High-Dimensional Multipartite Quantum Comm

• High-dimensional multipartite quantum communications are systems using qudits (d>2) that leverage expanded state spaces and enhanced noise resilience.
• They enable practical applications such as quantum secret sharing, conference key agreement, and layered key distribution through advanced entanglement and sequential encoding techniques.
• Implementations using multi-arm interferometry, integrated photonics, and quantum walks achieve high fidelity and reduced communication complexity across scalable networks.

High-dimensional multipartite quantum communications encompasses protocols, architectures, and physical implementations that leverage multipartite entangled and/or sequential quantum systems of local dimension $d>2$ (qudits; notably qutrits and ququads) for the transmission, distribution, and processing of quantum information across multiple parties. Exploiting both the expanded state space and enhanced noise robustness of high-dimensional systems, these schemes enable advanced primitives such as quantum secret sharing, layered key distribution, conference key agreement, and communication-complexity reduction, while providing systematic operational quantum advantages over classical strategies under equivalent resource constraints.

1. State Spaces and Foundational Protocols

The Hilbert space for high-dimensional multipartite communication is $\mathcal{H}_d^{\otimes N}$, with $d$ denoting the local dimension (qutrit: $d=3$; ququad: $d=4$; qudit: general $d$). A fundamental role is played by maximally entangled states, such as the $N$-party $d$-dimensional GHZ state,

$$|GHZ_N^d\rangle = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} |k\rangle^{\otimes N},$$

and more general superpositions tailored for application-specific tasks (e.g., the (4,4,2) state $$|\Psi_{442}\rangle = \frac{1}{2}(|000\rangle + |111\rangle + |220\rangle + |331\rangle)$$) (Hu et al., 2020). The communication primitives typically fall into the following categories:

• Secret sharing: The secret is distributed such that only joint action of a sufficient subset of parties can reconstruct it. Qudits increase the per-round key capacity and noise threshold (Smania et al., 2016, Hu et al., 15 May 2025, Hu et al., 2020).
• Detectable Byzantine agreement: The network achieves consensus even in the presence of potential faults or adversaries, with enhanced parameter regimes in high dimensions (Smania et al., 2016).
• Communication complexity reduction: A function of several private variables (across parties) is computed with sub-classical communication using phase-encoded high-dimensional states (Smania et al., 2016).
• Layered key distribution: High-dimensional multipartite states can support simultaneous extraction of keys shared across different subsets (“layers”) of the parties (Hu et al., 2020).

2. Physical Implementation Strategies

Implementations of high-dimensional multipartite communication rely on various degrees of freedom:

Encoding	Principal Implementation Approach	References
Path/temporal	Multi-arm interferometry, path identity, beam displacers	(Smania et al., 2016, Hu et al., 15 May 2025, Arlt et al., 12 Oct 2025)
Polarization+Path	Polarization-path hybrid, dimension-increasing modules	(Hu et al., 2020)
Integrated photonics	Soliton-induced dynamical Casimir effect, MW-optical hybrid circuits	(Dorche et al., 2020)
Quantum walks	Position-coin discrete-time quantum walks with engineered coin layers	(Nie et al., 2022)

Notable experimental setups include:

• Fiber-based single-qutrit sequential encoding: utilizes time/delay encoding and phase modulating elements in fiber interferometers with detection in the Fourier basis; allows prepare-and-measure protocols with detection efficiency scaling linearly rather than exponentially in the number of parties (Smania et al., 2016).
• Path-identity entanglement synthesis: creates high-dimensional GHZ (and generalized) states by coherent pumping of multiple heralded single-pair sources with post-selection, suitable for multipartite secret sharing and conference key agreement (Arlt et al., 12 Oct 2025, Hu et al., 15 May 2025).
• On-chip hybrid platforms: integrates Kerr soliton microresonators and coupled microwave resonators to produce persistent multipartite high-dimensional entangled states at microwave or THz frequencies, enabling quantum interfaces between superconducting and photonic platforms (Dorche et al., 2020).

3. Protocol Design, Scalability, and Layered Architectures

The principal design axes in high-dimensional multipartite communication protocols include:

• Sequential single-qudit protocols: A single qudit traverses $N$ parties in order, with each locally encoding information using commuting diagonal unitaries (primitive roots of unity): $U, V$ defined as $$U = |0\rangle\langle0| + \omega|1\rangle\langle1| + \omega^2|2\rangle\langle2|$$, $V = ...$, where $\omega = e^{2\pi i/d}$. Final measurement in the Fourier (mutually unbiased) basis extracts the global function or secret (Smania et al., 2016). Linear resource and loss scaling makes these protocols particularly advantageous versus entanglement-based strategies.
• Multipartite entanglement-based protocols: Prepare a high-dimensional multipartite state (e.g., $|\Psi_{442}\rangle$ or $|GHZ^3_3\rangle$) and distribute one subsystem to each party. Measurement in MUBs and suitable basis choices enable layered key agreement: a three-party key and simultaneous bipartite keys in different subspaces, as demonstrated in experiment for the (4,4,2) state (Hu et al., 2020). Path identity architectures similarly provide direct $N$-party high-dimensional distributions (Arlt et al., 12 Oct 2025, Hu et al., 15 May 2025).
• Quantum walk–engineered states: Discrete-time quantum walks with time/position-dependent coin unitaries can synthesize arbitrary $c$-party, $d$-dimensional states with circuit resources and communication costs that scale as $O(d^{c})$ (naive version) or $O(\log d)$ (parallelized, low-coherence-depth version) (Nie et al., 2022).
• Hybrid and on-chip entanglement generation: Dynamical Casimir platforms generate a continuous family of multipartite qudit entangled states with tunable dimension and mode structure by exploiting frequency-matched modulation and soliton dynamics; this allows Fock basis expansion into GHZ-like and graph states, supporting complex entanglement distribution with persistent on-chip coherence (Dorche et al., 2020).

4. Performance Metrics and Demonstrated Quantum Advantages

Key figures-of-merit and their typical empirical values in high-dimensional multipartite protocols include:

• Fidelity: Experimentally achieved state fidelities $F \gtrsim 0.85$ for nontrivial multipartite states (e.g., $|\Psi_{442}\rangle$, $|GHZ^3_4\rangle$) well above the genuine multidimensional entanglement threshold ($\frac{3}{4}$ for (4,3,2) rank states) (Hu et al., 2020, Hu et al., 15 May 2025).
• Quantum error/trit error rate (QTER): Sequential qutrit protocols yield $QTER \approx 6\% - 10\%$, with secret reconstruction requiring at least two collaborating parties (Smania et al., 2016).
• Success probability (SP): CCR protocols achieve $SP \sim 90–94\%$, surpassing the classical $7/9 \approx 77.8\%$ bound for comparable communication (Smania et al., 2016).
• Quantum-classical gap: Facet-based characterization reveals systematic quantum advantage for dimension- and distinguishability-bounded protocols, with explicit quantum strategies (e.g., projective measurements on simultaneous qubit or qutrit ensembles) exceeding the best classical polytopal bounds (Pandit, 5 Dec 2025). See table below for selected inequalities (with $d=2$):

Inequality	Classical Bound ($S_C$)	Best Quantum ($S_Q^{L}_2$)	SDP Quantum Upper Bound ($S_Q$)
I₁ (3,2,2)	2	2.4142	3
I₂ (4,2,2)	2	2.8284	4
I₄ (4,3,2)	10	13.3843	16

Ideal and near-ideal entropy and key rates can be asymptotically achieved in high-dimensional secret sharing using the relation $K \geq \log_2 d - h_d(e)$, with $h_d$ the $d$-ary entropy and $e$ the dit-error rate (Arlt et al., 12 Oct 2025).

5. Security, Robustness, and Error/Noise Tolerance

Security in high-dimensional multipartite protocols is subject to both intrinsic and extrinsic noise sources, as well as protocol architecture:

• Noise thresholds: $d=3$ GHZ protocols can tolerate up to $16\%$ error rates for secret sharing, compared to $11\%$ for qubit-based schemes; violations of dimension-sensitive Bell inequalities persist for $p>0.66$ (GHZ state visibility) (Hu et al., 15 May 2025).
• Device-independent characterization: Efficient two-measurement witnesses, as well as high-dimensional Bell inequalities, provide certification of genuine multipartite entanglement and can rule out qubit-based explanations (Hu et al., 15 May 2025).
• Side-channel and Trojan-horse countermeasures: Sequential interferometric protocols require active phase monitoring, polarization control, and may require extra security layers to counter extrinsic attacks (Smania et al., 2016).
• Error-correcting capabilities: Graph and GHZ qudit states can underlie topological error-correcting codes (e.g., qudit surface codes), enhancing fault tolerance and supporting higher thresholds (Dorche et al., 2020).

6. Resource Scaling, Implementation Complexity, and Outlook

Scalability and technological prospects are determined by the chosen protocol and physical architecture:

• Resource scaling:
  • Sequential single-qudit: Linear in party number ($N$) and dimension ($d$), requiring only a $d$-arm interferometer with $(d-1)$ phase modulators per site (Smania et al., 2016).
  • Entanglement-based (path-identity, walk-based): Sources and detectors scale as $O(Nd)$ or $O(d^c)$ for $N$ parties and $c$-partite target states (Arlt et al., 12 Oct 2025, Nie et al., 2022).
  • Quantum-walk-based preparation enables reduction of long-range CNOT cost from $O(d^c)$ (fully-controlled) to $O(c \log d)$ (parallelized) (Nie et al., 2022).
• Experimental overhead: For path-identity protocols, success probabilities are polynomial (rather than exponential) in $N$ and $d$ due to post-selection and absence of Bell-state measurements (Arlt et al., 12 Oct 2025), but fidelity is limited by mode-matching and indistinguishability.
• Integration and hybridization: On-chip DKS-DCE architectures offer monolithic, scalable routes to multimode, high-$d$ entanglement sources at both MW and THz frequencies, bridging superconducting and photonic domains for quantum networks (Dorche et al., 2020).
• Challenges and open directions: Extending to higher $d$ and $N$ while maintaining fidelity, closing all detection and locality loopholes in Bell certification, integrating with low-loss fiber platforms, and moving toward device-independent, dimension-agnostic self-testing of multipartite states remain frontier topics (Hu et al., 15 May 2025, Arlt et al., 12 Oct 2025).

High-dimensional multipartite quantum communication thus establishes a robust, resource-efficient, and extensible framework for foundational and applied quantum network protocols, enabling functionalities and performance unreachable by classical and two-dimensional quantum approaches. The combination of protocol innovation, advanced experimental architectures, and rigorous mathematical framework continues to drive the field towards complex, secure, and scalable quantum networks.