Weak Coherent Pulses in Quantum Communication

• Weak coherent pulses (WCPs) are attenuated, phase‐randomized optical pulses with a Poissonian photon number distribution used to approximate single-qubit states in quantum protocols.
• They support quantum communication and blind quantum computing by enabling remote state preparation and secure transmission despite device imperfections and photon loss.
• Protocol security is enhanced by quantifying state deviations through ε‐blindness and optimizing resource trade-offs using statistical checks and QND measurements.

Weak coherent pulses (WCPs) are phase-randomized optical pulses generated by attenuated lasers such that their photon number distribution is Poissonian with a mean far below unity. WCPs are a primary workhorse of quantum communication and distributed quantum information processing, given the practical infeasibility of on-demand, long-distance single-photon sources. Their controlled generation and predictable statistical structure allow WCPs to support quantum protocols—such as blind quantum computing—by serving as effective surrogates for ideal single-qubit states while requiring only simple laser technology. The use of WCPs underpins blind quantum computing, remote qubit preparation, and other protocols where device capabilities and photon loss must be explicitly managed without sacrificing security or composability.

1. Quantum State Preparation with Weak Coherent Pulses

In an ideal universal blind quantum computing (UBQC) protocol, the client is required to prepare and send random, separable single-qubit states of the form

$$|+_{θ}\rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{iθ}|1\rangle), \quad θ \in \{0, π/4, ..., 7π/4\},$$

to encode and “blind” the computation from the server. However, generating near-perfect single photonic qubits is technologically prohibitive in realistic scenarios, especially over long-distance quantum networks.

Instead, the client prepares weak coherent pulses: laser pulses attenuated until the mean photon number $\mu$ (typically $\mu \ll 1$) is sufficiently small. After randomizing their optical phase and encoding each pulse’s polarization in a randomly chosen angle $\sigma \in \{0, π/4, ..., 7π/4\}$, the prepared state assumes the form

$$\rho^{(\sigma)} = \sum_{k=0}^\infty p_k |k\rangle\langle k|_{(\sigma)}, \quad p_k = e^{-\mu} \frac{\mu^k}{k!},$$

where $|k\rangle_{(\sigma)}$ denotes a Fock state of $k$ photons in polarization $\sigma$. Phase randomization ensures decoupling between phase coherence and photon-number information, yielding a classical statistical mixture in the Fock basis.

This physical preparation step significantly reduces client hardware requirements, since only standard laser diodes with commonly available polarization controllers and intensity modulators are necessary. The optimal value for $\mu$ is chosen to match the channel transmittance $T$, such that $\mu = T$ maximizes the security-versus-efficiency tradeoff.

2. Formalizing Approximate Blindness with Imperfect States

The key security question is quantifying the deviation from perfect blindness introduced by imperfect state preparation. The concept of $\varepsilon$-blindness captures this rigorously. Consider:

• The ideal joint state between the client’s classical register (computation angles, one-time pad bits) and the server’s quantum system $\pi_{AB}^{(\mathrm{ideal})}$, in which the server’s marginal is independent of the computational secret.
• The actual joint state with imperfect, WCP-based preparations $\pi_{AB}^{(\{\rho^{(\theta_i)}\})}$, where each $\rho^{(\theta_i)}$ is not a pure $|+_{\theta_i}\rangle\langle+_{\theta_i}|$.

$\varepsilon$-blindness requires that there exists a completely positive, trace-preserving (CPTP) map $\mathcal{E}$ such that

$$\frac{1}{2} \| \pi_{AB}^{(\{\rho^{(\theta_i)}\})} - \pi_{AB}^{(\mathcal{E})} \| \leq \varepsilon,$$

where $$\pi_{AB}^{(\mathcal{E})} = (\mathbb{I}_A \otimes \mathcal{E}) \pi_{AB}^{(\mathrm{ideal})}$$, and $\|\cdot\|$ is the trace norm.

The error parameter can be bounded as the sum over preparation errors for each computational step: $$\epsilon_\mathrm{prep} = \max_{\theta} \frac{1}{2} \| \rho^{(\theta)} - \mathcal{E}(|+_\theta\rangle\langle+_\theta|) \|,$$

$$\frac{1}{2} \| \pi_{AB}^{(\rho)} - \pi_{AB}^{(\mathcal{E})} \| \leq S \cdot \epsilon_\mathrm{prep},$$

with $S$ the number of computational rounds/qubits. Thus, the overall deviation from ideal blindness is governed by the sum of the single-qubit preparation errors.

3. Protocol Mechanisms and Channel Verification

While WCPs are not true single photons, protocol security is enforced using a combination of statistical checks, causality constraints, and server-side postselection:

• Photon Number Resolution: The server, assumed to have advanced measurement capabilities, performs quantum nondemolition (QND) photon-number measurements on each incoming WCP. For each pulse, detection is stratified into three events:
  1. Vacuum $(k=0)$,
  2. Single photon $(k=1)$,
  3. Multi-photon $(k \geq 2)$.

• Cluster State “Purification”: When at least one single-photon event is obtained, the server performs an “Interlaced 1D Cluster” (I1DC) construction: detected photons are fused using controlled-phase (ctrl-Z) gates and Hadamard operations, with the sequence and polarization angles known to the client, culminating in a single-qubit state indistinguishable (modulo a CPTP map) from the desired $|+_\theta\rangle$.

• Client-Side Abort: To discourage or detect pathologically adversarial or lossy channels, the client monitors reported QND outcomes. The protocol aborts if the number of detected vacua among $N$ pulses exceeds approximately $N [e^{-T^2} + T^2/6]$ (with $T$ a lower bound on transmittance). This statistical test prevents the server from artificially highloss strategies or unreported measurement outcomes.
• Angle Computation: Knowing the set of polarizations $\{\sigma_\ell\}$ and the bit string $t$ encoding BSM outcomes, the final state’s angle is computed as $\theta = \sum_\ell (-1)^{t_\ell} \sigma_\ell$, ensuring client knowledge of the computation angle even after aggregation and error correction.

4. Quantitative Security and Efficiency Guarantees

Security and efficiency are quantitatively controlled via protocol parameters $N$ (number of pulses per qubit), $T$ (channel transmittance), and $S$ (number of steps):

• The failure and abort probabilities obey

$$p_\mathrm{fail},\,p_\mathrm{abort} \leq \exp\left(-\frac{N T^4}{18}\right).$$

• The global blindness parameter is bounded by

$$\varepsilon \leq S \exp\left(-\frac{N T^4}{18}\right).$$

• To guarantee $\varepsilon$-blindness, it suffices to set

$N \geq \frac{18 \ln(S/\varepsilon)}{T^4}.$

The design ensures that imperfect state preparations, channel losses, and multi-photon contributions can all be counterbalanced by increasing $N$ (at the cost of greater resource use), keeping the protocol within security thresholds.

5. Comparison to Prior and Related Techniques

The use of WCPs combined with aggregation and statistical analysis is conceptually adapted from quantum key distribution (QKD) protocols, where weak coherent pulses and decoy-state methods are standard. In the UBQC scenario, WCPs enable remote state preparation without the need for on-demand single-photon sources or quantum memories on the client, at the cost of an increased number of physical pulses and more stringent server-side verification.

The protocol bridges the gap between the theoretical minimum quantum requirement for blindness (single separable qubit states) and the practical limitations of quantum light sources. Approximate blindness is made composable and quantifiable by mapping physical imperfections directly to rigorous security parameters.

6. Practical Implications and Deployment Strategies

From a systems deployment standpoint, the protocol enables scalable and robust blind quantum computing over arbitrary distances and realistic lossy channels, with only minimal quantum resources on the client side. The main practical implications include:

• Resource reduction: The technological overhead for the client is reduced to a standard laser, polarization/audio phase modulator, and attenuator.
• Parameter tuning: All tradeoffs between security, efficiency, and resource cost can be explicitly calculated, enabling protocol parameter optimization for given experimental capacities and channel conditions.
• Server-side error detection: The protocol requires that the server be capable of QND measurements and cluster state generation; any deviation from the honest behavior is caught by client-side checks on reported vacuum events and angle statistics.
• Security composition: By introducing the $\varepsilon$-blindness definition, the protocol security becomes composable and robust under general attacks.

A plausible implication is the further adoption of WCP-based remote quantum state preparation in near-term experimental UBQC and delegated quantum computation schemes, especially given the continually increasing quality and controllability of commercial laser sources and stabilization components.

7. Summary Table: Protocol Elements

Protocol Element	Description	Quantitative Formula
State Preparation	Mean-photon-number $\mu$ phase-randomized WCP with polarization $\sigma$	$\rho^{(\sigma)} = \sum_k p_k |k\rangle\langle k|$
Blindness Criterion	Security/approximation to the ideal joint state	$$(1/2)\|\pi_{AB}^{(\rho)} - \pi_{AB}^{(\mathcal{E})}\| \leq \varepsilon$$
Failure probability	Max error of protocol not correctly producing qubit/abort	$$p_\mathrm{fail}, p_\mathrm{abort} \leq e^{-N T^4/18}$$
Angle Computation	Final state angle from measurement outcomes and polarization angles	$\theta = \sum_\ell (-1)^{t_\ell} \sigma_\ell$
Number of pulses needed	Resource scaling for $\varepsilon$-blindness	$N \geq 18 \ln(S/\varepsilon) / T^4$

8. Concluding Remarks and Reflections

Weak coherent pulses, when harnessed with carefully designed verification and state purification protocols, provide a concrete, experimentally feasible method for remote quantum state preparation and secure delegated quantum computing. The rigorous introduction of $\varepsilon$-blindness, as well as explicit bounds for resource requirements and failure rates, establishes a robust framework for blind quantum protocols that accommodate source imperfections, transmission losses, and adversarial servers. This approach defines a path for deploying large-scale secure quantum computation using standard photonics hardware and enables composable quantification of protocol security in realistic scenarios (Dunjko et al., 2011).