Relative Entropy of Coherence

• Relative entropy of coherence is a quantum measure quantifying the superposition content by comparing a quantum state with its decohered counterpart.
• It extends naturally to generalized frameworks like Rényi and sandwiched Rényi entropies, ensuring robust operational and mathematical properties.
• Applied in metrology and thermodynamics, it aids in coherence distillation, entanglement activation, and the analysis of quantum resource capabilities.

Relative entropy of coherence is a fundamental quantum resource quantifier at the intersection of quantum information theory, quantum resource theories, and quantum metrology. It provides an operationally meaningful, basis-dependent measure for the superposition content of a quantum state, under both finite- and infinite-dimensional settings. The measure extends naturally to generalized frameworks such as Rényi and sandwiched Rényi entropies and enjoys a suite of desirable mathematical and operational properties, positioning it as the standard figure of merit in a variety of quantum protocols and applications.

1. Formal Definitions

Let $\rho$ be a density operator on a finite- or infinite-dimensional Hilbert space, with a distinguished orthonormal reference (incoherent) basis $\{|j\rangle\}$. The set of incoherent states $\mathcal{I}$ consists of all density operators diagonal in this basis. The dephasing map is $$\Delta(\rho) = \sum_j |j\rangle \langle j| \rho |j\rangle \langle j|$$. The quantum relative entropy is $$S(\rho \| \sigma) = \operatorname{Tr}[\,\rho(\ln \rho - \ln \sigma)]$$. The relative entropy of coherence is

$$C_{\text{rel}}(\rho) = \min_{\sigma \in \mathcal{I}} S(\rho \| \sigma) = S(\Delta[\rho]) - S(\rho)$$

where $S(\rho) = -\operatorname{Tr}[\rho \ln \rho]$ is the von Neumann entropy. For pure states, $C_{\text{rel}}(|\psi\rangle) = -\sum_j p_j \ln p_j$, where $p_j = |\langle j|\psi\rangle|^2$ (Winter et al., 2015, Zhu et al., 2017).

Generalizing to Rényi and Tsallis entropies,

$$S^R_\alpha(\rho) = \frac{1}{1-\alpha} \log \operatorname{Tr} \rho^\alpha, \quad D^R_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1} \log \operatorname{Tr}(\rho^\alpha \sigma^{1-\alpha}),$$

the corresponding Rényi relative entropy of coherence is

$$C^R_\alpha(\rho) = \min_{\sigma\in \mathcal{I}} D^R_\alpha(\rho\|\sigma) = S^R_\alpha(\Delta_\alpha(\rho)) - S^R_\alpha(\rho)$$

with an appropriate nonlinear dephasing $\Delta_\alpha$ (Vershynina, 2022, Zhu et al., 2017). Sandwiched Rényi variants and related maximization/minimization constructions yield further parameterized families (Xu, 2018).

For general (not necessarily projective) measurements, let $E = \{E_j\}$ be a POVM on $H$. The POVM-based coherence is

$$C_r(\rho, E) = \min_{\sigma \in I_P} S(\rho\|\sigma) = \sum_j S(\sqrt{E_j}\rho \sqrt{E_j}) - S(\rho)$$

with $I_P$ the set of POVM-incoherent states (Xu et al., 2021).

2. Mathematical Properties

Relative entropy of coherence satisfies a suite of resource-theoretic axioms (Winter et al., 2015, Zhu et al., 2017, Xu, 2018):

• Faithfulness: $C_{\text{rel}}(\rho) \ge 0$, with equality iff $\rho \in \mathcal{I}$.
• Monotonicity: For any incoherent channel $\Lambda$, $$C_{\text{rel}}(\Lambda(\rho)) \le C_{\text{rel}}(\rho)$$.
• Convexity: $$C_{\text{rel}}(p\rho_1 + (1-p)\rho_2) \le pC_{\text{rel}}(\rho_1) + (1-p)C_{\text{rel}}(\rho_2)$$.
• Additivity: $$C_{\text{rel}}(\rho_1\otimes\rho_2) = C_{\text{rel}}(\rho_1) + C_{\text{rel}}(\rho_2)$$.

These properties extend to POVM-based coherence, Rényi and sandwiched-Rényi variants within specific parameter regimes and under appropriate classes of free operations (Zhu et al., 2017, Xu, 2018, Xu et al., 2021, Vershynina, 2022). For infinite-dimensional systems, well-definedness and finiteness are guaranteed under physically relevant constraints, such as finite mean energy or particle number (Zhang et al., 2015).

3. Operational Interpretations

The operational significance of relative entropy of coherence is manifest in several quantum information contexts:

• Coherence Distillation: For any state $\rho$, the maximal asymptotic rate $C_d(\rho)$ at which one can extract maximally coherent qubit states via incoherent operations equals $C_{\text{rel}}(\rho)$ (Winter et al., 2015, Char et al., 2021):

$C_d(\rho) = C_{\text{rel}}(\rho).$

• Entanglement Activation: If $\rho$ is allowed to interact with an incoherent ancilla, the maximum creation of relative-entropy entanglement is $C_{\text{rel}}(\rho)$ (Zhu et al., 2017).
• Bayesian Quantum Metrology: For a parameter-encoded ensemble $\{p_x, \rho_x\}$, the ensemble relative entropy of coherence quantifies the information lost to a measurement scheme and appears as the difference between the Holevo bound and accessible information, the so-called CXI equality:

$$C_M(\{p_x,\rho_x\}) = \chi(\{p_x,\rho_x\}) - I(\Phi;M)$$

where $M$ is the measurement, $\chi$ is the Holevo quantity, and $I$ is the extracted mutual information (Lecamwasam et al., 2024).

These interpretations extend to catalytic regimes (single-shot protocols using coherence catalysts) and collaborative tasks (e.g., assisted distillation under local operations and classical communication) (Char et al., 2021).

4. Relation to Other Quantifiers and Frameworks

Relative entropy of coherence is tightly related to other resource measures (Zhu et al., 2017, Xu, 2018):

• Logarithmic Robustness: The logarithmic robustness $C_\infty(\rho)$, the $\alpha\to\infty$ sandwiched Rényi case, upper-bounds $C_{\text{rel}}(\rho)$.
• Geometric Coherence: Geometric coherence, corresponding to $-\ln \max_{\sigma\in\mathcal{I}} F(\rho, \sigma)$, is realized as the $\alpha=1/2$ sandwiched-Rényi case.
• $l_1$-Norm of Coherence: $C_{l_1}(\rho) = \sum_{i\neq j} |\rho_{ij}|$ bounds and complements $C_{\text{rel}}(\rho)$.
• Entanglement Theory: For maximally correlated states, Rényi relative entropy of coherence coincides with the corresponding entanglement Rényi measure; thus, results on additivity, analytical formulas, and bounds translate between settings (Zhu et al., 2017).
• Quantum Addition Coherence: Reformulations based on quantum vs. classical mixing channels yield new monotones bounded by $C_{\text{rel}}(\rho)$ and lead to new uncertainty relations (Mukhopadhyay et al., 2018).

5. Extensions: Infinite-Dimensional Systems, Channels, and Generalized Operations

In bosonic and infinite-dimensional settings, provided the mean energy constraint $\operatorname{Tr}[\rho H] < \infty$ holds, $C_{\text{rel}}(\rho)$ remains well-defined, with coherent and squeezed-vacuum states serving as benchmarks (Zhang et al., 2015). In the multimode case, coherent resource increases strictly with the number of modes at fixed thermal-like energy.

The maximum relative entropy of coherence for quantum channels, $C_{\max}(\Phi)$, quantifies the channel’s convertibility to maximally coherent channels under various classes of superchannels (ISC, DISC, SISC) and exactly characterizes the discrimination advantage in sub-superchannel tasks. It is operationally equivalent to robustness and inherits faithfulness, monotonicity, strong monotonicity, and additivity (Jin et al., 2021).

For general measurements (POVMs), the resource theory and relative entropy of coherence extend through Naimark dilation, and the salient structural, monotonicity, and convexity properties are preserved (Xu et al., 2021, Lecamwasam et al., 2024).

6. Applications and Physical Significance

Relative entropy of coherence serves as the primary tool to dissect and quantify the role of superposition in:

• Thermodynamics: It quantifies the quantum (coherent) part of irreversible work in driven, non-equilibrium quantum systems, decomposing the excess work into coherent and incoherent contributions and fulfilling fluctuation theorems at trajectory level (Francica et al., 2017).
• Metrology: In Bayesian parameter estimation, the ensemble relative entropy of coherence quantifies precisely the gap between information accessible by a given measurement scheme and the optimal measurement, directly trading off metrological performance and potential quantum advantage (Lecamwasam et al., 2024).
• Quantum Control and Evolution: Quantifies cohering and decohering powers of quantum evolutions, forms a hierarchy of inequalities connecting coherence, discord, and entanglement, and directly bounds resource-generating capability of quantum channels (Xi et al., 2015).
• Direct Measurement: Enables practical protocols, bypassing full state tomography, by relating directly to measurable differences of Shannon entropies in prescribed experimental bases (Bernardo, 2018).

7. Generalizations, Limitations, and Current Research Directions

The parameterized Rényi and sandwiched-Rényi extensions enrich the resource-theoretic landscape, with Rényi relative entropy of coherence providing valid coherence monotones for $0<\alpha<1$, while Tsallis-based variants are monotones only under restrictive classes of operations (Vershynina, 2022, Xu, 2018). Linear rescalings are essentially the only allowed nontrivial functional compositions yielding valid measures in general dimensions (Xu, 2018).

A notable subtlety is that the operational content of Tsallis-based and some Rényi-based measures depends on compatibility with the full set of incoherent operations; e.g., Tsallis coherence fails strong monotonicity except under “α-GIO” operations commuting with nonlinear dephasing (Vershynina, 2022). Resource-theoretic generalizations, such as the extension to general POVMs and the embedding of coherence in broader frameworks (e.g., thermodynamic, metrological, and communication scenarios), remain active areas of research (Lecamwasam et al., 2024, Xu et al., 2021).