Phase Aggregated Smoothing (PAS)

• Phase Aggregated Smoothing (PAS) is a set of techniques that aggregate phase data across time or offsets to reduce estimation uncertainty.
• It leverages methods like Wiener and Kalman smoothing to achieve optimal quantum phase estimation with reduced mean-squared error.
• PAS also stabilizes temporal positional encoding in Video LLMs by aggregating phase offsets, enhancing attention consistency.

Phase Aggregated Smoothing (PAS) encompasses a set of methodologies for reducing uncertainty in phase-sensitive estimation and representation problems by leveraging noncausal (or multi-phase) smoothing operations. The core principle is aggregation—across time, phase offsets, or attention heads—to suppress high-frequency fluctuations and attain theoretically optimal or highly robust performance. In quantum metrology and adaptive phase estimation, PAS derives from optimal Wiener or Kalman smoothing, while in modern Video LLMs, it refers to phase-offset aggregation in temporal positional encoding to stabilize attention kernels. Contemporary research demonstrates that PAS yields unbounded or provably tight improvements in mean-squared error (MSE) and stability across these domains (Laverick et al., 2017, Roy et al., 2013, Sun et al., 14 Nov 2025).

1. Phase-Aggregated Smoothing in Quantum Phase Estimation

PAS in quantum settings addresses real-time estimation of a stochastic, time-varying phase $\phi(t)$ in coherent optical beams, with photon flux $\mathcal{N}$. The stochastic process is typically modeled as Gaussian with power-law spectrum $S_\phi(\omega) = \kappa^{p-1}/|\omega|^p$, $p>1$, where $\kappa$ is the fluctuation intensity and $p$ characterizes spectral roughness. The measurement process involves adaptive homodyne detection, which under linearized tracking approximations yields a linear Gaussian estimation problem (Laverick et al., 2017).

2. Quantum Cramér–Rao Bound and Wiener Smoothing

For time-varying phase estimation, the achievable accuracy is established through the Quantum Cramér–Rao Bound (QCRB), derived via Fisher information analysis for waveform estimation. The minimum attainable MSE for an unbiased estimator $\hat{\phi}(t)$ is:

$$\mathrm{MSE}_\mathrm{Q} = [p\sin(\pi/p)]^{-1} (4\mathcal{N}/\kappa)^{-(p-1)/p}$$

This QCRB can be exactly attained via noncausal Wiener smoothing—Phase Aggregated Smoothing—effectively combining all past and future information. In the frequency domain, the smoothed estimate leverages the optimal gain:

$$G_S(\omega) = \frac{S_\phi(\omega)}{S_\phi(\omega) + S_n}$$

where $S_n=1/(4\mathcal{N})$ is the measurement noise power (Laverick et al., 2017).

3. Comparison: Filtering vs. Smoothing and Unbounded Error Reduction

Standard filtering (e.g., optimal causal Kalman or Wiener filter) yields an MSE

$$\mathrm{MSE}_\mathrm{F} = [\sin(\pi/p)]^{-1}(4\mathcal{N}/\kappa)^{-(p-1)/p}$$

with identical scaling but a prefactor larger by $p$ compared to the QCRB. The ratio

$$\mathrm{MSE}_\mathrm{F} / \mathrm{MSE}_\mathrm{Q} = p$$

demonstrates that PAS achieves an unbounded improvement factor $p$ as $p\to\infty$, far exceeding the classical $2\times$ gain encountered with Brownian phase noise ($p=2$) (Laverick et al., 2017). This improvement persists except when measurement linearization fails at low photon flux. Monte Carlo simulations confirm convergence to these analytic limits under appropriate operating conditions (Laverick et al., 2017).

4. PAS in State-Space Filtering: Kalman and Rauch–Tung–Striebel Smoothers

In implementations involving Ornstein–Uhlenbeck phase noise, PAS is realized through a forward Kalman filter combined with a backward Rauch–Tung–Striebel (RTS) smoother. The steady-state error covariance expressions are:

• Filtering: $$P_f = \frac{-\lambda+\sqrt{\lambda^2+4\kappa|\alpha|^2}}{4|\alpha|^2}$$
• Smoothing: $$P_s = \frac{\kappa}{2\sqrt{\lambda^2+4\kappa|\alpha|^2}}$$ where $\lambda$ is the phase mean reversion rate, $\kappa$ the process noise, and $|\alpha|^2$ the photon flux (Roy et al., 2013). Smoothing yields a $\sim$3 dB reduction in MSE over filtering. Robust variants further reduce the worst-case MSE under parametric uncertainty using integral quadratic constraint (IQC) methods. However, the noncausal nature of smoothing necessitates access to future data, making PAS best suited for post-processing or moderate-latency applications (Roy et al., 2013).

5. PAS as a Temporal Stabilizer in Video LLMs

In Video LLMs, PAS refers to multi-phase aggregation applied to temporal rotary position encodings (RoPE). Video LLMs employing multimodal RoPE suffer from temporal inconsistency: small frame timing shifts introduce frame-level attention instabilities due to high-frequency ripples in the inverse Fourier kernel

$$m(\Delta t) = \frac{1}{m}\sum_{i=0}^{m-1}e^{j\omega_i \Delta t}$$

PAS mitigates this by partitioning attention heads and applying small, opposed phase offsets $\delta_h$ to the query embeddings. The aggregate over heads results in an effective kernel

$$m_\mathrm{eff}(\Delta t) = \sum_{h}a_h\,m(\Delta t+\delta_h)$$

which smooths temporal fluctuations and yields provable Lipschitz continuity of the attention logit in $\Delta t$. The method preserves per-head spectrum magnitude under Nyquist-valid sampling and adds negligible computational overhead. Under ablations, PAS stably improves mean diagnostic metrics by $1$–$2$ points and yields consistent gains in low frame rate and sub-Nyquist regimes (Sun et al., 14 Nov 2025).

6. Mathematical Mechanisms and Theoretical Guarantees

• In quantum estimation PAS recovers the QCRB by optimal fusion of all temporal data, with error scaling strictly determined by the properties of $S_\phi(\omega)$.
• In state-space settings, the combination of Kalman filter and RTS smoother achieves sub-Standard Quantum Limit accuracy, and robust design guarantees worst-case performance under modeling uncertainty.
• In Video LLMs, multi-phase head aggregation directly reduces the local slope $L_m$ of the temporal kernel, yielding tighter Lipschitz constants (Theorem 2 (Sun et al., 14 Nov 2025)); kernel variance and frequency-domain ripple attenuation are mathematically quantified (Theorems 3 and 4).

7. Practical Implications and Limitations

In quantum sensing, PAS offers a rigorous route to optimal phase tracking, especially for highly nonstationary or rough phase processes ($p>2$), as encountered in precision metrology. However, the requirement for future data and increased computational cost restrict real-time deployment. In Video LLMs, PAS provides a plug-and-play, training-free stabilizer for temporal encoding, with demonstrated empirical advantages and minimal marginal cost. The methodology is broadly applicable wherever phase-based temporal irregularity or noncausal smoothing is advantageous, subject to the requirements of the estimator and the nature of underlying process or attention kernel (Laverick et al., 2017, Roy et al., 2013, Sun et al., 14 Nov 2025).