Observational Ergotropy in Quantum Systems

• The paper introduces observational ergotropy by quantifying work extraction when only coarse-grained measurement data is available, addressing realistic quantum battery scenarios.
• It defines the replacement of full density operator characterization with observational entropy, linking coarse graining to maximum-entropy state reconstruction.
• The work explores extensions through measurement-assisted protocols, dynamic work extraction, and certification methods for energy extraction under incomplete information.

Observational ergotropy is an operational extension of ergotropy for quantum systems whose microstate is not fully known. In the formulation developed for unknown quantum sources, it quantifies the extractable work when the only accessible characterization is a single coarse-grained measurement and no further measurements are allowed once the source is used as a quantum battery; in the large-$N$ limit, the relevant state descriptor is the observational entropy rather than the full density operator (Šafránek et al., 2022). Closely related literature studies allied measurement-assisted notions under partial knowledge, including ancilla-assisted weak-measurement protocols, coarse-grained work extraction in chaotic dynamics, and certification of extractable work from incomplete data (Balkanlu et al., 2022, PG et al., 2024, Pagliaro et al., 19 Mar 2026).

1. Standard ergotropy, passive states, and the motivation for observational formulations

Standard ergotropy is the maximum work extractable from a quantum state $\rho$ with Hamiltonian $H$ by a cyclic, unitary protocol. For $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$ and $\rho=\sum_i r_i |r_i\rangle\langle r_i|$, it is

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$

The minimizing unitary rearranges the eigenvalues of $\rho$ onto the energy eigenbasis in the opposite order, producing the passive state $\pi$, so that

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$

A passive state is diagonal in the energy basis and ordered so that no further work can be extracted by unitaries (Šafránek et al., 2022).

The motivation for observational ergotropy is that standard ergotropy assumes perfect knowledge of $\rho$. In realistic settings, sources may be unknown or untrusted, and full quantum tomography is prohibitively costly because it scales exponentially with system size, requires many non-commuting measurements, long integration times, and may be physically inaccessible. The central question is therefore operational rather than purely kinematic: if access to the source is restricted to a single coarse-grained measurement, what is the realistic maximum work that can be guaranteed (Šafránek et al., 2022)?

This shift replaces state-complete characterization by measurement-limited characterization. As a consequence, observational ergotropy is not defined by the exact spectrum and eigenbasis of $\rho$0, but by the information retained after coarse graining.

2. Coarse graining and observational entropy

The coarse-grained measurement is specified by a family of projectors $\rho$1 satisfying $\rho$2. Each projector defines a macrostate of “volume”

$\rho$3

and for an unknown true state $\rho$4, outcome $\rho$5 occurs with probability

$\rho$6

The associated observational entropy is

$\rho$7

equivalently,

$\rho$8

It quantifies the uncertainty that remains after the fixed coarse-grained measurement (Šafránek et al., 2022).

In the work-extraction protocol, random unitaries are applied within each macrostate subspace $\rho$9, scrambling the microstate into the maximally mixed state on that subspace. Averaging over this randomness yields the coarse-grained density matrix

$H$0

This state is the effective object from which work is extracted when only the coarse-grained statistics are available (Šafránek et al., 2022).

A related coarse-grained reconstruction appears in chaotic-system studies. There, after a macrostate measurement $H$1 with $H$2, the maximum-entropy reconstruction is

$H$3

and the remaining uncertainty is measured by the observational entropy

$H$4

This suggests a close structural connection between observational ergotropy and maximum-entropy state reconstruction under partial information (PG et al., 2024).

3. Conditional and unconditional work extraction

For $H$5 identically prepared but individually unknown copies, the coarse-grained framework yields two asymptotic operational scenarios (Šafránek et al., 2022).

Scenario	Information retained	Asymptotic work per copy
Standard ergotropy	Full $H$6 known	$H$7
Boltzmann ergotropy	Outcome sequence stored	$H$8
Observational ergotropy	Only $H$9 known	$$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$0

In the conditional protocol, each copy is measured in the coarse graining $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$1, the full sequence of outcomes is recorded, random unitaries scramble each macrostate, and an optimal global unitary is then applied. The average extractable work per copy in the $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$2 limit is

$$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$3

where

$$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$4

Equivalently,

$$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$5

with $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$6 the Gibbs state at temperature $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$7 (Šafránek et al., 2022).

In the unconditional protocol, no further measurements are allowed once characterization is complete. One estimates only the distribution $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$8, applies the same macrospace randomization, and then the optimal global unitary that minimizes the final energy of $$H=\sum_j \epsilon_j |\epsilon_j\rangle\langle\epsilon_j|$$9. The large-$\rho=\sum_i r_i |r_i\rangle\langle r_i|$0 extractable work per copy is

$\rho=\sum_i r_i |r_i\rangle\langle r_i|$1

or equivalently

$\rho=\sum_i r_i |r_i\rangle\langle r_i|$2

where $\rho=\sum_i r_i |r_i\rangle\langle r_i|$3 is the Gibbs state implicitly fixed by $\rho=\sum_i r_i |r_i\rangle\langle r_i|$4 (Šafránek et al., 2022).

The distinction between the two scenarios is operationally significant. Boltzmann ergotropy assumes that the measurement outcomes for each copy are available during extraction, whereas observational ergotropy assumes that only the coarse-grained distribution is known. The former therefore carries, in principle, a Landauer erasure cost for resetting the measurement record, while the latter is the relevant figure of merit when no further measurements are performed once the source is in service (Šafránek et al., 2022).

Both quantities reduce to standard ergotropy when the coarse graining is fine, with $\rho=\sum_i r_i |r_i\rangle\langle r_i|$5 for every macrostate. In that limit, the gap between complete microscopic knowledge and coarse-grained operational knowledge disappears (Šafránek et al., 2022).

4. Measurement-assisted extensions and the role of correlations

A related branch of the literature studies measurement-assisted work extraction in bipartite settings. In the daemonic protocol, an ancilla $\rho=\sum_i r_i |r_i\rangle\langle r_i|$6 is measured projectively, yielding conditional states $\rho=\sum_i r_i |r_i\rangle\langle r_i|$7 on the system $\rho=\sum_i r_i |r_i\rangle\langle r_i|$8, and the optimal local unitary on $\rho=\sum_i r_i |r_i\rangle\langle r_i|$9 is then chosen conditional on the outcome. The daemonic ergotropy satisfies

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$0

For weak measurements on the ancilla, the non-selective protocol reproduces exactly the projective-measurement gain,

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$1

while a selective weak measurement can reveal strictly more extractable work than daemonic ergotropy,

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$2

The paper refers to this selective enhancement as “super ergotropy” (Balkanlu et al., 2022).

This literature broadens the operational landscape of observational ergotropy from coarse-grained ignorance about a single source to outcome-assisted extraction from correlated composite systems. For Bell-diagonal two-qubit states, the same analysis shows that projective measurement on the total system can increase both total and non-local extractable work, while there is no direct relationship between quantum correlation and non-local extractable work in those cases (Balkanlu et al., 2022).

Correlations also enter through a distinct thermodynamic identity. For bipartite systems with locally thermal marginals,

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$3

where $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$4 is the quantum mutual information and $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$5 is the quantum relative entropy. This immediately implies the bound

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$6

Using the decomposition $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$7, one can formally separate classical-correlation and quantum-correlation contributions, together with the penalty term $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$8. In the idealized limit where that penalty vanishes, the ergotropy becomes

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$9

so the work is harvested entirely from correlations (Touil et al., 2021).

Taken together, these results show that observationally constrained work extraction is not limited to ignorance about a single density matrix. It also encompasses measurement strength, outcome selectivity, and correlation structure. This suggests that the term “observational ergotropy” is used in more than one operational sense in current work.

5. Chaotic dynamics, coarse-grained reconstruction, and ergotropy backflow

Observational ergotropy has also been examined in explicitly dynamical settings where partial information is combined with unitary complexity. In the quantum kicked top and the kicked Ising chain, the system state is not assumed to be known exactly; instead, it is partially characterized by a coarse-grained measurement and reconstructed as $\rho$0. Two extraction protocols were studied. In Protocol 1 (“measure-then-average”),

$\rho$1

where conditional ancilla outcomes are retained. In Protocol 2 (“average-then-measure”),

$\rho$2

where ancilla outcomes are pre-averaged (PG et al., 2024).

The main dynamical result is a competition between entanglement-assisted information gain and coarse-graining-induced information loss. In Protocol 1, $\rho$3 is non-monotonic in chaos strength: for the kicked top there is a pronounced maximum around $\rho$4, and for the kicked Ising chain around $\rho$5–$\rho$6. This “sweet spot” coincides with a minimum in the averaged observational entropy $\rho$7, summarized by

$\rho$8

By contrast, in Protocol 2 no entanglement-assisted gain occurs and $\rho$9 falls monotonically with chaos strength, tracking a monotonic increase of $\pi$0 (PG et al., 2024).

A broader thermodynamic dynamical picture comes from ergodic channels with a passive fixed point. For channels of the form

$\pi$1

the ergotropy decreases monotonically under CP-divisible dynamics: $\pi$2 When the dynamics becomes indivisible, there are intervals with $\pi$3, and then

$\pi$4

The resulting ergotropy backflow is interpreted as memory-induced resource recovery and can be used as a non-Markovianity measure via

$\pi$5

This is not the same notion as coarse-grained observational ergotropy, but it places ergotropy-based observables within a wider program of operational diagnostics for open-system thermodynamics (Basu et al., 2024).

6. Certification and experimental access under partial information

A central practical problem is that even observational protocols require reliable inference from incomplete data. A general certification framework addresses this by lower bounding ergotropy from expectation values of a limited set of arbitrary observables in the finite-statistics regime. If $\pi$6 observables $\pi$7 are measured and $\pi$8 denotes the empirical mean from $\pi$9 shots, then with overall failure probability $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$0,

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$1

simultaneously for all $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$2. This defines a confidence region

$$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$3

A two-step semidefinite-programming protocol then produces a confidence-certified lower bound $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$4: first choose a trial $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$5 by minimizing the purity $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$6, then fix the corresponding unitary $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$7 and minimize the extracted work over all $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$8 (Pagliaro et al., 19 Mar 2026).

This framework was benchmarked on synthetic data and on measurements from an IBM quantum processor. For an IBM Perth experiment with a 4-qubit GHZ state, $$W_{\mathrm{erg}}(\rho,H)=\mathrm{Tr}(\rho H)-\mathrm{Tr}(\pi H).$$9 shots per Pauli string, and $\rho$0 up to 60 four-qubit Pauli constraints, the certified lower bound grows monotonically with $\rho$1; by $\rho$2, about $\rho$3–$\rho$4 of the true ergotropy is certified even under severe shot noise and device errors (Pagliaro et al., 19 Mar 2026).

Experimental access can also be indirect. For a dephasing qubit with $\rho$5, the dynamic phase

$\rho$6

depends solely on the incoherent ergotropy, while the geometric phase obeys the exact relation

$\rho$7

In the weak-coupling, long-time regime, the geometric phase becomes determined exclusively by the incoherent ergotropy, suggesting that the ergotropy of a two-level system could be inferred indirectly from geometric-phase measurements using standard techniques such as quantum state tomography (Lombardo et al., 1 Mar 2026).

These developments indicate two complementary experimental directions. One is protocol-specific, based on fixed coarse-grained measurements, macrospace randomization, and optimized unitaries as in observational ergotropy proper (Šafránek et al., 2022). The other is protocol-agnostic, based on certified lower bounds or interferometric inference under limited information (Pagliaro et al., 19 Mar 2026, Lombardo et al., 1 Mar 2026). A plausible implication is that future work will combine these approaches to make extractable-work quantification both operationally faithful and experimentally scalable.