Path-Encoded High-Dimensional Entanglement

• The paper demonstrates how assigning each qudit basis state to a unique spatial path enables both deterministic and probabilistic generation of high-dimensional quantum entanglement.
• Experimental techniques like multipath SPDC and integrated photonic circuits achieve record fidelities and scalability, with implementations reaching up to d=32 modes.
• High-dimensional state certification using partial tomography and entanglement witnesses confirms robust nonlocality and paves the way for advanced quantum key distribution protocols.

Path-encoded high-dimensional entanglement denotes quantum states in which a discrete set of spatial paths encodes high-dimensional quantum information, and multipartite entanglement is established across these modes. In such schemes, each basis state of a single qudit (or higher-level quantum particle) is assigned to a unique optical path or waveguide mode, allowing the direct mapping of computational basis states onto physical spatial channels. This paradigm enables scalable, robust, and efficient generation, manipulation, and distribution of high-dimensional entangled states—crucial for quantum communication, computation, and foundational studies in quantum mechanics.

1. Theoretical Foundation and State Construction

The core of path-encoded high-dimensional entanglement is the identification of each spatial path with a single-qudit logical state. For a $d$-dimensional system, the computational basis is formed as $|k\rangle$, where $k$ labels the path the photon occupies. The canonical two-photon maximally entangled state is: $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$ where $A$ and $B$ refer to two parties or subsystems. Both deterministic and probabilistic schemes for path entanglement exist, relying on controlling the coherences among spatial modes—frequently via beam displacers, Mach–Zehnder or multiport interferometers, and active phase stabilization (Erhard et al., 2019, Krenn et al., 2016, Thomas et al., 17 Oct 2025). The generalization to $N$-partite and arbitrary $d$ is

$$|\mathrm{GHZ}_{d}^{(N)}\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle^{\otimes N}$$

encompassing entire high-dimensional multipartite entanglement structures (Bell et al., 2022, Arlt et al., 12 Oct 2025).

A flexible family of path-encoded high-$d$ states is: $|k\rangle$0 allowing full tunability of amplitude and phase via pump distribution and optical settings (Erhard et al., 2019, Krenn et al., 2016).

2. Experimental Realizations and Techniques

Generation of path-encoded high-dimensional entanglement leverages both bulk and integrated photonic platforms. Key experimental approaches include:

• Multipath SPDC: Coherent splitting of a pump laser into $|k\rangle$1 spatially separated beams, each directed to an individual nonlinear crystal for spontaneous parametric down-conversion (SPDC), with paths recombined such that which-crystal information is erased, producing entanglement by path identity (Kysela et al., 2019, Krenn et al., 2016, Svozil, 2017).
• Integrated photonic circuits: On-chip implementations use arrays of microring resonators or nonlinear waveguides, enabling path encoding across multiple (often up to $|k\rangle$2) waveguides, with phase shifters for active stabilization and reconfigurability (Thomas et al., 17 Oct 2025, Zhang et al., 2021, Forbes et al., 17 Oct 2025).
• Bulk-optical beam displacers and wave plates: Cascades of beam displacers and half-wave plates expand path dimensionality and permit the implementation of arbitrary local unitaries for manipulation and measurement (Hu et al., 2020, Hu et al., 2020).

Advanced schemes enable hyperentanglement in path and frequency or transverse mode by introducing frequency shifters, hybrid encoding (e.g., path and TE mode), or quantum frequency combs (Majumdar et al., 2021, Forbes et al., 17 Oct 2025, Zhang et al., 2021). Experimental systems have demonstrated path-encoded entangled states of up to $|k\rangle$3 with record fidelities $|k\rangle$4 and entropy nearing the maximally entangled limit (Hu et al., 2020).

3. State Characterization and Entanglement Certification

Characterization employs both direct tomography and entanglement witnesses optimized for high-dimensionality. Standard approaches include:

• Partial tomography: Measuring key diagonal probabilities and select coherences is efficient for large $|k\rangle$5, drastically reducing total settings required compared to full tomography (Hu et al., 2020, Hu et al., 2020).
• Mutually unbiased bases: Complete reconstruction for $|k\rangle$6-dimensional systems via $|k\rangle$7 settings, tractable for small $|k\rangle$8 and used to certify genuine $|k\rangle$9-level entanglement entropy and state fidelity (Thomas et al., 17 Oct 2025).
• Witnesses and inequalities: Dimensionality witnesses compare measured fidelity against maximal possible overlaps for lower-$k$0-entangled or separable states, confirming irreducible high-dimensionality (e.g., Huber–de Vicente witness with threshold $k$1 for $k$2-type states in $k$3 (Hu et al., 2020)), and generalized Bell/Mermin/CGLMP inequalities for demonstration of nonlocality and device-independent security (Hu et al., 15 May 2025, Erhard et al., 2019, Kysela et al., 2019).

Certification metrics are summarized in the following table:

Metric	Definition/Threshold	Certification Purpose
Fidelity $k$4	$k$5	Overlap with target MES
Schmidt number $k$6	$k$7	Effective number of modes
Entanglement entropy $k$8	$k$9	Degree of entanglement
Witness threshold $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$0	Max overlap for lower Schmidt rank	GME certification
Bell/CGLMP/Other	Inequality violation	Nonlocality/data independence

For example, $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$1 was measured for a $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$2 tripartite state, well above the $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$3 threshold for non-genuine high-dimensional entanglement (Hu et al., 2020).

4. Multipartite and Layered High-Dimensional Entanglement

Multipartite states including GHZ structures and layered entanglement have been realized via combinations of path and polarization degrees of freedom. In the three-photon case, hybrid path-polarization encoding allows the generation of states such as: $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$4 with mode assignment derived from upper/lower path and $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$5 polarization (Hu et al., 2020). This enables the extraction of layered keys for quantum cryptography—e.g., a three-party key from joint binning across layers, or an additional two-party key independent of the third party’s outcome (layered QKD).

GHZ-type states in $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$6 and $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$7 across $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$8 photons have been generated in multi-pair sources with path identity and polarization-controlled exchange to effect coherent mixing and post-selection, verified via high-dimensional Mermin-type inequalities (Hu et al., 15 May 2025). Layered entanglement provides functional advantages in quantum networks, enabling multiple, independent key layers and enhanced parallelism.

5. Scalability, Integration, and Performance Limits

Path-based high-dimensional entanglement is distinguished by its scalability and compatibility with both bulk and integrated platforms:

• Scalability: The number of spatial modes can be increased nearly arbitrarily, with demonstrations up to $$|\Psi_d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle_A |k\rangle_B$$9 in bulk optics and $A$0 on silicon photonic chips (Hu et al., 2020, Thomas et al., 17 Oct 2025, Forbes et al., 17 Oct 2025).
• Integration: On-chip generation and manipulation employing microring resonators, thermo-optic shifters, and multiport MZIs are compatible with telecom wavelengths and standard fiber architectures; phase stabilization algorithms maintain coherence across all modes with minimal measurement rounds or hardware overhead (Thomas et al., 17 Oct 2025, Zhang et al., 2021).
• High fidelity and entropy: Experimental setups have realized fidelities $A$1 and entropy $A$2 (maximal) for high-$A$3 entangled states, with Schmidt numbers approaching the number of paths in use (e.g., $A$4 for $A$5 paths) (Hu et al., 2020).
• Error tolerance and loss: Path-based encoding exhibits strong robustness to channel noise, loss, and phase drift, with error thresholds and key rates increasing as $A$6 for quantum key distribution (Bell et al., 2022, Thomas et al., 17 Oct 2025, Erhard et al., 2019).

Limitations arise in the form of increased optical loss and greater complexity of phase stabilization for higher $A$7, but integrated feedback and compact photonic circuitry mitigate scaling overheads. Hybrid encoding—combining path with polarization, frequency, or transverse-mode DOF—multiplies accessible Hilbert space with little physical overhead (Forbes et al., 17 Oct 2025, Majumdar et al., 2021).

6. Applications and Future Directions

The proliferation of path-encoded high-dimensional entanglement underpins several emerging capabilities:

• High-capacity QKD: Encoded alphabets $A$8 offer $A$9 key bits per photon and enhanced resilience to noise (Thomas et al., 17 Oct 2025, Erhard et al., 2019).
• Device-independent protocols: Generalized Bell inequalities are violated over multiple dimensions for robust, loophole-free security (Hu et al., 15 May 2025).
• Dense coding, teleportation, and measurement-based quantum computing: Exploiting large local Hilbert space allows transfer of more classical bits per photon, efficient cluster-state engineering, and reduced overhead in logic/correction (Erhard et al., 2019, Zhang et al., 2021).
• Distributed networks and inter-chip quantum links: Multipath entanglement is compatible with multicore fibers and on-chip photonic routers, facilitating scalable quantum networking and distributed sensing architectures (Thomas et al., 17 Oct 2025, Hu et al., 2020).
• Hybrid and hyperentangled architectures: Simultaneous entanglement in multiple DOFs increases parallelism and enables advanced error-correcting and distillation protocols (Forbes et al., 17 Oct 2025, Majumdar et al., 2021).

Future work will address the extension to $B$0, larger multipartite topologies, real-time adaptive stabilization, and integration of sources, unitaries, and detectors on single chips. Full on-chip implementations utilizing passive and active elements (multiport MMIs, mode sorters, frequency shifters) are progressing toward universal, low-loss, and highly reconfigurable sources of multipartite high-dimensional entanglement (Zhang et al., 2021, Forbes et al., 17 Oct 2025).

7. Comparative Advantages and Fundamental Impact

Path-encoded entanglement combines high-fidelity, robust control, and integration-readiness unmatched by other DOFs (notably OAM or time bins). Advantages include straightforward implementation of arbitrary unitaries and measurements (via multiport interferometers), minimal loss for moderate $B$1, and direct mapping to photonic chip and fiber architectures. Experimental observations highlight significantly higher interference visibilities ($B$2) than typical OAM-based implementations ($B$3), and the path interface facilitates seamless conversion to transverse spatial/OAM encoding for long-distance transmission (Hu et al., 2020, Fickler et al., 2014, Hu et al., 2020). This versatility has established path-encoded high-dimensional entanglement as an indispensable resource for the next generation of photonic quantum technologies.