Daemonic Gain in Quantum Thermodynamics

• Daemonic gain is defined as the additional work extractable from a quantum system by optimally using measurement results on an ancilla, beyond standard ergotropy.
• It operationalizes the energetic value of correlations in bipartite systems, demonstrating that both quantum and classical correlations can enhance work extraction.
• The concept adapts to various measurement protocols—including projective, generalized, and weak measurements—and finds applications in quantum batteries and open-system dynamics.

Searching arXiv for recent and foundational papers on daemonic ergotropy and related uses of “daemonic gain”. Daemonic gain denotes the increase in extractable work that becomes available when a controller measures an ancilla correlated with a target system and then applies outcome-dependent unitaries, relative to the ordinary ergotropy of the reduced system. In the formulation introduced for closed bipartite quantum systems, the quantity links quantum non-equilibrium thermodynamics to quantum information theory by making the energetic value of correlations operational: it is nonnegative by construction, vanishes on product states, and is constrained by the structure of discord and entanglement (Francica et al., 2016).

1. Formal definition and baseline framework

The original setting considers a finite-dimensional system $S$ with Hamiltonian

$$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$

prepared in a state $\hat\rho_S$, and driven cyclically by unitary control. Standard ergotropy is the maximum work extractable from $\hat\rho_S$ by such unitaries,

$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$

Its minimum-energy endpoint is the passive state, obtained by placing the eigenvalues of $\hat\rho_S$ in decreasing order on the energy eigenbasis in increasing-energy order. Passive states satisfy $[\hat\rho_S,\hat H_S]=0$ and admit no further unitary work extraction (Francica et al., 2016).

Daemonic gain is defined after enlarging the setting to a closed bipartite state $\hat\rho_{SA}$, where $A$ is a non-interacting ancilla. A projective measurement $\{\hat\Pi_a^A\}$ is performed on $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$0, with outcome probabilities

$$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$1

and conditional states

$$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$2

If an optimal unitary $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$3 is then applied on $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$4 for each branch, the corresponding daemonic ergotropy for that measurement is

$$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$5

where $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$6 are the eigenvalues of $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$7 arranged in decreasing order. The daemonic gain is the optimized increment

$$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$8

A general lemma shows $$\hat H_S=\sum_k \epsilon_k\,|\epsilon_k\rangle\langle \epsilon_k|,$$9 for every projective measurement, so measurement and feedback never reduce extractable work. The original formulation treats $\hat\rho_S$0 as a raw resource quantity: a lower bound for the measurement-energy cost is given, but that cost is not included in the definition of the gain itself (Francica et al., 2016).

2. Correlations, discord, entanglement, and what the gain does not imply

If the initial state is uncorrelated,

$\hat\rho_S$1

then $\hat\rho_S$2 for all $\hat\rho_S$3, so $\hat\rho_S$4 and $\hat\rho_S$5. In the correlated case, measurement on $\hat\rho_S$6 resolves $\hat\rho_S$7 into an ensemble of conditional states whose individual non-passivity can exceed that of the average state. In this precise sense, daemonic gain quantifies the extra population reordering enabled by conditioning (Francica et al., 2016).

The main structural theorem of the original theory is asymmetric. Vanishing daemonic gain implies vanishing discord with respect to measurements on $\hat\rho_S$8,

$\hat\rho_S$9

but the converse does not hold, and zero discord on either side does not imply zero gain. Classically correlated states can therefore exhibit strictly positive daemonic gain. This directly rules out a common misconception: quantum correlations are sufficient for nonzero gain in the original direction-sensitive sense, but they are not necessary (Francica et al., 2016).

For pure bipartite states the characterization is sharper. If

$\hat\rho_S$0

is a Schmidt decomposition, then

$\hat\rho_S$1

and, when entanglement is present,

$\hat\rho_S$2

In that regime daemonic gain is a necessary-and-sufficient separability criterion and functions as a thermodynamic entanglement witness (Francica et al., 2016).

The two-qubit analysis makes the non-monotonicity of correlation–work relations explicit. For $\hat\rho_S$3, numerical exploration yields lower bounds

$\hat\rho_S$4

for fixed discord $\hat\rho_S$5 and concurrence $\hat\rho_S$6, respectively. At the same time, zero-discord states can reach arbitrarily large daemonic gain up to the absolute maximum $\hat\rho_S$7. The literature on Bell-diagonal and related states makes the same point more broadly: there is no direct relationship between quantum correlation and non-local extractable work, and more correlation does not automatically mean more daemonic advantage (Francica et al., 2016, Balkanlu et al., 2022).

3. Measurement structure: projective, generalized, weak, and Gaussian

A first generalization concerns the measurement class. For quantum-classical states, projective measurements on the classical ancilla basis are optimal, and classical correlations can attain the maximal daemonic ergotropy allowed by the Hamiltonian. More generally, if the optimal measurement happens to be projective, quantum correlations are not advantageous over classical correlations. However, generalized measurements can do better: a qutrit–qubit example exhibits a POVM with strictly larger daemonic ergotropy than any projective measurement, and a see-saw algorithm was proposed to search over measurements by alternating between POVM optimization and optimal conditional unitaries (Bernards et al., 2019).

A second generalization replaces projective measurements by weak measurements. Non-selective weak measurement on the ancilla reproduces the same average daemonic ergotropy as strong projective measurement for all measurement strengths,

$\hat\rho_S$8

whereas selective weak measurement can exceed standard daemonic ergotropy. In that setting the authors call the enhanced quantity “super ergotropy”, with

$\hat\rho_S$9

An explicit mixed-state example reaches a factor-of-two enhancement in the strong-measurement limit, showing that outcome-biased weak measurements can reveal additional work potential hidden from the standard protocol (Balkanlu et al., 2022).

Continuous-variable formulations extend the same logic. For Gaussian states under general-dyne detection, the single-mode ergotropy depends only on energy and purity,

$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$0

and the corresponding daemonic ergotropy becomes

$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$1

This makes measurement-induced purification the key mechanism: maximizing daemonic gain is equivalent to maximizing the conditional purity produced by the Gaussian measurement. For phase-invariant bipartite Gaussian states, heterodyne detection is optimal; for a two-mode squeezed thermal state it outperforms homodyne, and in a continuously monitored optical parametric oscillator the best daemonic strategy is the one that optimizes purity rather than the one that maximizes squeezing in a single quadrature (Kua et al., 27 Jun 2025).

4. Open systems, monitored environments, and quantum batteries

In open quantum batteries, the ancilla can be replaced by the environment itself. For continuously monitored open systems, the unconditional battery state $$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$2 is decomposed into conditional trajectories $$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$3, and the unravelling daemonic ergotropy is defined as the trajectory average

$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$4

It obeys the general bound

$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$5

The upper bound is achieved for an initial pure state and a perfectly efficient projective measurement on the environment, independently of the kind of measurement performed; for mixed initial states or inefficient monitoring, the gain depends on the measurement strategy. A driven two-level atom monitored by homodyne, heterodyne, or photo-detection provides the paradigmatic example (Morrone et al., 2023).

This framework has been transferred to quantum-battery platforms. In a continuously monitored collisional model implemented as a quantum circuit, the corresponding daemonic extracted work was experimentally simulated on IBM hardware. The measured values were reported to be close to the theoretical upper bound given by the daemonic ergotropy, and noise-aware optimization of the conditional extraction unitaries further improved performance, especially in the stronger-coupling regime (Elyasi et al., 2024).

A different battery architecture uses strong battery–charger correlations generated by thermalization. There the battery is measured indirectly through a memory that probes the charger, leaving the battery intact. The extracted energy becomes the sum of the battery’s daemonic ergotropy and the memory ergotropy, while the work cost includes disconnecting and reconnecting work, measurement work, and reset work. In a modified transverse spin-$$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$6 Ising chain, the efficiency of the measured protocol can surpass that of the unmeasured protocol for suitable temperature, coupling, and measurement choices (Satriani et al., 2024).

Recent charging analyses rephrased the same logic in terms of a daemonic gap and daemonic band. The daemonic gap is defined as the difference between the minimum daemonic ergotropy and the maximum ordinary ergotropy at the time where the latter is maximal; the daemonic band is the difference between the maximum and minimum daemonic ergotropy. For a harmonic-mode battery charged through a Jaynes–Cummings interaction with a qubit auxiliary, the gap can be closed for specific initial passive states, and repeated charging cycles can access the full daemonic band and achieve complete charging (Pushpan et al., 10 Nov 2025).

Noise with memory can also act as a resource. For classically correlated qubits subjected to amplitude damping, local non-unital noise generates quantum correlations and a positive daemonic gain. Introducing channel memory amplifies this effect by preserving system–ancilla correlations: standard ergotropy remains $$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$7, but $$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$8 stays positive over a much broader parameter region and, for sufficiently large memory parameter $$\mathcal W=\operatorname{Tr}[\hat\rho_S\hat H_S]-\min_U \operatorname{Tr}[U\hat\rho_SU^\dagger \hat H_S].$$9, over the full damping range (Hadipour et al., 1 Jun 2025).

5. Purity, bound energy, fluctuations, and risk-sensitive variants

A recent refinement places daemonic gain under a bound-energy constraint. For the reduced system state $\hat\rho_S$0 with ground energy $\hat\rho_S$1, the bound energy is

$\hat\rho_S$2

This yields a tight upper bound

$\hat\rho_S$3

which is saturated for globally pure system–ancilla states. Motivated by that relation, the paper introduces the purity-based gain

$\hat\rho_S$4

which equals $\hat\rho_S$5 for pure global states and qualitatively predicts it for mixed states without explicit optimization over measurements. In collective environments, dissipation can dynamically generate and stabilize finite daemonic gain through environment-induced correlations; in interacting systems, level crossings and spectral restructuring modify the attainable gain through their effect on accessible bound energy (Vigneshwar et al., 18 Jun 2026).

Another extension abandons risk neutrality. Instead of maximizing average extracted work, the demon maximizes expected utility of work fluctuations: $\hat\rho_S$6 For a risk-neutral agent, $\hat\rho_S$7 reduces to the standard daemonic ergotropy gain. For constant absolute risk aversion, zero daemonic expected-utility gain implies separability, while for general utilities entangled zero-gain states can exist. This establishes that the thermodynamic usefulness of correlations becomes utility-dependent once fluctuations are treated explicitly (Francica et al., 2024).

The experimental many-particle photonic Maxwell demon realizes the same theme in a bosonic setting. Single-photon measurements on one thermal mode and feedforward SWAP operations on two identical thermal light beams yield a deterministic increase in mean output energy, with maximum relative increases of approximately $\hat\rho_S$8 at $\hat\rho_S$9 for $[\hat\rho_S,\hat H_S]=0$0 and $[\hat\rho_S,\hat H_S]=0$1 at $[\hat\rho_S,\hat H_S]=0$2 for $[\hat\rho_S,\hat H_S]=0$3. The output also exhibits an improved stability metric $[\hat\rho_S,\hat H_S]=0$4, and the experiment verifies the generalized relation

$[\hat\rho_S,\hat H_S]=0$5

This suggests that, beyond average-work language, daemonic gain can also denote an information-enabled increase in energy and thermodynamic stability in a many-particle regime (Hloušek et al., 2024).

6. Operational meaning, lineage, and limits

Operationally, daemonic gain is the value of information carried by accessible correlations. In the original protocol, the demon measures the ancilla, learns which conditional state of the system is realized, and tailors the extracting unitary branch by branch. In open-system variants, the demon monitors the environment and conditions feedback on the resulting quantum trajectory. In Gaussian settings, the same principle reduces to measurement-induced purification. In battery settings, it becomes a charge-discharge advantage supplied by accessible side information rather than by additional coherent driving (Francica et al., 2016, Morrone et al., 2023).

The conceptual lineage is older than the term itself. In the deterministic Szilard-map formulation of Maxwell-demon dynamics, the rate of energy extraction from the heat bath is set by the Kolmogorov–Sinai entropy, while the statistical complexity quantifies the minimum memory required for the engine to function. This suggests that daemonic gain belongs to a broader family of information-powered thermodynamic effects in which memory, instability, and feedback jointly determine energetic advantage (Boyd et al., 2015).

Two limitations recur across the literature. First, daemonic gain is usually a statement about raw extractable work, not full net efficiency. The original formulation gives a lower bound on measurement-energy cost but omits it from $[\hat\rho_S,\hat H_S]=0$6; later cycle analyses reintroduce measurement, reconnecting, and reset costs explicitly (Francica et al., 2016, Satriani et al., 2024). Second, the gain is not a monotone of “how quantum” the correlations are. Classically correlated states can yield large positive gain, while discord, entanglement, purity, spectral gaps, and even risk attitude each control different aspects of what information can actually be converted into work (Francica et al., 2016, Francica et al., 2024, Vigneshwar et al., 18 Jun 2026).

In that sense, daemonic gain is best understood not as a single correlation measure, but as an operational thermodynamic increment: the part of work extraction that becomes available only after side information is acquired and used. The modern literature shows that this increment can be bounded by bound energy, reshaped by generalized measurements, stabilized or generated by collective dissipation, amplified by channel memory, and, in special cases, made exactly equal to the reduced system’s otherwise inaccessible energetic content (Bernards et al., 2019, Hadipour et al., 1 Jun 2025, Vigneshwar et al., 18 Jun 2026).