Coherence–Ergotropy Gap in Quantum Thermodynamics

• The coherence–ergotropy gap is a metric that quantifies the additional work extractable from off-diagonal elements in quantum states, serving as a key resource indicator.
• It distinguishes the coherent work extraction from incoherent population reshuffling, and is critical in protocols for quantum batteries and thermal machines.
• This concept underpins experimental designs by separating genuine quantum coherence contributions from classical effects, optimizing performance in various thermodynamic regimes.

The coherence–ergotropy gap is a central concept in quantum thermodynamics that quantifies the energetic resource associated with quantum coherence, specifically the extra work extractable from quantum states due to their off-diagonal elements in the energy eigenbasis. In composite systems and open quantum scenarios, this gap not only characterizes the fundamentally quantum component of work extraction but also signals limitations and design principles for quantum batteries, heat engines, and related devices (Çakmak, 2020, Li et al., 20 Jun 2025, Francica et al., 2020).

1. Formal Definition and Mathematical Structure

Given a quantum state $\rho$ and a Hamiltonian $H$ with spectral decompositions

$$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$

the ergotropy $W(\rho; H)$ is defined as the maximal work extractable by a cyclic unitary operation: $$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$ The minimization is achieved when $U$ permutes the eigenvalues of $\rho$ onto the lowest energy eigenstates, yielding the passive state $P_\rho$. The ergotropy admits a decomposition: $$W(\rho; H) = W_{\mathrm{incoh}}(\rho; H) + W_{\mathrm{coh}}(\rho; H)$$ where:

• $W_{\mathrm{incoh}}$ is the ergotropy of the dephased state $H$0
• $H$1 is the additional work due to coherence: $H$2

The coherence–ergotropy gap is defined as: $H$3 or equivalently,

$H$4

This gap quantifies the strictly quantum resource component, vanishing identically if $H$5 is diagonal in $H$6's eigenbasis (Li et al., 20 Jun 2025, Francica et al., 2020).

2. Operational Interpretation and Physical Regimes

The coherence–ergotropy gap has distinct operational significance in protocols for work extraction and energy storage:

• Purely population-based work ($H$7) can be extracted by population reshuffling via incoherent permutations in the energy basis.
• Coherence-driven work ($H$8) requires unitaries that exploit off-diagonal elements and thus access genuine quantum resources unavailable in classical or passive states (Çakmak, 2020, Niu et al., 2024).

In many-body or battery scenarios, such as in qubit arrays or spin chains collectively coupled to a thermal bath, the absence of population inversion (no negative temperature) means that all extractable work must come from coherence (Çakmak, 2020).

The gap depends crucially on the thermodynamic regime:

• High-temperature limit: For two-qubit collectively-dissipative systems, $H$9, where $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$0 is the $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$1-norm of coherence. Thus, in this special regime, the entire coherence is convertible into work (Çakmak, 2020).
• Finite and low temperature: The gap is strictly positive except at thermal or purely incoherent points, and grows as temperature decreases or system size increases (Çakmak, 2020).
• Open-system non-equilibrium dynamics: In driven-dissipative processes, noise can induce or preserve coherence, enhancing the gap compared to equilibrium settings but with a complex trade-off with efficiency (Sarmah et al., 2024, Hadipour et al., 2024).

3. Quantitative Evaluation and Scaling Behavior

Several explicit forms and scaling laws for the coherence–ergotropy gap have been established:

For Two-Qubit Collective Steady States

For the steady state

$$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$2

the $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$3-norm of coherence is

$$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$4

and the work/energy ratio in the high-temperature limit coincides with the coherence

$$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$5

indicating full convertibility of coherence into work at high temperature (Çakmak, 2020).

Scaling with System Size

For $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$6 qubits initialized in ground state, at high temperature,

$$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$7

so the gap $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$8 diverges exponentially with $$\rho = \sum_j r_j |r_j\rangle\langle r_j|\,, \quad H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|\,,\quad \epsilon_1 \leq \epsilon_2 \leq \ldots$$9. Most of the coherence is thus "locked" and thermodynamically inactive for large systems (Çakmak, 2020).

Summary Table: Gap Behavior

Regime / Scenario	Behavior of Gap $W(\rho; H)$0	Reference
High temperature, $W(\rho; H)$1	$W(\rho; H)$2, full coherence-to-work	(Çakmak, 2020)
$W(\rho; H)$3 (large system)	$W(\rho; H)$4 diverges, coherence "locked"	(Çakmak, 2020)
Low temperature	$W(\rho; H)$5, grows except at maximal $W(\rho; H)$6	(Çakmak, 2020)

4. Resource-Theoretic and Measurement-Theoretic Views

The resource-theoretic framework associates coherence with quantifiers such as the $W(\rho; H)$7-norm and the relative entropy of coherence,

$W(\rho; H)$8

where $W(\rho; H)$9 is the von Neumann entropy, and $$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$0 is quantum relative entropy (Francica et al., 2020, Niu et al., 2024). The gap is bounded as

$$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$1

for any Gibbs state $$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$2 (Francica et al., 2020).

Measurement-based definitions of ergotropy, e.g., in terms of observational ergotropy under POVMs, reveal that the coherence–ergotropy gap quantifies the quantum advantage achievable via coherent measurements or drivings compared to energy-diagonal (classical) procedures (Biswas, 26 Feb 2026).

5. Experimental Signatures and Protocols

Recent experimental reports have realized precise quantification of the coherence–ergotropy gap:

• In superconducting transmon qubits, the separation of ergotropy into incoherent and coherent components was achieved by explicit protocol design, confirming that higher state coherence leads to increased coherent ergotropy (Li et al., 20 Jun 2025).
• In single-spin (NV center) systems, ancilla-based measurements distinguished and directly measured both components, verifying the operational role of quantum coherence (Niu et al., 2024).

These studies demonstrated protocol dependence: for dephasing-dominated noise, store energy incoherently ($$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$3); for relaxation-dominated decoherence, store at maximal coherence ($$W(\rho; H) = \operatorname{Tr}[\rho H] - \min_{U} \operatorname{Tr}\left[U \rho U^\dagger H\right]$$4). Extractable work and protocol efficiency both critically depend on the balance between stored coherence and dephasing, as well as on technological gate costs (Li et al., 20 Jun 2025).

6. Thermodynamic Implications and Quantum Battery Design

The coherence–ergotropy gap implies a trade-off in engineering quantum batteries and thermal machines:

• To maximize usable work, one must operate in regimes where coherence is efficiently converted to ergotropy (high temperature and moderate system size) (Çakmak, 2020).
• In larger or low-temperature systems, most coherence becomes thermodynamically locked, leading to a large gap and severely reduced conversion efficiency.
• Charging protocols relying solely on collective dissipation (no population inversion, no external driving) can yield nonzero ergotropy purely by exploiting steady-state coherence, but closing the gap requires careful temperature and system-size control (Çakmak, 2020).
• In non-equilibrium environments or with noise-induced coherence, the coherence–ergotropy gap can be dynamically enhanced, but power and flux optimization occur at moderate values of the gap, not at its maximum (Sarmah et al., 2024, Hadipour et al., 2024).

7. Generalizations, Extensions, and Classical Analogues

The coherence–ergotropy gap has been extended to

• Multipartite systems, where local vs global coherence and ergotropy can differ, and entanglement/correlation resources further refine the decomposition (Niu et al., 2024, Wang et al., 26 Dec 2025).
• Classical systems: via the ergodic hypothesis, classical ergotropy can be split into incoherent and "coherent" parts, with the gap quantifying the extra work from phase-space shell nonuniformity, underlining that the resource interpretation of coherence in work extraction is not uniquely quantum (Campisi, 18 Aug 2025).

In summary, the coherence–ergotropy gap encapsulates the energetic role of quantum coherence beyond classical population reshuffling, establishes fundamental bounds on the quantum advantage in work extraction, and guides the optimal regime selection for quantum thermodynamic devices (Çakmak, 2020, Li et al., 20 Jun 2025, Francica et al., 2020, Biswas, 26 Feb 2026, Niu et al., 2024, Wang et al., 26 Dec 2025).