Coherence-Constrained Maximal Work (CCMW)

• CCMW is a framework in quantum thermodynamics that defines maximal work extraction from quantum states under fixed coherence limitations and basis-dependent operational constraints.
• It synthesizes various formulations, from observational ergotropy to single-shot work extraction, to highlight how measurement restrictions and coherence structure affect work output.
• The framework provides practical insights for optimizing quantum batteries and energy protocols by revealing coherence’s dual role as a resource and a limitation.

Searching arXiv for the cited papers to ground the article in current literature. arXiv Search Query: id:(Biswas, 26 Feb 2026) OR id:(Mondal et al., 22 Jul 2025) OR id:(Francica et al., 2020) OR id:(Rodrigues et al., 2023) OR id:(Kwon et al., 2017) OR id:(Korzekwa et al., 2015) OR id:(Shi et al., 2022) OR id:(Chen et al., 4 Aug 2025) OR id:(Mingo et al., 2018) OR id:(Díaz et al., 2020) OR id:(Nayak et al., 30 Jan 2026) OR id:(Ali et al., 2 Jul 2026) OR id:(Tirone et al., 2023) OR id:(Xu et al., 2020) Coherence-Constrained Maximal Work (CCMW) denotes a family of constrained work-extraction notions in quantum thermodynamics in which the attainable work is optimized under explicit restrictions involving quantum coherence. In the most explicit current formulation, CCMW is defined as the highest amount of work extractable via coherence-preserving unitaries, optimized over all quantum states with fixed coherence in a given dimension (Mondal et al., 22 Jul 2025). Closely related frameworks treat measurement-restricted observational ergotropy in closed systems (Biswas, 26 Feb 2026), coherence-corrected maximum-work theorems beyond linear response (Rodrigues et al., 2023), deterministic single-shot work from internal coherence (Kwon et al., 2017), and resource-accounted extraction of the coherent free-energy contribution (Korzekwa et al., 2015). Across these formulations, the central question is not merely whether a state contains coherence, but which coherence is thermodynamically relevant, in which basis it is defined, and under which operational constraints it can be converted into work.

1. Definitions and conceptual scope

The term “coherence-constrained maximal work” is used explicitly in “Isocoherent Work Extraction from Quantum Batteries: Basis-Dependent Response” (Mondal et al., 22 Jul 2025). There, the constraint is twofold: the initial state must have a fixed amount of $l_1$-coherence in a chosen basis, and the allowed unitaries must preserve that coherence. In other strands of the literature, the same conceptual problem appears under different names. “Observational ergotropy” studies how much work remains accessible when the state is known only through a restricted measurement (Biswas, 26 Feb 2026). “Coherent work” and “work from coherence” treat deterministic coherent energy transfer and the coherent contribution to extractable work (Mingo et al., 2018, Francica et al., 2020). Free-energy-based approaches replace the usual equilibrium maximum-work statement by coherence- and athermality-corrected bounds (Rodrigues et al., 2023).

These formulations are not interchangeable. Some optimize over states with fixed coherence; some fix the state and vary the admissible measurements; some quantify average unitary work, others free-energy-limited work with bath contact, and others deterministic single-shot work. What unifies them is the idea that coherence is not thermodynamically free, and that the maximal work must therefore be indexed by the coherence constraint itself rather than by energy alone.

Formulation	Operational setting	Defining quantity
Isocoherent CCMW	Fixed coherence, coherence-preserving unitaries	$\xi_d(\mathcal C)$
Observational CCMW-style restriction	Closed system, limited measurement access	$R(\rho,\mathbf M)$
Coherent contribution to ergotropy	Cyclic unitary control	$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$
Nonequilibrium coherent maximum work	Closed or open driven dynamics	$W_{\max}^{\rm ext}=-\Delta\mathcal F$
Single-shot deterministic coherent work	Thermal processes, preserved energy statistics	$W_{\rm coh}$

A plausible implication is that CCMW is best understood as a class of constrained optimization problems rather than a single scalar functional. The precise constraint may be isocoherence, energy-basis incoherence of allowed measurements, restriction to internal coherence, limited reference-frame quality, or exposure to dephasing.

2. Closed-system formulations: ergotropy, observational access, and measurement restrictions

In the closed-system setting of “Information and coherence as resources for work extraction from unknown quantum state and providing quantum advantages,” work is extracted from a finite-dimensional system in state $\rho$ and Hamiltonian $H$ by cyclic time-dependent driving $H(t)$ with $H(0)=H(\tau)\equiv H$, so that the induced unitary is

$\xi_d(\mathcal C)$0

With complete state knowledge, the maximal extractable work is the ergotropy

$\xi_d(\mathcal C)$1

where $\xi_d(\mathcal C)$2 is the passive state associated with $\xi_d(\mathcal C)$3 (Biswas, 26 Feb 2026). This is the unconstrained benchmark.

The distinctive CCMW issue enters when the agent lacks full knowledge of $\xi_d(\mathcal C)$4. The relevant quantity then becomes the measurement-dependent observational ergotropy

$\xi_d(\mathcal C)$5

where $\xi_d(\mathcal C)$6 is the passive state of the coarse-grained state inferred from measurement statistics alone (Biswas, 26 Feb 2026). The framework establishes three structural results. First, observational ergotropy is monotone under classical post-processing: $\xi_d(\mathcal C)$7 Second, if the admissible measurements are energy-incoherent POVMs, then

$\xi_d(\mathcal C)$8

Third, optimizing over all measurements restores the full ergotropy: $\xi_d(\mathcal C)$9

These statements provide an operational interpolation between incoherent and fully coherent work extraction. Fine-grained, coherence-sensitive measurements preserve enough information to recover standard ergotropy; energy-diagonal measurements cannot access the coherent part of the work resource. A natural CCMW-style definition suggested by this framework is therefore

$R(\rho,\mathbf M)$0

with $R(\rho,\mathbf M)$1 the restricted measurement class. This is an inference from the structure of the results rather than the paper’s explicit terminology, but it captures exactly the operational role of limited coherence access (Biswas, 26 Feb 2026).

The same closed-system perspective appears in “Quantum Coherence and Ergotropy,” which decomposes ergotropy into incoherent and coherent parts,

$R(\rho,\mathbf M)$2

with

$R(\rho,\mathbf M)$3

Here $R(\rho,\mathbf M)$4 is the dephased state in the energy eigenbasis (Francica et al., 2020). This decomposition isolates the additional maximal work attributable specifically to energy-basis coherence and is one of the cleanest closed-system precursors of CCMW.

3. Basis dependence and the explicit isocoherent definition

The explicit CCMW definition given in (Mondal et al., 22 Jul 2025) fixes a coherence basis and an $R(\rho,\mathbf M)$5-coherence value $R(\rho,\mathbf M)$6, then optimizes the extractable work over all $R(\rho,\mathbf M)$7-dimensional states with that coherence and over unitaries that preserve it. The central message is that the dependence of maximal extractable work on coherence is not universal: it depends on how the Hamiltonian looks in the chosen coherence basis.

For qubits, if the Hamiltonian in the coherence basis is

$R(\rho,\mathbf M)$8

the CCMW is

$R(\rho,\mathbf M)$9

(Mondal et al., 22 Jul 2025). Two limiting cases are decisive. If the coherence basis is the energy eigenbasis, then $$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$0 and

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$1

so CCMW decreases monotonically with coherence. If instead the Hamiltonian has equal or zero diagonal entries and nonzero off-diagonal entries in the coherence basis, then

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$2

so CCMW increases linearly with coherence (Mondal et al., 22 Jul 2025).

This basis dependence resolves a common misconception. There is no basis-independent rule that more coherence implies more extractable work. Coherence helps when the Hamiltonian itself is off-diagonal in the chosen basis, because the constrained optimization can exploit the off-diagonal energetic structure. The same coherence can hinder work when it restricts the allowed population imbalance in an energy-diagonal Hamiltonian.

Higher-dimensional results reinforce this point. For $$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$3, numerical optimization shows that when coherence is fixed in the energy eigenbasis and the Hamiltonian is diagonal, the maximizing states are pure and the CCMW decreases with coherence, vanishing at maximal coherence (Mondal et al., 22 Jul 2025). By contrast, for Hamiltonians with only off-diagonal entries in the coherence basis, the pure-state-restricted CCMW generally grows with coherence, with only a small downturn near maximal coherence for $$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$4 (Mondal et al., 22 Jul 2025).

The battery literature sharpens the distinction between local and global accessibility. “Entanglement, Coherence, and Extractable Work in Quantum Batteries” shows that the coherent contribution to free-energy extractable work is exactly

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$5

while coherence and entanglement inhibit the incoherent component through the diagonal entropy $$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$6 (Shi et al., 2022). “Locally Passive, Globally Charged Quantum Batteries” goes further: in its controlled-shift model, maximal charger coherence can make the battery locally passive while leaving the full charge globally extractable, with a qubit complementarity

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$7

and equality in the active regime (Ali et al., 2 Jul 2026). This suggests that CCMW is not only basis dependent but also access-structure dependent: local CCMW and global CCMW need not coincide.

4. Nonequilibrium and open-system generalizations

A broader thermodynamic version appears in “Nonequilibrium thermodynamics of quantum coherence beyond linear response,” which treats both closed and open driven systems and replaces the standard equilibrium maximum-work theorem by a coherence- and athermality-corrected relation (Rodrigues et al., 2023). Coherence is measured in the instantaneous energy eigenbasis by

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$8

and the diagonal population lag by

$$\mathcal E_c=\mathcal E(\rho)-\mathcal E(\Delta[\rho])$$9

The generalized fluctuation relation is

$W_{\max}^{\rm ext}=-\Delta\mathcal F$0

and Jensen’s inequality yields

$W_{\max}^{\rm ext}=-\Delta\mathcal F$1

Equivalently, the maximum extractable work is

$W_{\max}^{\rm ext}=-\Delta\mathcal F$2

with generalized free energy

$W_{\max}^{\rm ext}=-\Delta\mathcal F$3

(Rodrigues et al., 2023).

In this formulation, coherence is beneficial only when it is consumed in a way that overcomes any accompanying athermality cost. The necessary condition for extracting more work than the standard equilibrium amount is

$W_{\max}^{\rm ext}=-\Delta\mathcal F$4

The paper is explicit that coherence can also be detrimental: coherence generated during driving from an initially thermal state corresponds to quantum friction and raises the required work input rather than lowering it (Rodrigues et al., 2023).

The dynamical criteria are likewise restrictive. In the closed/unitary case, beneficial coherence requires nonadiabatic driving with protocol duration satisfying

$W_{\max}^{\rm ext}=-\Delta\mathcal F$5

In open systems, coherence-to-work conversion is effective only when extraction is faster than decoherence: $W_{\max}^{\rm ext}=-\Delta\mathcal F$6 (Rodrigues et al., 2023). This suggests that CCMW is intrinsically finite-time and far-from-equilibrium in the regime where coherence improves the maximum-work budget.

A concrete realization is given by “Work extraction from long-lived quantum coherence of a three-level system,” which studies a degenerate or nearly degenerate $W_{\max}^{\rm ext}=-\Delta\mathcal F$7-type atom coupled to a thermal bath (Chen et al., 4 Aug 2025). There, bath-induced excited-state coherence persists for degenerate excited states with aligned transition dipoles, and an optimized protocol converts that coherence into population asymmetry using energy-preserving unitaries, then extracts work through controlled level shifts and thermalization. The paper’s single-cycle quasistatic protocol saturates the free-energy benchmark

$W_{\max}^{\rm ext}=-\Delta\mathcal F$8

thereby showing that, under its symmetry and control assumptions, coherence can be fully converted into free-energy-limited work (Chen et al., 4 Aug 2025).

5. Single-shot theory, work locking, and the clock–work trade-off

Single-shot and resource-theoretic approaches impose a sharper distinction between kinds of coherence. In “Clock-work trade-off relation for coherence in quantum thermodynamics,” coherence splits into internal coherence, which lies within a fixed total-energy eigenspace and can contribute to deterministic work extraction, and external coherence, which connects different total energies and instead quantifies clock usefulness (Kwon et al., 2017). The deterministic coherent work content is

$W_{\max}^{\rm ext}=-\Delta\mathcal F$9

Only $W_{\rm coh}$0, which preserves internal coherence while removing external coherence, enters this quantity (Kwon et al., 2017).

This result directly constrains CCMW. A state may have substantial total coherence and yet zero deterministic work value if that coherence is entirely external, or if the energetically dominant sector lacks the relevant internal structure. The same paper proves a clock/work trade-off: increasing quantum Fisher information as a clock resource reduces the maximum deterministic work obtainable from coherence. For $W_{\rm coh}$1 two-level systems,

$W_{\rm coh}$2

and for arbitrary local dimensions,

$W_{\rm coh}$3

(Kwon et al., 2017). This is a genuine coherence trade-off within maximal-work theory: external coherence does not add deterministic work directly, but it constrains it indirectly through asymmetry.

“Extraction of work from quantum coherence” establishes the complementary resource-accounted picture (Korzekwa et al., 2015). Under strict thermal operations without an external coherence resource,

$W_{\rm coh}$4

and similarly in the single-shot regime,

$W_{\rm coh}$5

This is work locking: the coherent contribution to the nonequilibrium free energy,

$W_{\rm coh}$6

is present formally but inaccessible operationally without an additional coherent reference (Korzekwa et al., 2015).

The same work also shows that bounded repeatable thermal machines can extract work from coherence arbitrarily well, approaching the ideal coherent contribution $W_{\rm coh}$7, but finite resources never extract all of it exactly (Korzekwa et al., 2015). A plausible implication for CCMW is that the maximal work is a function not only of the state and Hamiltonian, but also of the coherence-processing power of the machine. In that sense, CCMW is machine-relative.

“Decomposable coherence and quantum fluctuation relations” adds another deterministic layer by defining coherent work processes

$W_{\rm coh}$8

and proving that nontrivial coherent work extraction exists iff the energy-measurement random variable of the initial pure state is decomposable (Mingo et al., 2018). Under such a process, the effective potential satisfies

$W_{\rm coh}$9

This does not define CCMW explicitly, but it supplies a natural coherence-aware work primitive beyond average energy: the coherent work output state $\rho$0 and its effective potential $\rho$1 (Mingo et al., 2018).

6. Measurement dependence, applications, and unresolved issues

Measurement protocol matters because coherence-sensitive and coherence-erasing work definitions need not agree. “Quantum work statistics with initial coherence” compares the two-point measurement (TPM) scheme with the Margenau–Hill (MH) scheme and shows that TPM depends only on the dephased initial state $\rho$2, whereas MH retains initial energy-basis coherence (Díaz et al., 2020). For cyclic processes,

$\rho$3

with exact saturation for qubits after optimization over $\rho$4 (Díaz et al., 2020). The same paper shows that average entropy production can become negative in the MH framework. This does not furnish a CCMW formula, but it demonstrates that coherence can alter inferred work statistics and even the apparent thermodynamic irreversibility, depending on the operational work definition.

“A single measurement scheme for quantum work statistics based on coherent or squeezing state” reaches a similar conclusion through an explicit detector model (Xu et al., 2020). There, coherent contributions to the measured work distribution are exponentially suppressed by detector resolution through factors of the form

$\rho$5

and the scheme reduces to TPM in the sharp-measurement limit (Xu et al., 2020). This suggests that any measurement-based CCMW must be indexed not only by state coherence but also by measurement visibility.

In open driven systems, the role of coherence can shift from average work to work precision. “Signatures of coherent initial ensembles on all work moments” finds that, for a dissipative qubit with coherence-less driving, initial ensemble coherence does not change the mean work but does change higher moments, and for monotonic driving the work fluctuations are maximal for coherence-less initial ensembles (Nayak et al., 30 Jan 2026). The generalized Jarzynski relation becomes

$\rho$6

leading to

$\rho$7

This indicates that in some models coherence is a resource for thermodynamic precision rather than for larger average work (Nayak et al., 30 Jan 2026).

Quantum-battery applications sharpen the operational distinctions. “Entanglement, Coherence, and Extractable Work in Quantum Batteries” shows that either battery coherence or battery–charger entanglement is necessary during charging to generate nonzero extractable work, that coherence promotes the coherent component of final work, and that both coherence and entanglement inhibit the incoherent component through the diagonal entropy (Shi et al., 2022). “Quantum work extraction efficiency for noisy quantum batteries: the role of coherence” shows that in noisy multi-cell batteries, coherent input states can improve the asymptotic work/energy ratio, while sufficiently strong dephasing erases that advantage (Tirone et al., 2023). “Locally Passive, Globally Charged Quantum Batteries” adds that coherence can control not only how much work is stored but where it resides: in its model, maximal coherence can lock the entire charge into correlations so that the battery is locally passive even though the total state remains globally active (Ali et al., 2 Jul 2026).

Several misconceptions are therefore excluded by the present literature. More coherence does not universally imply more work. Total coherence is not the correct resource notion when internal and external coherence behave differently. The same coherence can increase global work, decrease local work, or leave the mean unchanged while reducing fluctuations. Measurement restrictions, basis choice, reference-frame quality, and decoherence timescales are not secondary details but constitutive parts of the maximal-work problem itself.

A plausible synthesis is that CCMW should be treated as a structured family of quantities. In closed, fully controlled settings it interpolates between incoherent ergotropy and full ergotropy through measurement or coherence-preservation constraints (Biswas, 26 Feb 2026, Francica et al., 2020). In isocoherent battery problems it becomes an explicitly basis-dependent constrained optimum $\rho$8 (Mondal et al., 22 Jul 2025). In nonequilibrium thermodynamics it is governed by generalized free energies $\rho$9 rather than by $H$0 alone (Rodrigues et al., 2023). In single-shot thermodynamics it depends only on internal coherence and is limited by work locking and clock resources (Kwon et al., 2017, Korzekwa et al., 2015). The topic remains unified by a single principle: maximal work from quantum coherence is not an intrinsic function of coherence alone, but an operational quantity defined by which coherence is preserved, accessed, converted, or forbidden.