Coherent Ergotropy under Energy Constraints

• Energy-constrained coherent ergotropy is defined as the maximum work extractable from a quantum system via cyclic operations under strict energy limits and quantum coherence contributions.
• It demonstrates that, beyond population distributions, off-diagonal coherences in the energy basis play a crucial role in achieving nonzero work output in dissipative, collectively coupled systems.
• Scaling analyses reveal that while total coherence can grow exponentially with system size, the extractable work increases only linearly, posing design challenges for scalable quantum batteries.

Energy-constrained coherent ergotropy is a central concept in quantum thermodynamics, defined as the maximal work extractable from a quantum system by cyclic (unitary) operations, subject to explicit energy constraints and with particular attention to the contribution originating from quantum coherence. Its physical and operational structure departs substantially from classical notions of energy: not only is the extractable work limited by the population distribution over energy levels, but under many realistic constraints, the presence of coherence and correlations in non-equilibrium steady-states or during driven dynamics fundamentally influence both the theoretical bounds and practical protocols for work extraction.

1. Ergotropy, Coherence, and Steady-State Activation

Ergotropy $\mathcal{W}$ is formally the difference between the mean energy of a quantum state $\rho$ and that of its passive state $\rho_\text{p}$—the state obtained by rearranging populations in the energy basis in non-increasing order: $$\mathcal{W} = \mathrm{Tr}(\rho H) - \min_U \mathrm{Tr}(U \rho U^\dagger H) = \mathrm{Tr}[H(\rho - \rho_\text{p})],$$ where the minimization is over all unitaries $U$. In the spectral decomposition framework, $\rho = \sum_j r_j |r_j\rangle\langle r_j|$ and $$H = \sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|$$, the ergotropy is

$$\mathcal{W} = \sum_{j,i} r_j \epsilon_i \left(|\langle r_j | \epsilon_i \rangle|^2 - \delta_{ji}\right).$$

This formula shows that off-diagonal coherences $\langle r_j | \epsilon_i \rangle$ for $j \ne i$ can contribute to work extraction. In open quantum systems, especially with collective coupling of multiple qubits to a thermal bath, it is possible to reach steady states that possess significant coherence in the energy eigenbasis even if they are not population inverted. For a single qubit under such dissipative evolution, the state relaxes to a Gibbs state without coherence. With $N > 1$ qubits collectively coupled, steady-state coherences emerge, and nonzero ergotropy in such passive-population states is enabled exclusively by these coherences (Çakmak, 2020).

2. Analytical Results for Few-Qubit Steady States

For two qubits collectively coupled to a thermal bath, the steady state can be expressed as

$$\rho_\text{ss}(\beta, c) = (1-c)|\psi_-\rangle\langle\psi_-| + c Z_+^{-1}(\beta) \left[e^{-2\beta\omega}|\psi_1\rangle\langle\psi_1| + e^{-\beta\omega}|\psi_+\rangle\langle\psi_+| + |\psi_0\rangle\langle\psi_0|\right],$$

where

$$|\psi_0\rangle = |00\rangle, \quad |\psi_1\rangle = |11\rangle, \quad |\psi_+\rangle = \frac{|01\rangle+|10\rangle}{\sqrt{2}}, \quad |\psi_-\rangle = \frac{|01\rangle-|10\rangle}{\sqrt{2}},$$

and $c$ depends on the initial state. The $l_1$ norm of coherence in this state is $C_{l_1} = |c(Z/Z_+)-1|$; at high temperature ($\beta\to 0$), $Z_+ = Z = 4$ and the result simplifies to $C_{l_1}=|(4c/3)-1|$. The ergotropy similarly reduces to

$\mathcal{W} = \omega |(4c/3) - 1|,$

so that, strikingly,

$\frac{\mathcal{W}}{\omega} = C_{l_1}.$

Hence, in the high temperature limit, all the extractable work in the steady state is due to quantum coherence, as captured by the $l_1$ coherence norm (Çakmak, 2020).

3. Scaling with System Size and the Coherence–Ergotropy Gap

For $N$ collectively coupled qubits prepared in the ground state, the steady-state density matrix is block-diagonal, and the $l_1$ norm of coherence scales as

$C_{l_1} \approx \frac{2^N - N - 1}{N+1}$

in the high temperature regime, exhibiting exponential growth with system size $N$. However, the corresponding ergotropy only grows linearly,

$\mathcal{W} \approx \omega \frac{N(N-1)}{2(N+1)}.$

Thus, for large $N$, the overwhelming majority of steady-state coherence cannot be utilized for work extraction: the "coherence–ergotropy gap" increases, and only a sublinear fraction of the total coherence is usefully convertible into ergotropy. This exposes a fundamental scalability limitation for charging protocols based solely on dissipative, collectively coupled systems (Çakmak, 2020).

For random initial states, the mean coherence again grows rapidly with $N$, but the mean ergotropy displays a weaker, linear scaling; its distribution narrows with larger system sizes, establishing that simply increasing the system size does not guarantee proportional increases in extractable work.

4. Application to Quantum Batteries and Dissipative Charging

An immediate implication is for the design of quantum batteries. The steady states reached with collective dissipative charging possess no population inversion but display nonzero ergotropy that is entirely due to steady-state coherences induced by the bath. Importantly, after the system is decoupled from the bath, the stored work can be extracted by applying an appropriate global unitary. This protocol does not require the precise coherent control or isolation needed for charging via purely unitary means.

This mechanism provides a scalable, robust alternative to unitary battery charging—"charging" via engineered dissipative steady states—where the detailed balance is naturally satisfied, and only a global unitary operation is required for extraction (Çakmak, 2020). However, the trade-off between the exponential growth of bath-induced coherence and limited ergotropy scaling imposes an upper bound on performance as the number of battery cells increases.

5. Mathematical Formulation and Key Results

Table 1: Summary of Key Quantities in Steady-State Collective Bath Charging

Quantity	Expression	Scaling Regime / Comment
Ergotropy $\mathcal{W}$	$$\sum_{i,j} r_j \epsilon_i (|\langle r_j|\epsilon_i \rangle|^2 - \delta_{ji})$$	General formula
$l_1$ coherence $C_{l_1}$ (2 qubits)	$C_{l_1} = |(4c/3) - 1|$	High temperature limit
Ergotropy (2 qubits, high $T$)	$\mathcal{W} = \omega |(4c/3) - 1|$	$\mathcal{W}/\omega = C_{l_1}$
Coherence scaling (ground state, $N$)	$C_{l_1} \sim 2^N/(N+1)$	Coherence grows exponentially with $N$
Ergotropy scaling (ground state, $N$)	$\mathcal{W} \sim \omega N(N-1)/[2(N+1)]$	Ergotropy grows linearly with $N$

The above relationships emphasize that, specifically for dissipatively engineered steady states in weakly-coupled, collectively-coupled quantum systems, steady-state coherence can provide a precise quantifiable source of extractable work.

6. Broader Context and Relation to Quantum Thermodynamics

The finding that ergotropy in certain steady states can be entirely attributed to coherences even without population inversion is a sharp manifestation of the resource-theoretic perspective in quantum thermodynamics. The explicit separation into coherent and incoherent contributions, and the dimension-dependent ergotropy–coherence trade-off, have parallels and extensions in general formulations of work extraction, including resource theory analyses of coherence as a work resource (Francica et al., 2020), the tight second law in the presence of coherences (Łobejko, 2020), and protocols for fluctuating work and operational ergotropy in more general settings.

When designing quantum devices (e.g., batteries, heat engines, or energy transfer systems) in the energy-constrained regime, these results highlight that:

• Engineered steady-state coherences can be harnessed as usable work, under suitable protocols.
• There are fundamental limits—determined by system structure, coupling, and decoherence mechanisms—on how much of this coherence can be operationally converted into useful work.
• Scaling up the number of constituents in collective dissipative schemes requires balancing coherence growth and extractable ergotropy, as exponential increases in coherence do not translate into proportional increases in available work.

7. Implications, Limitations, and Future Directions

The identification and quantification of the role of coherence in steady-state ergotropy provide a framework for optimizing quantum batteries charged by dissipative baths. The findings imply that collective couplings (versus independent couplings) are necessary for steady-state coherence and thus ergotropy in open quantum systems (Çakmak, 2020). However, in the regime of large $N$, the gap between coherence and usable work grows, highlighting a subtlety in the design of scalable quantum energy storage systems.

Future research may aim to:

• Generalize these results to more complex Hamiltonian structures.
• Determine whether different coupling topologies, environmental engineering, or hybrid dissipative–unitary protocols can close the coherence–ergotropy gap.
• Extend the analysis to correlated batteries or those exploiting multipartite entanglement as a work resource.
• Incorporate operational constraints (e.g., limited control, restricted unitaries) in the design of practical work-extraction and charging protocols.

In summary, energy-constrained coherent ergotropy in open quantum systems demonstrates a clear, quantifiable link between steady-state quantum coherence and extractable work. In collective dissipative scenarios, coherence becomes not merely an indicator of system nonclassicality but a directly operational resource, subject to precise scaling limitations and offering new paradigms for quantum battery design.