Simultaneous Quantum-Classical Communication Protocols

• SQCC protocols are integrated communication schemes that transmit quantum and classical information simultaneously, leveraging entanglement and classical randomness for improved performance.
• They establish key resource trade-offs and capacity constraints, employing techniques like direct-sum theorems and message compression to optimize transmission efficiency.
• Practical applications include hybrid optical networks, remote state preparation, and efficient distributed computing, while also addressing challenges in noise tolerance and entanglement management.

Simultaneous Quantum-Classical Communication (SQCC) Protocols represent an integrated paradigm in communication complexity and quantum information theory, where both quantum and classical information are transmitted concurrently, often over the same physical channel or within unified communication resources. These protocols encompass theoretical models in communication complexity as well as practical implementations for cryptography, optical communication networks, and distributed computation. SQCC leverages the comparative strengths of quantum and classical channels—such as superposition, entanglement, classical randomness, and error-correcting codes—enabling new trade-offs and performance frontiers that are unattainable with purely classical or purely quantum schemes.

1. Formal Principles of Simultaneous Quantum-Classical Communication

SQCC protocols are defined by their simultaneous or integrated use of quantum and classical resources for the solution of communication tasks, often with the goals of efficiency, resource optimization, and security. In the communication complexity context, such protocols may operate in one-way, two-way, or simultaneous message-passing (SMP) models, with or without shared randomness or entanglement.

A central structural feature is the possibility for each participant to send both quantum and classical messages, potentially partitioned across independent instances or multiplexed within the same channel. In the channel coding setting (as in quantum Shannon theory), a pulse or channel use may encode both quantum information (e.g., for QKD) and classical information (e.g., for data transfer) via hybrid schemes that combine, for example, Gaussian modulation and coherent displacement.

The resource trade-offs central to SQCC can be summarized as follows:

• Direct Sum Theorems: For a function or relation $f$, the simultaneous cost to solve $m$ parallel instances satisfies optimal direct-sum lower bounds for both quantum and classical resources:

$$Q^{1, \text{pub}}(f^m) = \Omega( m Q^{1, \text{pub}}(f) ), \quad R^{1, \text{pub}}(f^m) = \Omega( m R^{1, \text{pub}}(f) ).$$

This result constrains the possibility of joint compression in SQCC protocols and precludes sublinear total cost scaling when solving multiple instances in parallel (0807.1267).

• Capacity Regions: In the presence of shared entanglement, the boundary of achievable classical ($C$), quantum ($Q$), and entanglement ($E$) rates over a quantum channel is characterized by inequalities such as:

$$\begin{aligned} C + 2Q &\leq I(AX; B)_\sigma, \ Q &\leq I(A\rangle BX)_\sigma + E, \ C + Q &\leq I(X; B)_\sigma + I(A\rangle BX)_\sigma + E, \end{aligned}$$

where $I(\cdot;\cdot)_\sigma$ and $I(\cdot\rangle\cdot)_\sigma$ denote Holevo and coherent informations of appropriate auxiliary variables. These characterize the feasible rate region of simultaneous transmission (0811.4227).

• Message Compression and Round Elimination: If a communicated message (quantum or classical) leaks only $k$ bits of mutual information about the sender's input, then it can be compressed to length $O(k)$, potentially enabling the elimination of protocol rounds and supporting efficient SQCC design (0807.1267).

2. Resource Trade-offs and Lower Bounds

SQCC protocols are shaped by fundamental trade-offs involving communication cost, privacy leakage, and quantum resources such as entanglement:

Protocol Feature	Quantum (Entangled)	Classical (Public-coin)
Direct sum scaling	Linear in $m$	Linear in $m$
Privacy trade-off	$$L(f, A,B)^{20} L(f,B,A) \geq Q^{1, A \to B, \text{pub}}_{\epsilon}(f)$$	analogous classical bound
Resource reduction	No black-box reduction of entanglement (Newman's technique inapplicable)	Public coins can be reduced via Newman's theorem

• The privacy trade-off result asserts that reducing privacy loss (mutual information leak) on one side necessarily increases the communication cost or privacy loss for the other side—a balance that is particularly significant in hybrid protocols.
• The impossibility of black-box entanglement reduction establishes that, unlike classical public randomness, prior entanglement cannot be "compressed" to logarithmic size via black-box constructions when entanglement is crucial for protocol performance. SQCC protocols must thus explicitly provision for entanglement when required (0807.1267).

3. Simultaneous Encoding, Protocol Architectures, and Message Specification

A diversity of architectures realize SQCC, from channel codings to network protocols and formal programming frameworks:

• Channel Coding: The classically-enhanced father protocol achieves the simultaneous coding of classical and quantum information over a noisy channel, outperforming time-sharing strategies and unifying prior quantum protocols. The resource inequality for a channel $\mathcal{N}$ is

$$\langle \mathcal{N}^{A' \to B} \rangle + \frac{1}{2} I(A;E|X)_\sigma [qq] \geq \frac{1}{2} I(A;B|X)_\sigma [q \to q] + I(X;B)_\sigma [c \to c].$$

This establishes the potential for joint classical and quantum mode utilization as opposed to splitting channel uses between modes (0811.4227).

• Programming Frameworks: Formal languages for specifying and proving SQCC protocols model quantum and classical channels explicitly, partitioning process variables and guaranteeing non-cloning and causal semantics. For instance,

$$P\ ||_\psi\ Q = \psi := (U_P \otimes U_Q) \psi \ \land\ c' = c + \Delta c \ \land\ q' = q + \Delta q,$$

where $U_P, U_Q$ represent local unitaries, $\Delta c, \Delta q$ account for communication increments (0907.5162).

• Network Coding: It is possible to quantumly simulate classical (including nonlinear) network coding protocols by mapping classical encoding functions to local unitaries, disentangling ancillas via Fourier-basis measurement, and correcting induced phases via local operations, while confining the required classical communication to adjacent nodes. This results in efficient SQCC schemes for multi-source, multi-destination scenarios (Kobayashi et al., 2010).

4. Quantum-Classical Complexity Separations and Protocol Efficiency

Sharp separations have been demonstrated between the efficiency of SQCC protocols (especially in SMP and related models) and their purely classical counterparts:

• Entangled SMP Protocols: There exist partial functions that can be computed by simultaneous, entangled quantum message protocols with $O(\log^2 n)$ qubits, while any classical protocol (even two-way) requires polynomial communication cost (Gavinsky, 2016). The Shape problem is a canonical example.
• Public Randomness Replacement: Quantum SMP models can eliminate the need for classical public coins by using fingerprinting and SWAP tests; carefully designed compilers translate public-coin classical protocols with modified equality queries into pure quantum protocols without any shared randomness, affording exponentially better efficiency in the private-coin regime (Gall et al., 11 Dec 2024).
• Entanglement-Communication Trade-offs: Explicit partial functions exist where reducing shared entanglement from $\tilde{\Theta}(k^5 \log^3 n)$ qubits to $O(k)$ qubits exponentially increases (classical or quantum) communication requirements, underscoring the need to allocate sufficient entanglement resource in high-efficiency SQCC protocol design (Arunachalam et al., 2023).

5. Optical, CV-Based, and Hybrid SQCC Implementations

Practical SQCC protocol implementations frequently employ optical coherent states and continuous-variable protocols, exploiting the physical compatibility of classical coherent communication and CV-QKD:

• Integrated Pulse Encoding: Classical information (e.g., via BPSK or QPSK displacement) and quantum information (via Gaussian modulation) are encoded on the same weak coherent pulse; Bob distinguishes both by sequential discrimination and re-displacement (Qi, 2016, Qi et al., 2017).
• Noise and Security Modeling: Accurate modeling of noise sources, including the distinction between trusted (locally calibratable) and untrusted (eavesdropper-accessible) phase noise, enhances the tolerance of CV-SQCC schemes to practical imperfections. Security analysis is supported by Monte Carlo simulation and covariance matrix renormalisation, ensuring composable security in both asymptotic and finite-key regimes (Zaunders et al., 6 May 2025).
• Channel Resource Trade-offs: For infinite-dimensional (optical) states, communication resource scaling is governed by a tradeoff between the mean photon number $\mu$ and number of time-bin modes $m$:

$$\min\{\mu\log m,\ m\log(1+\mu/\delta)\}\ \geq\ \Omega(\log D(f))$$

This sets physical lower bounds for energy and latency in optical SQCC implementations (Marwah et al., 2020).

6. Applications, Network Integration, and Future Prospects

SQCC protocols underpin applications across secure communications, distributed computing, and quantum cryptography:

• Hybrid Optical Networks: Simultaneous transmission of quantum and classical signals over shared fiber or free-space optical infrastructure is feasible through careful spectral allocation, filtering, and digital signal processing, offering significant cost, hardware, and routing efficiencies (Lukens et al., 11 Feb 2025).
• Cryptographic Functionality with Classical Clients: Advanced remote state preparation (RSP) protocols allow a fully classical sender to force a quantum receiver to generate BB84 states via exclusively classical interaction, enabling classical clients to participate in unclonable encryption, quantum copy-protection, and verifiable delegated quantum computation, assuming only post-quantum security (Gheorghiu et al., 2022).
• Performance Enhancements via Software Post-Processing: Gaussian post-selection enables SQCC systems to dynamically optimize the modulation variance in software, post channel-estimation, boosting key rates and transmission distances robustly in variable channels such as free-space and satellite links, with no hardware changes (Erkilic et al., 15 Oct 2025).
• Quantum-Classical Separation in Multiparty and NOF Models: Quantum protocols in the simultaneous number-on-forehead (NOF) model can achieve exponential reductions in communication cost relative to classical randomized schemes, highlighting the unmatched efficiency possible with quantum simultaneity (Yang et al., 20 Jun 2025).

7. Technical Challenges and Theoretical Frontiers

SQCC protocols face several design and analytical challenges:

• Entanglement Management: Efficient distribution and maintenance of large-scale shared entanglement remains a bottleneck, especially in networked or high-rate scenarios (0807.1267, Arunachalam et al., 2023).
• Compression and Privacy Trade-offs: Compression strategies must respect lower bounds on privacy loss, and attempts to trade rounds for message length are constrained by round-elimination lemmas and information-theoretic inequalities (0807.1267).
• Measurement Non-Cloning and State Reuse: In quantum SMP settings, reusing quantum fingerprints without collapse requires careful use of gentle measurements and sequential processing, with error amplification and quantum union bounds to preserve soundness (Gall et al., 11 Dec 2024).
• Noise Tolerance and Real-World Implementation: Practical CV-SQCC protocols must account for detector imperfections, phase reference instability, and fluctuating transmission—addressed via refined security modeling and post-selection (Qi et al., 2017, Erkilic et al., 15 Oct 2025, Zaunders et al., 6 May 2025).

The field continues to evolve with the development of protocols exploiting new coding schemes, more refined security analyses, advances in hybrid integration, and ever-stronger evidence of quantum-classical separations in communication complexity. As classical and quantum communication become increasingly intertwined within networked infrastructures, the principles and limitations established for SQCC protocols form a foundational guide for the secure, efficient, and scalable communication systems of the future.