Coherence Surplus

• Coherence surplus is defined as the excess coherence over expected thresholds, measured by the gap between the cost to form a state and its distillable coherence in quantum resource theories.
• In multipartite systems and quantum channels, surplus coherence captures the difference between global coherence and the sum of local coherences, reflecting irreversibility and simulation costs.
• Beyond quantum mechanics, coherence surplus informs applications in network synchrony, category theory, linguistics, and risk management by delineating operational constraints amid noise and disorder.

Coherence surplus refers to the existence, quantification, emergence, or utilization of “extra” coherence—beyond a minimal or expected baseline—across diverse scientific domains. This concept manifests distinctively in quantum resource theory, dynamical systems, network science, category theory, risk management, and natural language processing. Its technical characterization is highly context-dependent but universally involves the identification, distribution, limitations, or operational deployment of an excess of coherence relative to a canonically “free” or reference scenario.

1. Coherence Surplus in Quantum Resource Theories

In the resource theory of quantum coherence, surplus coherence is often operationally defined as the difference between the cost to create a state (coherence of formation) and the rate at which pure coherence can be distilled from it (distillable coherence) under incoherent operations. For a quantum state $\rho$, these quantities are given by

$$C_r(\rho) = S(\Delta(\rho)) - S(\rho)\,, \quad C_f(\rho) = \min_{\{p_i, |\psi_i\rangle\}} \sum_i p_i\, S(\Delta(|\psi_i\rangle\langle\psi_i|))\,,$$

where %%%%1%%%% is the decohered (diagonal) version of $\rho$ and $S(\cdot)$ is the von Neumann entropy. Surplus is then

$\mathrm{Surplus} = C_f(\rho) - C_r(\rho)\,.$

A nonzero surplus signals irreversibility: more coherence is required to prepare $\rho$ than can be distilled. This asymmetry is generic for mixed states and underpins the resource overheads in quantum protocols involving coherence consumption and extraction (Winter et al., 2015).

2. Coherence Surplus in Multipartite Quantum Systems

In multipartite settings, surplus can also refer to the amount by which global system coherence exceeds the sum of magnitudes attributable to its marginal reductions. This is captured via an “additivity” (or monogamy-type) relation: $$\delta_C(\rho) = C(\rho_{AB_1\ldots B_n}) - \sum_{k=1}^n C(\rho_{AB_k})\,.$$ For certain classes of states and with appropriate (unnormalized) coherence measures, $\delta_C \geq 0$, indicating surplus global coherence relative to the sum over marginals. However, the normalized versions can yield $\delta_C < 0$, reflecting subtleties in resource distribution and the importance of the coherence quantifier choice (Kumar, 2015).

The interplay between coherence and mixedness is encoded in basis-independent bounds: $$\frac{C_{l_1}^2(\rho)}{(d-1)^2} + M_l(\rho) \leq 1\,,$$ with $$M_l(\rho) = (d/(d-1)) [1 - \operatorname{Tr}(\rho^2)]$$ the normalized linear entropy (“mixedness”). The trade-off constrains surplus coherence in the presence of noise or disorder (Zhang et al., 23 May 2024).

3. Dynamical and Channel-Based Surplus

Quantum operations and channels have distinct resource profiles compared to states: the cost to simulate a channel (using coherence-resources under incoherent operations) can strictly exceed its capacity to generate coherence from incoherent inputs. For a channel $T$, let

• $C_\text{gen}(T)$: coherence generating capacity (asymptotic rate for creating pure coherence using $T$ and free operations),
• $C_\text{sim}(T)$: cost to simulate $T$ (rate of maximally coherent states required for faithful simulation).

Then $C_\text{sim}(T) \geq C_\text{gen}(T)$. The gap $C_\text{sim}(T) - C_\text{gen}(T)$ constitutes a channel's “coherence surplus” (or bound coherence), reflecting fundamental irreversibility and resource inaccessibility for maps that can be simulated only at a nonzero cost, despite generating zero distillable coherence (Dana et al., 2017).

In dynamical resource theories, a channel’s distance to the set of classical (fully dephasing) channels (e.g., via log-robustness or channel-divergence) similarly quantifies the “dynamical coherence surplus”—the extent to which coherence can be distributed or preserved beyond classical evolutions (Saxena et al., 2019).

4. Surplus Coherence and Operational/Measurement Contexts

The relative entropy of coherence not only quantifies resourcefulness in state manipulation but also has a direct operational meaning in the context of measurement precision. In Bayesian quantum metrology, the “CXI equality” rigorously relates the ensemble coherence to the gap between the optimal (Holevo) information $\chi$ and the mutual information $I$ extracted by a given measurement $M$: $$C_m(\mathcal{E}_\Phi) = \chi(\mathcal{E}_\Phi) - I(\Phi; M)\,.$$ Here $C_m$ is the ensemble coherence, so the surplus quantifies information “locked away” in superpositions not accessed by $M$. This surplus coherence thus directly measures the informational advantage attainable through collective measurements (Lecamwasam et al., 29 Jan 2024).

5. Surplus Structure, Coherence, and Category Theory

In gauge theory and the category-theoretic analysis of surplus structure, “coherence surplus” refers to the necessity of seemingly redundant data (e.g., gauge potentials and their automorphisms) for the ability to coherently glue together local information into globally nontrivial (topologically charged) solutions. The presence of “surplus” morphisms in a groupoid (e.g., $C_A$ for $U(1)$ connections) is mathematically redundant for local observables, but essential for representing the full spectrum of global gauge field models. This structural surplus ensures “coherence” by supporting the compatibility and rich assembly of local data—quantifying structural resources necessary for locality and global consistency (Nguyen et al., 2017).

6. Classical and Applied Notions of Surplus Coherence

In non-quantum contexts, coherence surplus describes measurable degrees of global or systemic coherence exceeding minimal thresholds. For instance, in computational linguistics, a “coherence surplus” describes texts or discourse that, by virtue of extra or especially strong cohesion, consistency, or relevance, achieve statistically higher evaluation scores in automatic coherence assessment tasks (Maimon et al., 2023). In insurance mathematics, surplus sharing schemes leverage coherent utility/risk measures so that any net positive outcome (insurance surplus) is distributed in proportion to fair capital allocation, with the “coherent” methodology ensuring fairness and structural rationality (Coculescu et al., 2018).

7. Surplus and Trade-offs: Environmental Noise and Resource Limitations

Environmental decoherence or noise imposes intrinsic, basis-independent constraints on the achievable quantum coherence. Quantitatively, one finds: $$\frac{d}{d-1} (C_{l_2}^{\text{max}}(\rho))^2 + M_l(\rho) = 1\,,$$ or, for the maximal relative entropy of coherence,

$C_r^{\text{max}}(\rho) + S(\rho) = \ln d\,.$

Such constraints generalize previous basis-dependent results and precisely delineate the limits to surplus coherence: as mixedness increases under noise, the surplus coherence diminishes, setting operational constraints for quantum technologies (Zhang et al., 23 May 2024, Kumar, 2015).

---

Summary Table: Key Manifestations of Coherence Surplus

Domain	Technical Manifestation	Reference
Quantum resource theory	Gap between formation cost and distillation rate (irreversibility)	(Winter et al., 2015)
Multipartite systems	Global–local coherence gap (additivity/monogamy relations)	(Kumar, 2015)
Quantum channels	Simulation cost minus coherence capacity (bound coherence)	(Dana et al., 2017)
Quantum metrology	Holevo information minus accessible mutual information (CXI equality)	(Lecamwasam et al., 29 Jan 2024)
Network science	Increased global synchrony from enhanced connectivity (proportional to links)	(Pereira et al., 2013)
Gauge theory	Category-theoretic necessity of redundant morphisms for global “coherence”	(Nguyen et al., 2017)
NLP coherence assessment	Exceeding threshold on cohesion, consistency, or relevance metrics	(Maimon et al., 2023)
Insurance/risk management	Premium surplus divided via coherent utility/risk measures	(Coculescu et al., 2018)

The unifying structural feature is that coherence surplus is rigorously delimited by operational constraints, trade-offs with disorder or noise, or categorical requirements for global compatibility. It can be computed or bounded by resource-theoretic quantities, dynamical divergence, or optimization problems, and has concrete implications in quantum information, complex networks, physical modeling, linguistics, and economics.