For example, (Niu et al., 2016) establishes

$$H_{\min}^\epsilon(\mathbf{X}_A|E) + H_{\max}^\epsilon(\mathbf{Y}_A|\mathbf{Y}_B) \geq -n \log_2[c(\delta, \beta_D)]$$

where $c(\delta,\beta_D)$ quantifies basis incompatibility.

• Composable security: Security is defined by trace distance between the actual and ideal key–eavesdropper joint states. Universal composability is achieved by summing correctness and secrecy errors: $\epsilon=\epsilon_c+\epsilon_s$ (Niu et al., 2016, Kanitschar et al., 6 May 2025).
• Decoy-state analysis: Finite-key security and defense against photon-number-splitting attacks require decoy-state methodology, bounding single-photon yields and noise via data from multiple source intensities (Bunandar et al., 2014, Niu et al., 2016).
• Finite-key protocols: Explicit finite-key bounds express the extractable secret key as

$$\ell \geq H_{\min}^\epsilon(\mathbf{X}_A|E) - \text{leak}_{EC} + \log_2(\epsilon_s^2\epsilon_c)$$

with all terms directly rooted in experimental data (Niu et al., 2016, Kanitschar et al., 6 May 2025).

Even under realistic, fluctuating channels (e.g., turbulent satellite or free-space links), variable-length protocols can significantly enhance expected key rates and reduce abort probability (Kanitschar et al., 6 May 2025).

4. Information Reconciliation and Error Correction

Efficient reconciliation is crucial to HD-QKD throughput and effective key rate:

• Nonbinary LDPC codes: Codes over GF($d$) or higher, optimized via density evolution, achieve reconciliation efficiency approaching the Slepian–Wolf bound on $d$-ary symmetric channels (Mueller et al., 2023).
• Cascade protocol (HD-Cascade): Adaptations of the Cascade algorithm for $d$-ary alphabets use bit-plane grouping, random interleaving, and self-cascading. HD-Cascade matches or exceeds nonbinary LDPC efficiency for $d$ up to 32, with practical communication overhead (Mueller et al., 2023).
• Efficiency impact: Switching from bitwise to truly $d$-ary reconciliation yields up to 10% longer secure distance and increased secret key rates under otherwise fixed system parameters.

5. Experimental Performance and Key Rate Scaling

Key rates, photon information efficiency, and maximal tolerable error all improve with dimension $d$:

• Per-photon capacity: Ideal BB84-style HD-QKD with $d$-dimensional encoding delivers a raw information rate $\log_2 d$ bits per sifted photon. Practical protocols have demonstrated up to 3.3 bits per photon at zero distance (Niu et al., 2016).
• Noise tolerance: The threshold quantum bit error rate (QBER) before $r_d(Q)=0$ increases with $d$: e.g., 11% ($d=2$), 18.9% ($d=4$), 24% ($d=8$) (Zhang et al., 12 Dec 2025).
• Long-distance operation: Time–energy HD-QKD yields positive key rates at 100–200 km with state-of-the-art detectors, outperforming qubit BB84 and CV-QKD in both range and per-photon efficiency (Niu et al., 2016).
• Robustness: Self-healing spatial modes (BG vs. LG) exhibit enhanced resilience to partial obstructions (Nape et al., 2018). OAM-based systems benefitted from adaptive optics or integrated mode sorters.

Table 2 shows achievable bits per photon vs. $d$:

Dimension $d$	Ideal info. per photon (bits)	QBER threshold (%)
2	1.00	11
4	2.00	18.9
8	3.00	24.0
16	4.00	$\sim$27–32

6. Protocol Variants and Architectures

HD-QKD encompasses both entanglement-based and prepare–measure paradigms, with a spectrum of protocol innovations:

• Time–energy HD-QKD: Exploits SPDC sources and dispersive optics; composable finite-key security established (Niu et al., 2016).
• Fourier-qubit (F-qubit) protocols: Use only 2-level superpositions with $d$ phases; experimentally demonstrated with OAM modes for $d=4$ (Scarfe et al., 4 Apr 2025).
• Self-healing spatial modes: Hybrid spin–orbit Bessel–Gaussian modes offer resilience against physical obstructions in free space (Nape et al., 2018).
• Composable entanglement-based satellite QKD: Finite-size, variable-length key protocols address atmospheric fluctuations and noise (Kanitschar et al., 6 May 2025).
• Resource-efficient detection: Single-detector/basis HD-BB84 using the temporal Talbot effect provides simplified X-basis analysis with $d=4$ (Ogrodnik et al., 2024), albeit requiring careful accounting of basis-detection asymmetry in the security proof.
• Round-robin differential-phase shift (RRDPS) generalizations: HD-RRDPS achieves hybridized security and scaling with dimension (Stasiuk et al., 2023).
• Integrated photonics for multicore fiber: Chip-scale MUB manipulation in $d=4$ demonstrated, albeit with high-efficiency loss and thermal tuning constraints (Ding et al., 2016).

7. Practical Challenges, Limitations, and Outlook

Despite demonstrated advantages, HD-QKD faces several implementation and security challenges:

• Mode crosstalk and loss: Higher $d$ often means increased crosstalk (fiber/fiber-dispersion, free-space turbulence) and per-mode loss (Zhang et al., 12 Dec 2025).
• Scalability: While some protocols achieve $d=25$ (e.g., using programmable MPLCs), increasing $d$ demands more complex, low-loss, and stable high-dimensional measurements (Lib et al., 2024).
• Device modeling in security proofs: As security depends on detection efficiency symmetry and precise experimental models, resource-efficient or passive schemes require adapted proofs that account for realistic hardware asymmetries (Ogrodnik et al., 2024).
• Integrated platform development: Recent progress in inverse-designed chips, metasurface OAM sources, and multiplexed detection is promising for scalable, field-deployable HD-QKD systems (Otte et al., 2023).
• Finite-key and composability frontiers: Ongoing work extends composable proofs to arbitrary $d$ and minimizes memory and syndrome leakage in reconciliation (nonbinary LDPC vs. HD-Cascade) (Mueller et al., 2023, Kanitschar et al., 6 May 2025).

HD-QKD is positioned to play a foundational role in next-generation quantum networks, maximizing photon-limited links (including satellites and fiber) and supporting high-throughput, composably secure cryptography over noisy, dynamic channels (Niu et al., 2016, Kanitschar et al., 6 May 2025, Zhang et al., 12 Dec 2025).