High-Dimensional Spectral Qudits

• High-dimensional spectral qudits are quantum states defined in discrete, multi-level Hilbert spaces constructed from spectrally separated frequency bins, enabling robust encoding of quantum information.
• State engineering relies on nonlinear interferometry, spectral phase control, and pulse shaping to manipulate joint spectral amplitudes and achieve high-fidelity entanglement.
• Applications span quantum key distribution, metrology, and integrated photonic networks, with innovative inverse design methods addressing challenges like spectral crosstalk and loss.

High-dimensional spectral qudits are quantum states residing in discrete, multi-level Hilbert spaces constructed from spectrally defined modes—most commonly realized in the form of entangled photon pairs or single-photon states partitioned into frequency bins. Leveraging the spectral degree of freedom for encoding d-level quantum information enables substantial advantages in quantum communication, metrology, and computation, with experimental schemes advancing both control and scalability. State engineering of such qudits exploits nonlinear optical processes (e.g., spontaneous parametric down-conversion, four-wave mixing) along with interferometric phase control, spectral pulse shaping, microresonator-based photonic integration, and numerically assisted inverse design (Payne et al., 17 Jan 2026, Bernhard et al., 2013, Jin et al., 2016, Borghi et al., 2023, Rozenberg et al., 2021, Yang et al., 2023).

1. Physical Foundations and Spectral Encoding

Discrete spectral eigenmodes, commonly termed "frequency bins," constitute the logical basis of spectral qudits as $$\{|k\rangle\equiv\hat{a}^{\dagger}(\omega_0 + k\,\delta)|{\rm vac}\rangle;\,k=0,\ldots,d-1\}$$, with $\delta$ the bin spacing. Under type-II or type-0 SPDC, the broadband joint spectral amplitude (JSA) of biphotons can be shaped such that projection onto these bins yields a high-dimensional entangled state; for instance, $$|\psi_d\rangle=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}|k\rangle_s|k\rangle_i$$ is a maximally entangled d-dimensional state (Bernhard et al., 2013, Jin et al., 2016, Borghi et al., 2023). The effective Hilbert-space dimensionality is quantified by the Schmidt number $K=1/\sum_n\lambda_n^2$ (via Schmidt decomposition of the JSA), with practical values varying from $K=3-4$ (SLM shaping (Bernhard et al., 2013)) to $K\sim14$ (telecom-comb engineering (Jin et al., 2016)).

High-dimensional spectral qudits are generated also using solid-state systems, such as spin-$3/2$ silicon-vacancy ensembles in SiC. The four spin sublevels ($m_S={\pm3/2,\pm1/2}$) produce a four-level manifold, accessible by selective microwave excitation and spectrally resolved detection (Soltamov et al., 2018).

2. State Engineering: Nonlinear Interferometry and Spectral Phase Control

Advanced protocols exploit nonlinear interferometry with controlled spectral phases to manipulate the JSA and realize spectral qudit states in photonics. The four-crystal NLI setup, with adjustable delay lines for pump, signal, and idler between crystals, enables coherent superposition among emission events from each crystal (Payne et al., 17 Jan 2026). By imposing linear spectral phase shifts $\phi_j^{(\mu)}(\omega_j)=\omega_j\tau_j^{(\mu)}$ and engineering the set of delays $\{\tau_{p,s,i}^{(1..4)}\}$, two broad classes of states are realized:

• Grid states, with modulation $$\beta_{\text{grid}}(\omega_s,\omega_i)\sim\cos[(\omega_s+\omega_i)\tau/4]\cdot\cos[(\omega_s-\omega_i)\tau/4]$$, result in a discrete frequency lattice. Heralded detection yields a single-photon spectral qudit of dimension $d$ set by the number of grid sites spanned ($d_{\text{grid}}=8\sqrt{2}\pi/\tau$).
• High-dimensional entangled (HDE) states, with $$\beta_{\text{HDE}}(\omega_s,\omega_i)\propto\cos^2[(\omega_s-\omega_i)\tau/4]$$, manifest as islands along the anti-diagonal $\omega_s+\omega_i=\text{const}$; they directly encode bipartite entanglement over $d$ frequency bins.

Loss mechanisms and imperfect spectral overlap are modeled as attenuating the amplitudes $A_\mu$ from each crystal. The fringe visibility and state fidelity degrade predictably: for low interface loss ($X\lesssim1$ dB) and moderate $d\leq10$, visibility $V>80\%$ and fidelity $F>0.9$ are maintained (Payne et al., 17 Jan 2026).

3. Pulse Shaping, Spatial-Spectral Mapping, and Integrated Sources

Spectral qudit preparation via amplitude/phase pulse shaping utilizes spatial-light modulators (SLM) aligned to a dispersed spectrum (Bernhard et al., 2013). Frequency bins are defined by nonoverlapping spectral windows; SLM pixels modulate amplitude and phase within each bin. Maximally entangled states are obtained by equalizing detection rates across all bins ("Procrustean filtering").

Spatial-spectral mapping implements frequency-bin assignment via pump beam segmentation. The crystal's angle-dependent phase-matching maps spatial pump bins to distinct spectral loci in the JSA; e.g., a multi-slit mask creates a three-frequency entangled state, with further extension to $d\gg1$ enabled by substituting with high-resolution SLMs (Yang et al., 2023).

Silicon photonic platforms incorporate micro-ring resonators cascading frequency bins. Multiple rings are simultaneously pumped, each contributing a frequency bin, with programmable amplitude/phase via Mach-Zehnder interferometers and thermal phase shifters (Borghi et al., 2023). This architecture enables on-chip control of bin spacing and dimensionality ($D=2-4$ per party), achieving Bell-state fidelities $>84\%$ in $D^2=16$ Hilbert spaces.

4. Characterization and Tomography

Quantum state tomography is performed by projective measurements onto a tomographically complete set (single-bin and two-bin superpositions), followed by density matrix reconstruction via maximum-likelihood optimization. Figures of merit include fidelity with the target maximally entangled state, $$F_d=\mathrm{Tr}[\sqrt{\sqrt{\rho_\text{me}}\rho_d\sqrt{\rho_\text{me}}}]$$, and purity $P_d=\mathrm{Tr}[\rho_d^2]$ (Bernhard et al., 2013, Borghi et al., 2023). Correlated spectral intensity (CSI) maps and time-of-arrival histograms characterize joint spectral features. In multimode HOM interference, comb structures in the CSI reveal the number of accessible entangled modes (up to $d=14$ after 15 km fiber transmission with fidelity $>0.9$) (Jin et al., 2016).

Bell-type nonlocality quantification uses the CGLMP parameter $S_d$; SLM-based shaping demonstrates violation ($S_3=2.32$) certifying three-dimensional entanglement in frequency (Bernhard et al., 2013, Borghi et al., 2023).

5. Inverse Design and Theoretical Modeling

The SPDCinv framework enables inverse quantum-optical design by integrating the SPDC Hamiltonian’s evolution in the frequency domain and optimizing the pump spectral envelope and crystal nonlinear profile to output a desired spectral qudit state (Rozenberg et al., 2021). The loss function balances fidelity to target density matrices or coincidence maps with physical constraints (energy, bandwidth, phase-matching). Differentiable propagation and automatic gradient computation (e.g., JAX) permit high-dimensional optimization. Realization of explicit maximally entangled $$|ψ_{\text{target}}\rangle=(1/\sqrt{d})\sum_{n=1}^d|ω_n\rangle_s|Ω_p−ω_n\rangle_i$$ states is demonstrated.

6. Ramsey Interferometry and Spin Qudit Spectroscopy

Ramsey interferometry with qudits, extended to SU(2) Wigner-Majorana systems, multiplies spectral resolution by $d$ over qubit-based protocols (Ilikj et al., 8 Sep 2025). The multi-level sequence, with $\pi/2$ rotations mediated by $J_x$ and phase accumulation under $J_z$, is analytically modeled for $d=2$–5. The resolution–contrast index $RCI_d$ scales favorably with $d$ for qutrits ($RCI_3\approx 2\times RCI_2$), less so for higher $d$ (contrast degrades). No additional resolution is gained by replacing $\pi/2$ pulses with quantum Fourier transforms.

In spin-ensemble qudits (SiC $S=3/2$ centers), selective saturation and probe with sub-MHz microwave fields resolve spectral holes $\sim0.25–0.5$ MHz wide inside a 10–20 MHz line (Soltamov et al., 2018). Ramsey fringe analysis achieves 30 kHz resolution, deploying these systems for absolute DC magnetometry immune to thermal and strain noise with nT/$\sqrt{\mathrm{Hz}}$ sensitivity.

7. Applications, Scalability, and Limitations

High-dimensional spectral qudits realize increased channel capacity ($\log_2d$ bits per photon pair), enhanced secret-key rates in QKD, and stronger nonlocality tests against local realism (Bernhard et al., 2013, Jin et al., 2016, Borghi et al., 2023). Integrated sources and fiber distribution protocols retain high fidelity over telecom lengths, supporting multiplexed quantum networking.

Technical challenges include spectral crosstalk, finite shaping resolution, thermal drifts in integrated platforms, spectral filtering precision, and losses affecting interference contrast. Scalability depends on spectral resolution (comb/SLM pixel count) and the pump/crystal parameters determining the total number of accessible modes. Inverse-designed sources and spatial-spectral mapping provide programmable control over these degrees of freedom.

Prospective directions involve integrating active electro-optic modulators for bin mixing and quantum gate implementation, scaling device architectures to $d>4$ via parallel resonator arrays, and employing feedback-stabilized phase control for robust operation in practical quantum communication and computation systems (Borghi et al., 2023, Rozenberg et al., 2021, Payne et al., 17 Jan 2026).