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Physica Scripta

Optimal states and correlation effects in ergotropy under operational constraints

Jiangyang Pu, Tao Feng, Tao Zhou

Published Oct 5, 2026• 0 citations
Publisher

Abstract

Abstract Operational constraints fundamentally affect the ergotropy of bipartite quantum systems. We compare optimal input states under global unitary (GU) and parallel local unitary (PLU) operations. Under GU operations, maximization over the input state attains the spectral width of the Hamiltonian, whereas under PLU constraints the optimum also depends on the operator structure of the Hamiltonian and on the transformations accessible through product unitaries. We then analyze the effect of correlations by comparing a correlated state with the product of its marginals. For noninteracting Hamiltonians, correlations do not change the parallel ergotropy. In the presence of interactions, correlations can either enhance or suppress both global and parallel ergotropy. By recasting the problem as a constrained variational optimization, we derive general bounds and one-sided sign criteria. Under GU operations, the relevant extrema are governed by eigenvalue ordering, whereas under PLU operations they are controlled by the accessible alignment between the correlation matrix and the interaction matrix in operator space.

DOI
10.1088/1402-4896/aeb079

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Abstract Operational constraints fundamentally affect the ergotropy of bipartite quantum systems. We compare optimal input states under global unitary (GU) and parallel local unitary (PLU) operations. Under GU operations, maximization over the input state attains the spectral width of the Hamiltonian, whereas under PLU constraints the optimum also depends on the operator structure of the Hamiltonian and on the transformations accessible through product unitaries. We then analyze the effect of correlations by comparing a correlated state with the product of its marginals. For noninteracting Hamiltonians, correlations do not change the parallel ergotropy. In the presence of interactions, correlations can either enhance or suppress both global and parallel ergotropy. By recasting the problem as a constrained variational optimization, we derive general bounds and one-sided sign criteria. Under GU operations, the relevant extrema are governed by eigenvalue ordering, whereas under PLU operations they are controlled by the accessible alignment between the correlation matrix and the interaction matrix in operator space.
Maximum quantum work extraction is generally defined in terms of the ergotropy functional, no matter how experimentally complicated is the implementation of the optimal unitary allowing for it, especially in the case of multipartite systems. In this framework, we consider a quantum battery made up of many interacting subsystems and study the maximum extractable work via concurrent local unitary operations on each subsystem. We call the resulting functional parallel ergotropy. Focusing on the bipartite case, we first observe that parallel ergotropy outperforms work extraction via egoistic strategies, in which the first agent A extracts locally on its part the maximum available work and the second agent B, subsequently, extracts what is left on the other part. For the agents, this showcases the need of cooperating for an overall benefit. Second, from the informational point of view, we observe that the parallel capacity of a state can detect entanglement and compare it with the statistical entanglement witness that exploits fluctuations of stochastic work extraction. Additionally, we face the technical problem of computing parallel ergotropy. We derive analytical upper bounds for specific classes of states and Hamiltonians and provide recipes to obtain numerical upper bounds via semidefinite programming in the generic case. Finally, extending the concept of parallel ergotropy, we demonstrate that system's free-time evolution and application of local unitaries allow one to saturate the gap with the ergotropy of the whole system.
The presence of correlations in the input state of a noninteracting many-body quantum system can lead to an increase in the amount of work we can extract from it under global unitary processes (ergotropy). The present work explores such effect on translationally invariant systems relying on the matrix product operator formalism to define a measure of how much they are correlated. We observe that in the thermodynamic limit of large number of sites, complete work extraction can be attained for relatively small correlation strength (a reduction of a 2 factor in dB units). Most importantly, such an effect appears to not be associated with the presence of quantum correlations (e.g., entanglement) in the input state (classical correlation suffices), and to be attainable by only using incoherent ergotropy. As a byproduct of our analysis, we also present a rigorous formulation of the heuristic typicality argument first formulated by Alicki and Fannes [R. Alicki and M. Fannes, Phys. Rev. E 87, 042123 (2013)], which gives the maximum work that is extractable for a set of many identical quantum systems in the asymptotic limit.
We put forth a notion of optimality for extracting ergotropic work, derived from an energy constraint governing the necessary dynamics for work extraction in a quantum system. Within the traditional ergotropy framework, which predicts an infinite set of equivalent pacifying unitaries, we demonstrate that the optimal choice lies in driving along the geodesic connecting a given state to its corresponding passive state. Moreover, in a practical scenario where unitaries are inevitably affected by environmental factors, we refine the existing definition of ergotropy and introduce the notion of operational ergotropy. It enables the characterization of work extraction in noisy scenarios. We find that for certain typical noise models, the optimal choice which governs the Schrodinger part of the dynamics, aligns with the optimal drive in the unperturbed scenario. However, we demonstrate that such optimality is not universal by presenting an explicit counterexample. Additionally, within this generalized framework, we discuss the potential for faster work extraction from quantum systems in the presence of noise.
We investigate the problem of finding the local analog of the ergotropy, which is the maximum work that can be extracted from a system if we can only apply local unitary transformation acting on a given subsystem. In particular, we provide a closed formula for the local ergotropy in the special case in which the local system has only two levels, and we give analytic lower bounds and semidefinite programming upper bounds for the general case. As nontrivial examples of application, we compute the local ergotropy for an atom in an electromagnetic cavity with Jaynes-Cummings coupling and the local ergotropy for a spin site in an XXZ Heisenberg chain, showing that the amount of work that can be extracted with a unitary operation on the coupled system can be greater than the work obtainable by quenching off the coupling with the environment before the unitary transformation.
For finite-dimensional bipartite systems with additive endpoint Hamiltonians, the ergotropic gap measures the work advantage of global over independent local cyclic unitary control. We express this gap through mutual information and the Gibbs-relative entropies of the local and global passive states. The balance retains local passive-state corrections for arbitrary marginals and reduces to the established locally thermal ergotropy identity when those corrections vanish. Dropping the global correction gives a separability-independent upper bound whose exact slack is the global passive-state free-energy excess. We show that optimizing the reference temperature is equivalent to matching the Gibbs entropy to the joint-state entropy. The optimized result is precisely the equal-entropy Gibbs bound on global ergotropy minus the local ergotropies, rather than a new thermodynamic bound. We characterize saturation and temperature endpoints, including degenerate ground spaces. Werner states yield a unique implicit crossover with a separability-based majorization bound and a small-correlation bound-to-gap ratio approaching 3/2. Structured and random two-qubit comparisons illustrate complementary performance. We also specify pointwise quantum-battery applicability and the obstruction from persistent interactions.
Quantum systems may contain underlying correlations which are inaccessible to computationally bounded observers. We capture this distinction through a framework that analyses bipartite states only using efficiently implementable quantum channels. This leads to a complexity-constrained max-divergence and a corresponding computational min-entropy. The latter quantity recovers the standard operational meaning of the conditional min-entropy: in the fully quantum case, it quantifies the largest overlap with a maximally entangled state attainable via efficient operations on the conditional subsystem. For classical-quantum states, it further reduces to the optimal guessing probability of a computationally bounded observer with access to side information. Lastly, in the absence of side information, the computational min-entropy simplifies to a computational notion of the operator norm. We then establish strong separations between the information-theoretic and complexity-constrained notions of min-entropy. For pure states, there exist highly entangled families of states with extremal min-entropy whose efficiently accessible entanglement in terms of computational min-entropy is exponentially suppressed. For mixed states, the separation is even sharper: the information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal. Under the cryptographic Learning with Errors hardness assumption, we construct efficiently preparable states exhibiting both separations, even when their preparation circuits are fully known. Overall, our results demonstrate that computational constraints can fundamentally limit the quantum correlations that are observable in practice.