High-Dimensional Photonic Path States

• High-dimensional photonic path states are quantum states encoded in discrete spatial paths, enabling robust multi-level entanglement for quantum information processing.
• They are generated through coherent splitting and SPDC in integrated optical systems, ensuring scalable, high-coherence qudit implementations.
• Applications span quantum communication and computation, leveraging programmable optical circuits, advanced tomography, and hybrid architectures for enhanced robustness.

High-dimensional photonic path states are quantum states of light encoded in the discrete spatial paths available to photons propagating through optical systems. Such states are fundamental in photonic implementations of qudits (d-level quantum systems, with d > 2), as their dimensionality is determined by the number of spatial modes (paths) over which photons are coherently distributed. Path encoding provides a versatile framework for entanglement generation, state manipulation, and measurement in integrated and fiber-based quantum photonic networks, offering compatibility with on-chip quantum circuits, efficient interfacing with other degrees of freedom, and enhanced robustness in quantum information tasks.

1. Generation of High-Dimensional Path-Entangled States

The creation of high-dimensional path-entangled photonic states fundamentally relies on engineering indistinguishability and controlled coherence among photons traveling in different spatial modes. A prevalent method involves coherent splitting of a pump laser into N arms (paths) using a fiber or chip-based 1×N beam splitter, with each arm directing the pump into a nonlinear crystal waveguide, such as a periodically poled lithium niobate (ppLN) fiber-integrated chip. Type-I spontaneous parametric down-conversion (SPDC) in each waveguide generates photon pairs at telecom wavelengths; due to quantum indistinguishability, the event of photon-pair creation in one of the N arms leads to a coherent superposition state of N spatially separated creation events:

$$|\Psi\rangle = \sum_{i=1}^N a_i e^{i\phi_i} |i_A, i'_B\rangle$$

where $a_i$ are the amplitudes (tunable via splitting ratios) and $\phi_i$ are relative phases. For N=2 (qubit case), this yields a maximally entangled Bell state over two paths; for N > 2, entangled quNits (qudits) are created (Schaeff et al., 2012).

Device-level implementations span standard fiber components (e.g., DWDMs for wavelength separation, polarization controllers, and delay lines for mode matching), monolithic nonlinear photonic crystals engineered for concurrent quasi-phase-matched SPDC in different reciprocal lattice directions (Jin et al., 2013), and on-chip silicon photonic superlattices that localize topological edge/interface modes (Zakeri et al., 17 Sep 2025). Other state-of-the-art techniques include programmable spatial light modulator (SLM) arrays to define arbitrary transverse-path–OAM correlations (Pabón et al., 2019), quantum-dot sources for deterministic time-bin path W-states (Lee et al., 2017), and programmable pulse-shaping of PDC pump spectra to define high-dimensional temporal-mode path entanglement (Serino et al., 7 Jun 2024).

2. Mathematical Framework and Scalability

Path-entangled states span a composite Hilbert space of dimension $d^2$ for bipartite systems, with $d$ the number of spatial modes. The general two-quNit state is described as

$$|\Psi\rangle = \sum_{i=1}^{d} a_{i}e^{i\phi_i}|i_A, i'_B\rangle$$

for arbitrary amplitudes $a_i$ and phases $\phi_i$.

Scalability is linear: the number of required optical elements (beam splitters, nonlinear sources, wavelength filters) increases proportionally with $d$, and commercial multiport splitters with $d>100$ ports are available (Schaeff et al., 2012). Waveguide arrays or multiwaveguide ppLN chips further reduce complexity by integrating several sources on a single chip. The approach robustly generates arbitrary high-dimensional path states, limited mainly by losses, mode crosstalk, and the challenge of maintaining phase coherence across all paths (Erhard et al., 2019).

Component integration is facilitated by available multi-channel chips and fiber arrays. In integrated photonic circuits, arbitrary unitary operations on path-encoded qudits ($U \in U(d)$) are realized with multiport interferometric networks, classically decomposed (e.g., using Reck or Clements schemes (Chen et al., 2022)) into a mesh of beam splitters and phase shifters acting on the spatial modes.

3. Characterization and Tomographic Methods

Efficient characterization of high-dimensional path states is essential for quantum information protocols. Standard quantum state tomography, which requires $O(d^2)$ measurements, becomes impractical for large $d$. Methods designed for path qudits include:

• Rotating 1D optical Fourier transform (OFT): By imaging the momentum-space interference pattern (using cylindrical lenses rotated to resolve all unique path-pair separations), both populations (diagonal elements) and coherences (off-diagonal) are extracted via discrete Fourier analysis. This approach achieves high-fidelity state reconstruction (average fidelity $\geq 0.985$ for $d = 6$) and scales efficiently (Curic et al., 2019).
• Integrated minimum POVM tomography: Symmetric informationally complete (SIC) POVMs implemented via compact photonic circuits in extended Hilbert spaces, using modular interferometer sectors with precisely calculated beam splitters and phase shifters. These designs generalize straightforwardly to arbitrary $d$ via 3D waveguide architectures and equidistant state constructions (Cardoso et al., 2019).
• Direct scan-free measurement: Weak measurement in one domain (momentum) is combined with parallel, strong measurement in position to directly reconstruct the entire high-dimensional state vector in a single setting, demonstrated for $d \sim 10^6$ (Shi et al., 2015).

These methods optimize for experimental resource efficiency and compatibility with integrated photonic platforms.

4. Manipulation and Universal Transformations

Universal manipulation of path-encoded qudits is achieved with programmable linear optical circuits. Arbitrary unitary transformations $U \in U(d)$ can be decomposed into sequences of two-mode rotations, implemented in rectangular or triangular mesh architectures of beam splitters and phase shifters.

Recent optimized schemes leverage dual encodings by combining path and polarization degrees of freedom. In hybrid architectures, half the unitary layers act only on polarization (implemented with local wave plates), thereby reducing interferometric requirements by half. Full-polarization schemes employ high-dimensional X gates to transfer path-based operations to local polarization rotations within each path, further reducing optical depth and robustly maintaining the symmetry of the operation (i.e., equal loss/optical traversals across all modes) (Chen et al., 2022).

Such architectures are essential for scaling up high-dimensional quantum computation, as they reduce system complexity and experimental losses for large $d$.

5. Integration with Other Degrees of Freedom and Topological Protection

High-dimensional path states can be interfaced with other photonic degrees of freedom, including orbital angular momentum (OAM), polarization, time bins (temporal modes), and frequency. Mode sorters transform spatial path entanglement into OAM entanglement suitable for free-space or fiber-based communications (Fickler et al., 2014). More generally, SLM-based methods can generate arbitrary $d \times m$ "bipartite" spatial–OAM states with independently programmed amplitude and phase control (achieving fidelities above 95%) (Pabón et al., 2019).

Topological photonic superlattices provide another robust foundation. Carefully designed silicon photonic waveguide arrays support multiple, co-localized, topologically protected interface modes; nonlinear four-wave mixing (DFWM) in such structures generates energy–time entangled photon pairs occupying a superposition of up to five topological modes. The resulting states exhibit resilience to nanofabrication disorder, and the topological protection mechanism ensures consistent entanglement dimensionality and minimal degradation (Zakeri et al., 17 Sep 2025). This design supports scalable, fault-tolerant photonic quantum architectures.

6. Applications in Quantum Information Processing

High-dimensional path-entangled photonic states form the backbone of several advanced quantum protocols and systems:

• Quantum communication: Path-entangled qudits enable higher information capacity per photon and improved security in quantum key distribution (QKD) schemes, particularly when integrated with robust platforms (fiber, photonic chips) and combined with other DoFs for hyper-entanglement (Erhard et al., 2019).
• Quantum computation: Resource-efficient measurement-based photonic quantum computing benefits from encoding multiple qubits or qudits onto single photons using high-dimensional path modes, thereby circumventing exponential detection bottlenecks typical of conventional schemes. This method has enabled the generation and measurement of cluster states equivalent to more than nine qubits at 100 Hz, with linearly scaling circuit requirements for measurement and feedforward (Lib et al., 2023).
• Quantum networks and memories: Multi-channel integrated quantum memories with random access (e.g., 11-channel laser-written waveguide arrays in rare-earth–doped crystals) demonstrate storage and retrieval of five-dimensional path-encoded quantum states with retrieval fidelities above 96%, supporting scalable high-dimensional quantum networking (Ou et al., 27 Aug 2025).
• Quantum simulation, metrology, and foundational tests: High-dimensional path states facilitate stronger fundamental tests of quantum mechanics, high-precision sensor protocols, and simulations of complex quantum systems.

7. Current Challenges and Future Directions

Despite the scalability and versatility of path encoding, several challenges limit practical implementations:

• Source and integration complexity: Engineering sources that maintain indistinguishability and coherence over large numbers of spatial paths places demands on stability, alignment, and phase control. Fully integrated source-to-circuit-to-detector platforms are under active development.
• Loss and interference: As $d$ increases, interferometric stability and loss management become increasingly demanding, especially in multi-photon settings.
• Characterization at scale: Even with new tomographic schemes, verification and certification of very high-dimensional states (e.g., $d > 100$) remain an active area.

Future work will likely include extended on-chip integration of sources, switches, and detectors; error-corrected topological or hybrid qudit encodings; resource-efficient cluster state generators; and advanced interfaces between spatial path and other DoFs for future quantum networks and distributed computation.

This ensemble of methods and architectures establishes the centrality of high-dimensional photonic path states as both a theoretical and technological platform within contemporary quantum optics and quantum information science.