Phase-Encoded Quantum Communication

• Phase-encoded quantum communication is a method that encodes information in the phase of quantum states, enabling secure key distribution and encryption.
• It leverages discrete and continuous phase-shift coding in coherent, single-photon, or bosonic modes to enable high-rate, low-error transmission over various channels.
• Innovative protocols integrate QKD, digital signatures, and multiparty computation, enhanced by advanced error correction and phase compensation techniques.

Phase-encoded quantum communication exploits the phase degree of freedom of optical or matter-wave quantum states to encode, transmit, and process information. This modality leverages discrete or continuous sets of phase-shifted quantum states—typically coherent, single-photon, or bosonic modes—with protocols tailored for quantum key distribution (QKD), direct encryption, signature schemes, and multiparty computation. Precision phase encoding and decoding enable robust, high-rate secure communications, quantum error correction against dephasing, and unique information-theoretic security properties in both fiber and free-space channels.

1. Encoding Principles and State Constructions

Phase-encoded communication is most fundamentally realized by preparing quantum states whose information is embedded in their optical phase, with symbols $k$ from an $N$-ary alphabet mapped to phase-shifted coherent or single-photon states: $$|\psi_k\rangle = |\alpha\,e^{i\theta_k}\rangle, \quad \theta_k = 2\pi k/N,\, k=0, \ldots, N-1,$$ where $|\alpha|^2$ is the mean photon number. This "constellation" $C(\alpha,N)$ is symmetrically distributed on the complex plane, and information is encoded solely in the relative phases, not amplitude (Papanastasiou et al., 2018).

In practical protocols:

• Time-bin and OAM phase encoding map phase differences onto temporally or spatially separated single-photon or multimode states, e.g., $$(|1\rangle_{1}|0\rangle_{2} + e^{i\varphi}|0\rangle_{1}|1\rangle_{2})/\sqrt{2}$$ or $$(|{\rm LP}_{11a}\rangle + e^{i\varphi}|{\rm LP}_{11b}\rangle)/\sqrt{2}$$ (Alarcón et al., 2021).
• Advanced schemes simultaneously encode in phase, time, and intensity domains ("compact coding") to increase the classical information per signal (Hajj et al., 2016).
• Direct phase shifts are applied via unitary operators $U(\phi) = \exp(i\phi \hat n)$ on coherent states (Chan et al., 2023).

Passive state preparation is also feasible: a gain-switched dual-laser architecture yields random but measurable relative phases between pulse pairs, which, once measured, are used to label the output quantum state as one of a discrete desired set (e.g., BB84 basis) (Shakhovoy, 29 Jul 2025).

2. Channel Models and Physical Transmission

The primary communication channel is a thermal-loss bosonic channel, modeled as a beamsplitter (transmissivity $\eta$) mixing the input mode with a thermal environment (mean photon number $n_{\rm th}$) (Papanastasiou et al., 2018). The channel maps an input coherent state to a displaced thermal state, degrading the phase information via loss and noise.

In addition to standard single-mode fibers, scheme-specific platforms include:

• Few-mode or multimode fibers, enabling spatial-division multiplexing and eliminating post-selection losses (Alarcón et al., 2021).
• Multi-mode delay interferometers for free-space/multimode-fiber phase-encoded links, where well-chosen beam waists suppress diffraction-induced visibility loss (Tretiakov et al., 10 Apr 2025).
• Heterogeneous free-space and fiber cascades, where turbulence-induced phase drifts between time-bins can be compensated due to their long timescale relative to pulse separation (Sun et al., 6 Feb 2026).

Accurate phase references are sometimes circumvented entirely by exploiting phase noise memory: information is encoded in the relative phases of correlated blocks, allowing quantum and entanglement-assisted capacities to be achieved even without shared phase stabilization (Zhuang, 2020).

3. Key Protocols and Security Mechanisms

Quantum Key Distribution (QKD)

Phase-encoded QKD protocols utilize alphabets as small as $N=4$, achieving practical rates and metropolitan-scale distances:

• In direct and reverse reconciliation, the rate $R = I(A:B) - \chi_E$ (Devetak–Winter) is computed, where $I(A:B)$ is the classical mutual information and $\chi_E$ is Eve's Holevo information (Papanastasiou et al., 2018).
• For $N=4$ and $|\alpha|=1$, reverse reconciliation tolerates up to $15\,\mathrm{dB}$ of loss with rates $\approx 4 \times 10^{-3}\ \mathrm{bits/use}$ at $\epsilon=0.01$ excess noise (Papanastasiou et al., 2018).
• Heralded, narrow-band single photons with phase applied via direct waveform modulation eliminate passive loss from beam splitters and boost the sending efficiency $N$-fold in $N$-slot DPS QKD (Yan et al., 2010).

Quantum Encryption

Physical-layer encryption uses random phase masks seeded by pre-shared keys to apply unitary phase shifts, transforming coherent optical channels into noncoherent ones for eavesdroppers. The result is that the bit-error rate for an attacker saturates at 0.5, even for complex modulation formats (Chan et al., 2023). The security argument relies on the number–phase uncertainty relation and the infeasibility of phase estimation without knowledge of the key-stream.

Quantum Digital Signatures (QDS)

Sequences of randomly phased, low-amplitude coherent pulses serve as quantum signatures. Recipient multiport interference and phase-matched detections enforce both unforgeability and non-repudiation, with security parameters (forging and repudiation probabilities) exponentially suppressed in signature length (Clarke et al., 2013).

Quantum Ciphers and Phase-Encoded Queries

Quantum ciphers encode classical bits into the pattern of phase inversions within superpositions, protected by a one-time pad derived from QKD. Decryption involves reversal of controlled-phase flips and Hadamard operations (Menon et al., 2018).

Phase-encoded queries enable privacy-preserving computation in secure multiparty protocols. Boolean or geometric predicates are embedded as phase flips (typically $\pi$), with the total circuit cost polynomial in the bit-length of the inputs (Li et al., 2023).

4. Robustness, Compensation, and Error Correction

Efficiency and robustness of phase-encoded protocols are affected by several physical and technological factors:

• Phase drift in fibers, arising from thermal, acoustic, and mechanical perturbations, can be mitigated using space-division multiplexed phase compensation. By correlating phase drift in adjacent (“signal” and “reference”) fibers, active feedback extends the usable interferometric gate time by orders of magnitude, even in aerial and urban environments ($t_{0.03}$ improved by factors of $18$–$180$; $R > 0.99$) (Maruyama et al., 2024).
• Multimode/diffraction management: Phase-encoded signals can tolerate spatial multimode reception by optimizing beam geometry to maintain high-visibility interference ($V > 0.9$), with direct QBER impact (Tretiakov et al., 10 Apr 2025).
• Quantum error correction codes based on number–phase (NP) lattices correct both photon loss (number-shift errors) and dephasing by mapping errors onto diagnosable phase “vortices.” Canonical phase measurement provides error syndromes, and correction thresholds enable one-way channel key rates and fidelities across long distances (Hu et al., 17 Aug 2025).

5. Measurement, Receivers, and Capacity

Detection of phase-encoded information is fundamentally limited by the measurement model:

• Ideal (canonical) phase receivers use projective measurements onto phase eigenstates, achieving near-optimal channel capacity. Heterodyne and covariant receivers are more practical but incur a $\pi/4$ penalty in the quantum regime (Trapani et al., 2015).
• Quantum receivers combining optomechanical transduction and variational quantum circuits have been shown experimentally to outperform classical individual-detection receivers for phase-encoded codewords under realistic noise and loss, provided transduction efficiency $\eta_{\rm tr} \gtrsim 0.9$ and added noise $\bar n_{\rm tr} \lesssim 0.01$ (Crossman et al., 2023).
• Joint detection (across several pulses) is required to achieve superadditive capacity and access the ultimate bosonic channel limits. Block encoding and block measurements are essential for full exploitation of collective phase correlations, especially in noisy or memory channels (Zhuang, 2020).

6. Advanced Protocols and Future Directions

Recent advances explore modular variable formalisms, allowing fault-tolerant embedding of logical qubits into infinite-dimensional phase space using GKP or modular code states. Logical Pauli and Clifford operations, as well as measurement and error correction, are performed via periodic functions of quadrature operators (Ketterer et al., 2015). This approach unifies discrete and continuous paradigms and supports scalable, fault-tolerant phase-based quantum information processing.

Multi-domain ("compact") coding in phase, time, and intensity, as well as spatial and frequency multiplexing, offer new avenues for boosting secure capacity and integrating quantum protocols into heterogeneous classical networks (Hajj et al., 2016, Maruyama et al., 2024).

7. Applications, Implementations, and Scalability

Phase-encoded quantum communication underpins a diverse set of deployed and proposed applications:

• Metropolitan- and wide-area QKD networks leveraging robust phase encoding for long-range, high-rate key distribution (Papanastasiou et al., 2018, Yan et al., 2010).
• Free-space quantum communication and satellite-to-ground links, where phase encoding now achieves $>99\%$ visibility and sub-3% QBER over km-scale links, aided by turbulence-resilient architectures and effective feedback (Sun et al., 6 Feb 2026).
• Quantum digital signatures, privacy-preserving quantum computation, and direct quantum encryption (cipher) protocols, exploiting the physical non-clonability and indistinguishability arising from random phase encoding (Clarke et al., 2013, Menon et al., 2018, Li et al., 2023).
• Integrated quantum-classical networks, where phase-based cryptographic primitives coexist with classical coherent optical flows and WDM/DWDM networking (Chan et al., 2023, Maruyama et al., 2024).

Scalability in fiber networks is facilitated by space-division multiplexed compensation and multiplexed lantern/fiber architectures, supporting high-dimensional encodings and massive parallel quantum channels without duplicative stabilization (Alarcón et al., 2021, Maruyama et al., 2024).

---

In summary, phase-encoded quantum communication provides a flexible, efficient, and secure paradigm for quantum information transfer, with protocols adaptable across channel types, encoding dimensions, and application domains. The continued evolution of phase control, compensation, and error correction, as well as integration with quantum memories and networks, are central drivers of ongoing research and deployment (Papanastasiou et al., 2018, Yan et al., 2010, Chan et al., 2023, Hu et al., 17 Aug 2025, Maruyama et al., 2024, Sun et al., 6 Feb 2026).