Minimum Ergotropy and Anti-Ergotropy Limits

• Minimum ergotropy and anti-ergotropy define the minimal extractable work and chargeable energy from a quantum state with fixed mean energy by optimizing over passive and antipassive states.
• The analysis utilizes linear programming and state diagonalization to efficiently determine these bounds, ensuring guaranteed performance even with incomplete state information.
• Practical protocols, such as population reversal and random-unitary strategies, demonstrate how to saturate these bounds, setting universal benchmarks for quantum thermal machines.

Minimum ergotropy and anti-ergotropy describe, within quantum thermodynamics, the fundamental limits of work extraction and charging capability for quantum states—particularly under scenarios with only limited information about the state. These notions are essential for quantifying guaranteed performance in realistic protocols, where full state tomography is infeasible and controls are restricted to fixed, universal operations. The recent work (Canzio et al., 10 Sep 2025) gives a complete analytic and algorithmic characterization of the minimal extractable work (minimum ergotropy) and minimal energy required to maximally charge (anti-ergotropy) quantum states of known average energy, and presents practical strategies for saturating these bounds in a range of relevant systems. This formalism extends the foundational ideas of passivity and antipassivity, showing how knowledge constraints determine the accessible thermodynamic resources of an unknown quantum system.

1. Formal Definitions: Passivity, Ergotropy, and Anti-Ergotropy

Let $H$ be a finite-dimensional Hamiltonian with eigenvalues $\{\epsilon_i\}$ (ordered non-decreasingly) and eigenstates $\{|e_i\rangle\}$. For a density matrix $\rho$ with eigenvalues $\{\lambda_i\}$, define:

• Passive state $$\rho^\downarrow = \sum_i \lambda_i |e_{d+1-i}\rangle\langle e_{d+1-i}|$$ (eigenvalues $\lambda_i$ permuted so largest populate lowest energy).
• Antipassive state $$\rho^\uparrow = \sum_i \lambda_i |e_i\rangle\langle e_i|$$ (smallest eigenvalues to lowest energy).
• Ergotropy: $$\mathcal{E}(\rho) = \operatorname{Tr}[\rho H] - \operatorname{Tr}[\rho^\downarrow H]$$, the maximum extractable work via unitaries.
• Anti-ergotropy: $$\mathcal{A}(\rho) = \operatorname{Tr}[\rho^\uparrow H] - \operatorname{Tr}[\rho H]$$, the maximum chargeable energy via unitaries.

These quantities measure, respectively, the furthest possible decrease and increase in energy accessible by unitary maps, given the spectrum of $\rho$.

2. Minimum Ergotropy and Anti-Ergotropy with Fixed Mean Energy

Suppose only the average energy $E = \operatorname{Tr}[\rho H]$ of an initial state is known, but not its full density matrix. The set of density matrices compatible with this information is $$\sigma_E = \{\rho : \operatorname{Tr}[\rho H] = E\}$$.

Define the minimal and maximal possible ergotropy and anti-ergotropy over this set:

$$\begin{aligned} \mathcal{E}_{\min}(E) &= \min_{\rho\in\sigma_E} \mathcal{E}(\rho), \qquad &\mathcal{E}_{\max}(E) &= \max_{\rho\in\sigma_E} \mathcal{E}(\rho) \ \mathcal{A}_{\min}(E) &= \min_{\rho\in\sigma_E} \mathcal{A}(\rho), \qquad &\mathcal{A}_{\max}(E) &= \max_{\rho\in\sigma_E} \mathcal{A}(\rho) \end{aligned}$$

• $\mathcal{E}_{\min}(E)$ gives the worst-case (minimum) extractable work, i.e., a universal upper bound on what could certainly be extracted without detailed state knowledge.
• $\mathcal{A}_{\min}(E)$ gives the minimum “chargeable” work (i.e., the state hardest to charge), setting a worst-case upper bound on anti-ergotropy protocols.

A key result is that these minima are always realized by states that are passive or antipassive: the minimizer is either diagonal in $H$ with eigenvalues monotonic (decreasing for passivity, increasing for antipassivity). Thus, the minimum ergotropy for given $E$ is always achieved for a passive state in energy shell $E$; similarly for anti-ergotropy.

3. Linear Programming Characterization

The set of passive (antipassive) states in a fixed energy shell forms a simplex in the space of diagonal density matrices. The vertices of this simplex (extremal points) are:

• For passivity: $$\rho_k^\downarrow = \frac{1}{k} \sum_{i=1}^k |e_i\rangle\langle e_i|$$,
• For antipassivity: $$\rho_k^\uparrow = \frac{1}{d+1-k} \sum_{i=k}^d |e_i\rangle\langle e_i|$$,

with $d$ the dimension. Consequently, minimizing or maximizing ergotropy or anti-ergotropy for a given $E$ is a linear programming problem: one searches over convex combinations of at most two adjacent simplex vertices to realize the target $E$. This yields efficient $O(d^2)$-scalable algorithms.

Operational implication: Determining the absolute minimal extractable or chargeable work compatible with knowledge of $E$ is computationally practical for moderate system size.

4. Coherent Ergotropy and its Bounds

The energy-constrained coherent ergotropy is the additional amount of extractable (or chargeable) work over the diagonal part:

$$\mathcal{E}_c(\rho) = \mathcal{E}(\rho) - \mathcal{E}(\Delta(\rho))$$

where $\Delta(\rho)$ is $\rho$ dephased in $H$'s eigenbasis.

The minimax analysis provides that ($E = \operatorname{Tr}[\rho H]$):

$$\mathcal{E}_{c,\max}(E) = \mathcal{E}_{\max}(E) - \mathcal{E}_{\min}(E)$$

with an analogous relation for anti-ergotropy. This fully determines the range of possible coherent contributions to work extraction or charging, knowing only the mean energy.

5. Universal Protocols and Saturation of Bounds

The paper identifies three distinct settings in which the bounds for minimum ergotropy and anti-ergotropy are saturated by explicit protocols:

(a) Antisymmetric Hamiltonians

A Hamiltonian is antisymmetric if $\epsilon_k + \epsilon_{d+1-k} = \text{const}$ for all $k$. In this case, simple population-reversal unitaries $$U_\mathrm{rev} = \sum_j |e_{d+1-j}\rangle\langle e_j|$$ attain the optimal worst-case work extraction and charging for all $E$:

$$\widetilde{\mathcal{E}}(E) = \begin{cases} 0 & \text{if } E \leq \epsilon_\text{mean} \ 2(E - \epsilon_\text{mean}) & \text{if } E \geq \epsilon_\text{mean} \end{cases}$$

with $\epsilon_\text{mean}$ the arithmetic mean of energy levels.

(b) Energy-Diagonal States or Access to Decoherence

If states are guaranteed to be diagonal (purely classical), or if decoherence is available to destroy quantum coherences, minimum ergotropy and anti-ergotropy bounds are saturated by mappings optimized for the diagonal subspace.

(c) Random Unitary Strategies

In absence of knowledge about coherences, random-unitary protocols (convex combinations of unitaries) reliably wash out unpredictable advantageous off-diagonal elements, ensuring saturation of the minimax bounds in all cases.

6. Qutrit Protocols: Case Study

For a three-level system (qutrit) with levels $\epsilon_1=0$, $\epsilon_2 = \epsilon (1+\delta)$, $\epsilon_3 = 2\epsilon$ (with $-1 < \delta < 0$), the bounds and optimal protocols can be illustrated numerically:

• The protocol based on strict population reversal (antisymmetric case or $U_{\text{rev}}$) guarantees the bound wherever applicable.
• For non-antisymmetric Hamiltonians, a protocol optimized for diagonals (via linear programming) further improves performance if the state is diagonal—but may underperform if coherences are significant and random.
• Analysis demonstrates that mishandling coherences can result in suboptimal, or even counterproductive, energy extraction/charging.

This concretely illustrates the necessity of protocol design adapted to the available information and system properties.

7. Broader Consequences and Significance

• Operational guarantees: The minimax framework quantifies in advance the minimal guaranteed (or maximal unavoidable) work that can be extracted or charged into any state of given mean energy, independent of details.
• Universality: The results hold for all quantum systems, regardless of dimension, so long as only $E$ is known and unitaries are the only allowed controls.
• Linear programming reduction provides an efficient, non-combinatorial route to evaluate worst-case thermodynamic capabilities for diverse classes of systems.
• Coherence management: While quantum coherences usually enhance ergotropy, in the regime of minimal information they can be detrimental; the optimal protocol is often one that is coherence-independent (e.g., randomization, decoherence, or population reversal).
• Design principle: In practical situations where full control is impossible, the minimal ergotropy and anti-ergotropy set universal, protocol-independent figures of merit for quantum thermal machines and batteries.

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In conclusion, the minimum ergotropy and anti-ergotropy for states of known average energy are achieved by passive or antipassive states (i.e., minimizers are always fully ordered in energy), and can be rigorously and efficiently determined via linear programming over the energy simplex. The analysis not only produces actionable upper bounds but also identifies protocols saturating these bounds for a host of physically relevant scenarios, establishing these metrics as universal limits for quantum work extraction and charging processes under restricted state knowledge (Canzio et al., 10 Sep 2025).