Coherence-Sensitive Quantum Measures

• Coherence-sensitive measures are quantification frameworks that capture the degree of quantum coherence—via off-diagonal structure and interdependence—using generalized entropies.
• Rényi-based measures provide robust, monotone quantifiers that align with operational protocols, ensuring reliable resource accounting in quantum systems.
• Tsallis-based measures offer analytical tractability but often fail to meet strong monotonicity under general operations, motivating further methodological refinements.

A coherence-sensitive measure is any quantification framework or metric that captures the degree of coherence—interpreted as superposition, off-diagonal structure, or set interdependence—in a mathematical or physical system, with sensitivity to the structural or resource-theoretic aspects of coherence. In quantum information and related disciplines, such measures extend standard coherence quantifiers by introducing sensitivity either to operational classes (e.g., allowed transformations), resource-type monotonicity, or alternative entropic or geometric interpretations. Recent research has explored generalizations beyond traditional von Neumann entropy and $l_1$-norm approaches, assessing whether alternative entropies such as Tsallis and Rényi generate faithful, operationally meaningful coherence measures. This entry focuses on the structure, advantages, limitations, and implications of these Tsallis and Rényi-based coherence-sensitive measures (Vershynina, 2022).

1. Generalized Entropy-Based Coherence Measures

Traditional quantification of coherence employs the relative entropy of coherence, given by the difference between the von Neumann entropy of the "dephased" (fully incoherent) state and the entropy of the original state: $C(\rho) = S(\Delta(\rho)) - S(\rho)$ where $S$ is the von Neumann entropy and $\Delta$ removes all off-diagonal elements in a fixed basis. Extensions replace the von Neumann entropy with generalized entropies, leading to coherence measures of the form: $$C^{(2)}_T(\rho) = S^T_\alpha(\Delta(\rho)) - S^T_\alpha(\rho)$$

$$C^{(2)}_R(\rho) = S^R_\alpha(\Delta(\rho)) - S^R_\alpha(\rho)$$

where $S^T_\alpha$ and $S^R_\alpha$ denote the Tsallis and Rényi entropies: $$S^T_\alpha(\rho) = \frac{\operatorname{Tr}[\rho^\alpha] - 1}{1 - \alpha}, \quad S^R_\alpha(\rho) = \frac{1}{1 - \alpha} \log \operatorname{Tr}[\rho^\alpha]$$ This formal structure aims to capture the entropy increment induced purely by "decohering" the state, thereby generalizing the sensitivity of the coherence measure to spectral distributions of $\rho$.

2. Closest-Incoherent-State Formalism

A crucial conceptual refinement involves defining coherence not with respect to the dephased state $\Delta(\rho)$ but rather the closest incoherent state $\Delta_\alpha(\rho)$ in the sense of Tsallis or Rényi relative entropy: $$S^T_\alpha(\rho\,||\,\delta) = \frac{1}{\alpha - 1} \left( \operatorname{Tr}[\rho^\alpha \delta^{1-\alpha}] - 1 \right)$$

$$S^R_\alpha(\rho\,||\,\delta) = \frac{1}{\alpha-1} \log \operatorname{Tr}[\rho^\alpha \delta^{1-\alpha}]$$

For both Tsallis and Rényi, the unique optimizer $\Delta_\alpha(\rho)$ has explicit form: $$\Delta_\alpha(\rho) = \left(\sum_j \langle j|\rho^\alpha|j\rangle^{1/\alpha} \right)^{-1} \sum_j |j\rangle\langle j| \langle j|\rho^\alpha|j\rangle^{1/\alpha}$$ Consequently, two measure families arise:

• Distance-based: $$C^{(1)}_R(\rho) = S^R_\alpha(\rho\,||\,\Delta_\alpha(\rho))$$ and analogously for Tsallis.
• Entropy difference: $$C_R(\rho) = S^R_\alpha(\Delta_\alpha(\rho)) - S^R_\alpha(\rho)$$. For Rényi, $C^{(1)}_R(\rho) = C_R(\rho)$.

The distinction between using the dephased state $\Delta(\rho)$ and the closest incoherent state $\Delta_\alpha(\rho)$ is critical for monotonicity and additivity properties.

3. Monotonicity, Continuity, and Additivity Properties

The operational merit of any coherence measure depends on monotonicity under relevant classes of quantum operations (e.g., Dephasing-Covariant Incoherent Operations, DIO; or Genuinely Incoherent Operations, GIO). The paper shows:

• Rényi coherence measures. $$C^{(1)}_R(\rho) = S^R_\alpha(\Delta_\alpha(\rho)) - S^R_\alpha(\rho)$$ are bona fide coherence monotones for all $\alpha$, satisfying strong monotonicity and asymptotic continuity. This ensures faithful resource accounting and operational soundness.
• Tsallis coherence measures. $C^{(1)}_T(\rho)$ does not, in general, generate a genuine coherence monotone unless operations are strictly limited (so-called “$\alpha$-GIO”—mixtures of diagonal unitaries and operations commuting with the $\alpha$-dependent dephasing). The entropy-difference Tsallis measure, $$C^T_\alpha(\rho) = S^T_\alpha(\Delta_\alpha(\rho)) - S^T_\alpha(\rho)$$, may actually increase under GIOs, violating strong monotonicity.

All these measures are shown to be asymptotically continuous. If two states have trace distance $\varepsilon$, the induced coherence measures $C_f$ associated with general $f$-entropy satisfy

$|C_f(\rho) - C_f(\sigma)| \leq 2 H(\varepsilon)$

where $H(\varepsilon) \to 0$ as $\varepsilon \to 0$.

Additivity for subspace-independent (block-diagonal) states holds for the “improved” Rényi-based and corresponding Tsallis-type measures constructed from the closest incoherent state, ensuring composability of the quantifier for multipartite and statistical mixtures.

4. Comparison of Tsallis and Rényi Coherences

The theoretical distinction between Rényi and Tsallis-based coherence measures is pronounced:

• Rényi entropy difference or distance-based versions: Provide robust, physically meaningful coherence quantification, with good monotonicity under both DIO and GIO.
• Tsallis entropy difference: Typically fails to provide a genuine coherence monotone except under extremely restrictive operations. This is due to the lack of joint convexity and other functional properties of the Tsallis relative entropy, which breaks monotonicity under general quantum operations.

In addition, the paper establishes an ordering relation between the entropy-difference and distance-based Tsallis coherence: $$C_T(\rho) \geq C^{(1)}_T(\rho) \;\text{for}\; 0 < \alpha < 1;\quad C_T(\rho) \leq C^{(1)}_T(\rho) \;\text{for}\; 1 < \alpha < 2$$ The choice of parameter $\alpha$ thus crucially affects the sensitivity and resource-theoretic acceptability of the resulting measure.

5. Continuity Estimates and Analytical Accessibility

The explicit structure of $\Delta_\alpha(\rho)$ yields tractable analytic expressions for both Rényi and Tsallis coherence measures. The continuity bounds, e.g., for $C^{(1)}_R$, state that for two pure states $\rho, \sigma$ with fidelity $F$ or trace distance $\varepsilon$, the difference in their Rényi coherence satisfies

$$|C^{(1)}_R(\rho) - C^{(1)}_R(\sigma)| \leq \frac{\alpha}{|1-\alpha|} \cdot \{\text{logarithmic terms in}\; \varepsilon, d, \alpha\}$$

For finite-dimensional systems, this ensures that small perturbations to the state cannot cause sharp jumps in the coherence quantifier.

6. Extensions and Future Directions

The limitations of Tsallis-based difference measures motivate additional definitions leveraging operator perspective techniques. For example, the unnormalized operator

$$\tilde{\Delta}_\alpha(\rho) = \sum_j |j\rangle \langle j| \rho^\alpha |j\rangle^{1/\alpha}$$

with new measures $$C^1_\alpha(\rho) = \operatorname{Tr}|\tilde{\Delta}_\alpha(\rho) - \rho|$$ and $$C^2_\alpha(\rho) = \operatorname{Tr}|[\tilde{\Delta}_\alpha(\rho)]^\alpha - \rho^\alpha|^{1/\alpha}$$, is shown to yield proper coherence quantifiers satisfying the additivity property for block-diagonal states.

Potential directions include:

• Operational interpretations of the new Rényi-type coherence measures in physical protocols (distillation, discrimination, metrology).
• Analysis of their performance under broader operational frameworks than DIO and GIO.
• Connections with coherence distillation rates especially in single-shot regimes or in resource interconversion settings.

7. Summary Table: Tsallis vs. Rényi Coherence Measures

Coherence Measure	Satisfies Monotonicity?	Additivity?	Closed-Form Closest Incoherent State?
Rényi (entropy difference/distance-based)	Yes (for any $\alpha$)	Yes	Yes
Tsallis (entropy difference)	No (except restrictive)	No (for general ops)	Yes
Tsallis (distance-based)	Partly, parameter-dependent	Parameter-dependent	Yes

Key abbreviations: DIO = Dephasing-Covariant Incoherent Operations; GIO = Genuinely Incoherent Operations.

8. Conclusion

Coherence-sensitive measures based on generalized entropies—particularly of the Rényi type—extend the traditional framework of quantum resource theory with analytically tractable, operationally meaningful quantifiers that preserve monotonicity and composability. While Tsallis-based measures fail to universally meet resource-theoretic demands, the development of improved and perspective-operator-based definitions recovers the essential sensitivity requirements. These findings offer a route to refined benchmarks for coherence as a quantum resource, and suggest future research into tailored quantification for diverse quantum information protocols (Vershynina, 2022).