Coherence & Robustness Enhancing (CRE)

• CRE is a framework that quantifies and preserves coherence using resource-theoretic monotones and optimization techniques across quantum, AI, and signal processing domains.
• It employs methodologies such as semidefinite programming, reinforcement learning, and physical adjustments to sustain coherence under noise, loss, or adversarial conditions.
• The approach enhances practical applications by enabling robust quantum communication, improved speech recognition, and resilient narrative generation in large language models.

Coherence and Robustness Enhancing (CRE) refers to a set of resource-theoretic, algorithmic, and experimental strategies designed to quantify, preserve, and optimize the coexistence of high coherence and maximal robustness under noise, loss, or adversarial conditions across quantum systems, signal processing pipelines, and advanced AI models. CRE protocols are built around precise monotones (such as the robustness of coherence and related resource measures), reinforcement-based or architecture-guided training, and physically grounded interventions to mitigate decoherence or narrative breakdown. CRE appears in diverse domains: continuous-variable quantum optics, quantum channel simulation, coherence-based speech enhancement, low-noise quantum hardware tuning, and logical structure induction in neural LLMs.

1. Formal Resource-Theoretic Foundations of Coherence and Robustness

The foundational structure of CRE is the quantification of coherence as a resource and its robustness against noise and other operations. In the finite-dimensional quantum setting, the central monotone is the robustness of coherence (RoC), defined as

$$R_C(\rho) := \min\{s \geq 0\,|\, \exists\, \tau\;\text{s.t.}\; ( \rho + s\,\tau )/(1+s) \in \mathcal{I} \}$$

where $\mathcal{I}$ is the set of incoherent (diagonal) states in the reference basis. RoC is a full monotone: it is non-negative, vanishes only for incoherent states, is convex, and monotonic under incoherence-preserving operations (Napoli et al., 2016, Piani et al., 2016). The operational meaning is explicit: $1+R_C(\rho)$ equals the maximal (multiplicative) advantage in phase discrimination tasks over all possible incoherent state strategies.

RoC is efficiently computable via semidefinite programming (SDP), and any Hermitian ‘coherence witness’ $W$ satisfying $\Delta(W)\geq0$ and $W\leq I$ provides lower (and in some cases, saturating) bounds via $R_C(\rho)\ge\max\{0,-\operatorname{Tr}(W\rho)\}$. For pure or X–type states, $R_C$ reduces exactly to the $\ell_1$ norm of coherence, and the extremal range is $0\le R_C(\rho)\le d-1$ for $d$-dimensional systems (Napoli et al., 2016, Piani et al., 2016). These properties make $R_C$ and its channel liftings the backbone of any CRE procedure in discrete quantum systems.

In continuous-variable (Gaussian) quantum systems, coherence is defined by the relative entropy between a Gaussian state and its associated incoherent (thermal) reference, computable via symplectic spectra and covariance matrices (Kang et al., 2021). For a Gaussian state $\rho(\bar{x}, V)$, the coherence measure is

$C_G(\rho) = S[\rho_{\text{th}}] - S[\rho]$

where $S[\cdot]$ denotes the von Neumann entropy and $\rho_{\text{th}}$ is the thermal reference matching the local marginals.

2. CRE in Quantum Dynamics: Robustness Under Thermal and Dephasing Channels

CRE strategies exploit the superior survival of quantum coherence relative to other resources (such as squeezing or entanglement) under noisy quantum channels. For Gaussian channels with transmissivity $\eta$ and classical noise $\delta$, the evolution is

$V \mapsto \eta V + (1-\eta)(\delta+1) I_2$

For a single-mode squeezed input, coherence does not vanish at any finite noise or loss,

$$C_G(\eta, \delta) > 0 \;\text{for any}\; \delta < \infty, \eta > 0,$$

while squeezing or EPR entanglement extinguish much more rapidly (Kang et al., 2021).

Experimentally, this robustness ensures observable coherence persists after entanglement has been destroyed in scenarios with extreme loss or thermal mixing. Such persistence suggests that error correction, QKD, metrology, and continuous-variable computing can be engineered to exploit local coherence, with architectures and operations that preserve off-diagonal structure even as traditional nonlocal quantum resources are lost.

Under dephasing noise, robustness is even more pronounced: in contextuality-based discrimination, arbitrary small coherence budget suffices to certify nonclassicality for properly chosen measurement bases. One can tolerate dephasing arbitrarily close to complete decoherence and still violate noncontextuality inequalities with vanishing coherence (Rossi et al., 2022). CRE design in such settings entails minimal coherent state encoding and measurement realignment, outperforming any protocol based on fragile entanglement or relying on depolarization-resistant nonclassicality.

3. Channel and Process-Level CRE: Minimal and Amortized Coherence Costs

As CRE is applied at the level of channels (CPTP maps), the robustness of coherence of a channel $R_C(\mathcal{E})$ provides a direct quantification of process coherence requirements. One has

$$1 + R_C(\mathcal{E}) = \min\{\lambda\mid \exists\,\mathcal{M}\in \mathrm{MIO} : \mathcal{E} \leq_{CP} \lambda\,\mathcal{M}\}$$

where $\mathrm{MIO}$ is the set of maximally incoherent operations (Díaz et al., 2018).

This yields (i) an exact formula for the minimal cosdit rank required for simulation, (ii) an additive log-robustness for amortized cost with coherence recycling, and (iii) additivity for zero-error, many-copy simulations: $C_{LR}(\mathcal{E}) = \log(1 + R_C(\mathcal{E}))$ These results govern both exact and approximate protocols, as well as hybrid simulations with arbitrary (non-maximal) resource states. A remarkable consequence is that in systems of dimension $d>2$, any pure state (however weakly coherent) can be used to implement some coherent unitary channel via transformation to a ‘flagpole’ state, indicating the universality of coherence in CRE scenarios (Díaz et al., 2018).

4. CRE Beyond Quantum: Coherence and Robustness in LLMs and Speech Processing

CRE methodologies generalize to classical signal and information processing, notably:

• In speech recognition, coherence-based spectral enhancement uses the coherent-to-diffuse power ratio (CDR) as an SNR surrogate. Applying a spatially-informed spectral postfilter derived from CDR estimations yields significant reductions in word error rates under realistic, reverberant, and noisy conditions. Both DOA-dependent and DOA-independent estimators are used, with relative WER improvements of up to 11% on challenge benchmarks. The pipeline comprises STFT analysis, beamforming, recursive coherence estimation, multi-channel CDR fusion, and adaptive Wiener filtering (Barfuss et al., 2016).
• In neural LLMs, architectural CRE is realized by embedding logical structure and resilience explicitly into the neural context window, using reinforcement learning with custom rewards for coherence, structural consistency, and robustness. Hierarchical encoders build sentence and paragraph embeddings, context alignment modules optimize long-range dependencies, and output regularization penalizes incoherent transitions. Empirical results show 10–15 point gains in coherence metrics, highly robust narrative generation even under adversarial noise (maintaining >88% logical coherence in stress tests), and reduced compute overhead due to architectural modularization (Irvin et al., 20 Jan 2025, Huang et al., 8 Dec 2025).
• In timeline-constrained narrative agents, CRE is implemented via dual-level story-time-gated retrieval grounded in a diegetic knowledge graph, systematically excluding future or out-of-domain knowledge during model decoding. This yields near-perfect scores in both timeline coherence (TT ≥ 87%) and robustness (RT ≥ 73.5%) in ablation studies versus flat RAG baselines (Huang et al., 8 Dec 2025).

5. Experimental Protocols and Empirical Results in CRE

Experimentally validated CRE approaches span physical, algorithmic, and hybrid domains, with comprehensive protocols:

Domain	CRE Protocol	Robustness Quantifier	Empirical Metric	Cited Results
Quantum optics	Gaussian coherence under loss/noise	$C_G(\rho)$ (rel. entropy)	Survives to $\delta=10$, $L=0.8$ (CG$>0$)	(Kang et al., 2021)
Quantum hardware	DC-field tuning of TLS	$T_1$ enhancement (qubit)	23% average $T_1$ gain, up to 85% cases	(Lisenfeld et al., 2022)
Speech processing	Postfilter via CDR	CDR estimate (spatial)	7–11% reduction in WER	(Barfuss et al., 2016)
LLMs/NLP	NCRF w/ RL + hierarchical attention	Coherence/structural reward	+10–15 pts coh. gain, 2–3% noise drop	(Irvin et al., 20 Jan 2025)
Narrative agents	Story-time-gated retrieval	Timeline/robustness scores	TT = 87–90%, RT = 60–73%	(Huang et al., 8 Dec 2025)

In superconducting qubits, DC-electric field biasing detunes problematic two-level system (TLS) defects, yielding persistent $T_1$ improvements across many frequencies, with practical in-situ and scalable implementation options (Lisenfeld et al., 2022). In continuous-variable optics, relative entropy measures confirm that coherence outlasts entanglement or squeezing by orders of magnitude under practical loss and noise, implying new fault-tolerance and error-mitigation architectures (Kang et al., 2021). Neural CRE protocols achieve state-of-the-art logical structure and resilience with only moderate overhead, and retrieval-constrained LLMs attain near-complete restriction of knowledge to allowed narrative timelines (Irvin et al., 20 Jan 2025, Huang et al., 8 Dec 2025).

6. Implications, Limitations, and Future Directions

CRE protocols fundamentally shift the paradigm in resource-oriented quantum and classical signal processing by privileging basis-dependent, operational monotones that can withstand realistic noise and loss profiles. This enables:

• Robust quantum communication and metrology strategies favoring coherence preservation over entanglement.
• Efficient simulation and synthesis of quantum channels using non-maximal, possibly recycled coherence resources.
• Classical-quantum analogues in speech and language, where coherence-inspired constraints and regularization impose global consistency, minimize error under adversarial perturbations, and are resource efficient.
• Structural and narrative agent design in AI, where hard architectural excitations (story-time gating) combine with soft learned penalties for long-term sustainability and robustness.

Notably, CRE effectiveness is noise-model dependent: dephasing preserves coherence-based nonclassicality far better than depolarization, and the benefit of CDR-based postfiltering may be diminished if prior stages already strongly suppress diffuse noise (Rossi et al., 2022, Barfuss et al., 2016).

Open avenues include hybridizing coherence monotones with other resource measures, dynamically optimizing encoding/decoding for CRE under online noise characterization, and generalizing CRE algorithms to non-canonical bases and high-dimensional systems. In multi-modal AI systems, architectural enforcement of coherence/robustness may extend to cross-domain embeddings, supporting further advancement in generative and interactive applications.