Coherence Measure: Quantum and Topic Perspectives

Updated 28 November 2025

Coherence measure is a quantitative metric that assesses the mutual consistency of system elements based on their information structure.
In quantum information theory, measures follow strict axioms like nonnegativity, monotonicity, and convexity to quantify superposition resources.
Different classes such as the l₁-norm for quantum states and UMass for topic models offer practical insights and trade-offs tailored to specific applications.

A coherence measure quantitatively assesses the degree of mutual consistency, relatedness, or “hanging together” of elements within a set, relative to the information structure in question. While the term is used in both quantum information theory (for quantum superposition resources) and in classical probabilistic or information-theoretic contexts (e.g., topic modeling, knowledge bases), the technical construction and interpretation of a coherence measure are highly domain specific. Below, two core research lines illustrate the landscape: (i) quantum coherence as a physical resource in quantum mechanics and (ii) topic coherence in computational linguistics.

1. Coherence Measures in Quantum Information Theory

In the resource theory of quantum coherence, a coherence measure is a scalar functional $C(\rho)$ assigning to every quantum state $\rho$ a non-negative number that quantifies “quantum superposition” with respect to a fixed basis. Critical formalizations are based on the framework of Baumgratz, Cramer, and Plenio (“BCP framework”), which requires any valid coherence measure to satisfy:

(C1) Nonnegativity and faithfulness: $C(\rho) \geq 0$ , and $C(\rho) = 0$ iff $\rho$ is incoherent (diagonal).
(C2) Monotonicity under incoherent operations: $C(\Lambda[\rho]) \leq C(\rho)$ for any incoherent CPTP map $\Lambda$ .
(C3) Strong monotonicity: $C$ does not increase on average under selective incoherent operations.
(C4) Convexity: $C(\sum_i p_i \rho_i) \leq \sum_i p_i C(\rho_i)$ .

Several mathematically and operationally distinct coherence measures have been introduced, distinguished by their structural properties and operational interpretations.

Major Classes of Quantum Coherence Measures

Measure Family	Defining Expression / Principle	Main Features / Context
$l_1$ -norm	$C_{l_1}(\rho) = \sum_{i \neq j} \| \rho_{ij}\|$	Simple, amplitude-sensitive, admits closed form for general $\rho$ ; basis dependent (Qi et al., 2016)
Relative entropy	$C_r(\rho) = S(\rho_{\rm diag}) - S(\rho)$	Distance to closest incoherent state; "gold-standard" for resource quantification (Yuan et al., 2015, Vershynina, 2021)
Convex-roof (intrinsic randomness, coherence concurrence, fidelity)	$C(\rho) = \min_{\{p_i, \psi_i\}} \sum_i p_i C(\psi_i)$ with $C(\psi_i)$ basis-dependent	Captures minimal resource cost, e.g., for coherence distillation; strong analogy to entanglement of formation (Yuan et al., 2015, Qi et al., 2016, Liu et al., 2017)
Sandwiched Rényi/Generalized entropic	$C(\rho) = \min_{\delta \in \mathcal{I}} D^S_\alpha(\rho \\| \delta)$	Tunable via parameter $\alpha$ , strong-data-processing properties; recovers $C_r$ for $\alpha \to 1$ (Xu, 2018)
Quasi-relative entropy	$C_f(\rho) = S_f(\Delta(\rho)) - S_f(\rho)$	Generalizes $C_r$ ; supports monotonicity under restricted classes of operations (Vershynina, 2021)
Logarithmic coherence number	$C_\ell(\rho) = \min_{\{p_i, \psi_i\}} \sum_i p_i \log_2 R_c(\psi_i)$	Discrete, rank-based, parallels entanglement Schmidt measure (Xi et al., 2018)
Fidelity/geometric	$C_F(\rho) = \min_{\delta\in \mathcal{I}} \sqrt{1-F(\rho,\delta)}$	Geometric: robustness against dephasing; operationally informative for single-qubit transformations (Liu et al., 2017)
Polynomial/G-concurrence	$C_p(\psi) = \|P_h(\psi)\|^m$ (homogeneous in amplitudes)	Only qubit case yields nondegenerate faithful measures (Zhou et al., 2017)
Tsallis operator or parameterized	Based on perspective-mapped Tsallis/α-Rényi functions, e.g. $C^{T}_{\alpha,\beta}$	Unified treatment, parameter tunability, systematic construction (Guo et al., 2023)
Epsilon-smooth	$C_\epsilon(\rho) = \inf_{\sigma: D(\sigma, \rho) \leq \epsilon} C(\sigma)$	"One-shot" scenario, robust to state-preparation uncertainty; links to coherence distillation (Xi et al., 2018)

Operational and Structural Consequences

Resource interpretation: Measures such as the relative entropy of coherence or the max-relative entropy of coherence directly quantify the operational advantage in subchannel discrimination, distillation cost, or conversion rates under free (incoherent) operations (Bu et al., 2017, Yuan et al., 2015, Qi et al., 2016).
Monotonicity under free operations: Much effort is dedicated to identifying under which classes of quantum operations (ICPTP, SIO, DIO, GIO, block-incoherent, etc.) a given coherence measure is (strongly) monotonic, with implications for universality and resource convertibility (Vershynina, 2021, Yadin et al., 2018).
Basis dependence and basis independence: Most measures are defined with respect to a fixed reference basis, but basis-independent alternatives (e.g., $C_{\rm IBIQC}(\rho) = \log d - S(\rho)$ ) have been constructed to quantify intrinsic, not reference-frame contingent, quantumness (Wang et al., 2017).
Composite, multipartite, and channel coherence: Measures have been extended to multipartite systems (distribution and monogamy relations), ensembles, and channels (dynamical coherence), using refined axiomatic generalizations to e.g., block or POVM incoherence, and superchannel monotonicity (Xu et al., 2020, Vershynina, 13 Jan 2024, Bu et al., 2017, Patra et al., 2021).
Ordering and comparison: For qubits, many coherence measures induce the same ordering; discrepancies appear in higher dimensions or for non-convex monotones (Guo et al., 2023).

2. Coherence Measures in Topic Modeling and Information Retrieval

In topic modeling and related natural language processing applications, a coherence measure provides a quantitative assessment of how interpretable or meaningfully related a set of words (topic) is, relative to observed data statistics.

Core Classes of Topic Coherence Measures

Family	Definition	Complexity	Reference
UMass	$C_{\rm UMass}(T) = \sum_{m=2}^M \sum_{l=1}^{m-1} \log \frac{D(w_m, w_l)+1}{D(w_l)}$	$O(M^2)$	(Rosner et al., 2014)
UCI/PMI	$C_{\rm UCI}(T) = \frac{2}{M(M-1)} \sum_{i<j} \log \frac{p(w_i,w_j)}{p(w_i)p(w_j)}$	$O(M^2)$	(Rosner et al., 2014)
NPMI	Like UCI, with normalization by $-\log p(w_i,w_j)$	$O(M^2)$	(Rosner et al., 2014)
One-all	Averages difference $p(\{w\}\mid W^) - p(\{w\})$ over $w\in W$ , $W^=W\setminus\{w\}$	$O(M)$	(Rosner et al., 2014)
One-any, any-any	Extends to arbitrary support between subsets, e.g., $\left(W',W^\right)$ pairs, averages $d(W',W^)$	$O(2^M)$ (exponential)	(Rosner et al., 2014)

Key variables: $D(w)$ is the document frequency, $D(w_i,w_j)$ the co-document frequency in corpus of size $D$ ; $T$ is the word set (topic); $p(w)$ empirical frequency.

Motivations and Empirical Findings

Pairwise measures (e.g., UMass, UCI) rely on word co-occurrence statistics, capturing association but sometimes missing higher-order topicality (e.g., two subtopics averaged together can appear pairwise-coherent).
Subset-based measures (one-all, one-any, any-any) from philosophical theories of confirmation impose stricter, more comprehensive checks that every subset is “supported” by the rest, thereby better aligning with human interpretability judgments (Rosner et al., 2014).
Experiments indicate one-any and any-any coherence measures achieve the best correlation with human ratings of topic interpretability, outperforming all pairwise methods especially on challenging, beam-search–generated sets (Rosner et al., 2014).

Practical Application

For topic evaluation or model selection, use one-any or any-any difference-coherence when feasible. For large-scale or time-critical applications, normalized PMI (NPMI) is an efficient and reasonable surrogate.
All measures rely on a background corpus; smoothing is used to avoid instabilities with rare events or zero co-documents. Careful normalization is essential.

3. Structural and Axiomatic Requirements

Quantum Case

A valid quantum coherence measure $C$ typically must satisfy:

Faithfulness (vanishes only on incoherent states)
Monotonicity and strong monotonicity under chosen free operations
Convexity (resource cannot be created by statistical mixing)

Depending on operational context, additional properties (additivity, computability, operational interpretability) may be imposed.

Topic Model Case

A valid topic coherence measure generally is:

Monotonic with respect to interpretable topics (higher score $\implies$ higher human interpretability)
Robust to rare/irrelevant co-occurrences
Scalable to large candidate sets

Subset-based measures further satisfy philosophical criteria such as global support (all subsets substantiate the whole).

4. Examples of Explicit Coherence Measures

Domain	Measure	Explicit Formula	Citation
Quantum	$l_1$ -norm	$C_{l_1}(\rho) = \sum_{i \neq j} \|\rho_{ij}\|$	(Qi et al., 2016)
Quantum	Relative entropy	$C_r(\rho) = S(\rho_{\textrm{diag}}) - S(\rho)$	(Yuan et al., 2015)
Quantum	Coherence concurrence	$C(\|\psi\rangle) = 2\sum_{j<k}\|\psi_j\psi_k\|$ for pure states, convex-roof for mixed	(Qi et al., 2016)
Topic model	UMass	$\sum_{m=2}^M \sum_{l=1}^{m-1} \log \frac{D(w_m, w_l)+1}{D(w_l)}$	(Rosner et al., 2014)
Topic model	One-any coherence	$\langle d(\{w\}, W^) \rangle_{w, W^}$	(Rosner et al., 2014)

5. Limitations and Open Problems

Quantum: Most coherence measures are basis dependent; basis-independent but resource-theoretically interpretable measures are more challenging (Wang et al., 2017). Many convex-roof and generalized entropy measures are computationally intractable for large $d$ . There is no universal total ordering on states induced by a coherence measure for $d>2$ (Guo et al., 2023, Xu, 2018).
Topic modeling: Subset-based measures are theoretically superior but exponentially hard for large sets; efficient proxies and truncations are an active area. Measures are sensitive to corpus statistics; domain selection and preprocessing (lemmatization, stopword filtering) impact robustness.

6. Extensions and Generalizations

Recent research extends coherence measures to subspaces (block coherence), general measurement contexts (POVMs), dynamical setting (channels), multipartite and ensemble settings, and parameterized entropic forms (e.g., sandwiched Rényi, Tsallis operator-entropy), opening new frontiers in resource quantification (Xu et al., 2020, Bu et al., 2017, Guo et al., 2023, Vershynina, 13 Jan 2024, Hazra et al., 11 Mar 2024).

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