---
title: Relative Entropy of Coherence
url: https://www.emergentmind.com/topics/relative-entropy-of-coherence
type: topic
---

# Relative Entropy of Coherence

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 $p_j = |\langle j|\psi\rangle|^2$ [1506.07975, 1706.00390].

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 $\Delta_\alpha$ [2208.06840, 1706.00390]. Sandwiched Rényi variants and related maximization/minimization constructions yield further parameterized families [1808.04662].

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 [2109.14323].

## 2. Mathematical Properties

Relative entropy of coherence satisfies a suite of resource-theoretic axioms [1506.07975, 1706.00390, 1808.04662]:

- **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 [1706.00390, 1808.04662, 2109.14323, 2208.06840]. For infinite-dimensional systems, well-definedness and finiteness are guaranteed under physically relevant constraints, such as finite mean energy or particle number [1505.05270].

## 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 $C_{\text{rel}}(\rho)$ [1506.07975, 2111.14645]:
  \[
  C_d(\rho) = C_{\text{rel}}(\rho).
  \]
- **Entanglement Activation:** If $\rho$ is allowed to interact with an incoherent ancilla, the maximum creation of relative-entropy entanglement is $C_{\text{rel}}(\rho)$ [1706.00390].
- **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 [2401.16020].

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) [2111.14645].

## 4. Relation to Other Quantifiers and Frameworks

Relative entropy of coherence is tightly related to other resource measures [1706.00390, 1808.04662]:

- **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 [1706.00390].
- **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 [1803.06982].

## 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 [1505.05270]. 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 [2106.16110].

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 [2109.14323, 2401.16020].

## 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 [1707.06950].
- **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 [2401.16020].
- **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 [1510.06473].
- **Direct Measurement:** Enables practical protocols, bypassing full state tomography, by relating directly to measurable differences of Shannon entropies in prescribed experimental bases [1812.04027].

## 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 [2208.06840, 1808.04662]. Linear rescalings are essentially the only allowed nontrivial functional compositions yielding valid measures in general dimensions [1808.04662].

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 [2208.06840]. 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 [2401.16020, 2109.14323].

Source: https://www.emergentmind.com/topics/relative-entropy-of-coherence