Quantum Work Capacitances: ultimate limits for energy extraction on noisy quantum batteries

This statement concerns our revision of the manuscript entitled Quantum work capacitances, that we submitted to SciPost on 7 December 2022. We thank the Referees for their careful reading, their questions and their comments, which have enabled us to substantially improve the manuscript. We have modified and revised the manuscript accordingly, with the main changes being listed below: Please also find a pdf of the new version with major changes highlighted in blue.

• We have modified the title of the manuscript.
• We have significantly rewritten the introduction following the Referees' suggestions.
• We have added several new comparisons of our work with the existing literature.
• We have clarified numerous expressions by adding physical meaning to them.
• We have corrected the typos and we have streamlined the notation.

Reply to Referee 1 We thank Referee 1 for their careful reading of our manuscript and for providing constructive feedback. The referee raises several important points, which we have addressed by revising the manuscript substantially.

The main criticism given by Referee 1 is that the manuscript lacks physical motivation and discussion of the results. Agreeing that this was a deficiency, and we have added significantly more physical context and intuition throughout the paper. Specifically: - The introduction now provides motivation and examples of how the proposed measures characterize useful properties like charging power, storage capacity, and noise resilience. - The work capacitances are explained in simple terms and related to the energy density per cell. - The physical meaning of key mathematical results are stated explicitly. For instance, we now highlight that Eq. (60) shows entanglement provides no advantage against dephasing noise. - More comparisons are provided with existing literature, explaining how our approach offers new insights. Other changes: - Notation is streamlined and confusing acronyms removed. - Typos and unclear passages identified by the referee are corrected.

Regarding the specific points raised by the Referee:

Question: The title is about "quantum work capacitances" and the abstract promises that this work will introduce them. However, the definition of the work capacitance is only presented in terms of formal mathematics in section IIIA, with no physical explanation. The introduction gives no hint of what they are, or if similar quantities have been studied before. It took me some time to realise that it can be explained in simple words as follows; it is defined as the maximum capacity of a quantum battery to do work when it stores an energy E in an infinite number of systems, so each system only needs to store an infinitesimal amount of energy. Answer: In the revised version we added a new part of the introduction, in which we aim to provide physical intuition about the figure of merit we are describing.

Question: When I googled for keywords like "quantum battery dissipation", I found a paper on the subject that was not cited by this manuscript; "Dissipative dynamics of an open quantum battery", M Carrega, A Crescente, D Ferraro and M. Sassetti, New Journal of Physics, Volume 22, August 2020 Answer: We have added this reference to the paper.

Question: The manuscript needs to place itself in the context of earlier works. It currently simply says that refs [5,13-21] ansd [22,23] worked on the same subject. The manuscript should summarize what is already known about loss of work capacity in quantum batteries due to decoherence and relaxation processes. What is the same or different in this manuscript compared to all these earlier works (and also the work of Caregal cited above)? For example, what measures did those earlier works use, and are the measures proposed in this work different or better? Answer: In the new introduction we compare the setting of our work to previous literature. However, we stress that the references mentioned by the Referee either analyzed specific physical models of quantum batteries, or general properties of the quantum batteries different from the ones we are studying in our paper (in particular, their charging power). In contrast, our manuscript tackles the general conceptual task of energy storage in presence of noise. To our knowledge no previous figures of merit were proposed to characterize the ability of quantum batteries to withstand noise.

Question: The manuscript provides various mathematical proofs for the "quantum work capacitances", but does not explain what these proofs mean in terms of the physics of real systems. They have simple examples in term of two level systems, but even then theystay in the language of formal mathematics, and do not say in simple words what they have proven about the physics of such two-level systems. Here are just two examples: (a) Eq. (60) seems to implies a clear physical consequence, that the battery provide the same maximum amount of work irrespective of whether there is entanglement between its quantum systems, and irrespective of whether one limits the extraction to local unitaries or not. BUT this is NOT explicitly stated.

(b) It looks from Eqs. (57,60,63) that we can only extract epsilon of work by unitary rotations, but you can extract MORE by coupling the battery to a bath, because then we can extract espsilon + Log Z/beta where Log Z is positive (as seen from the definition of Z below Eq (6)). BUT this is also NOT explicitly stated. Actually, this result SURPRISES ME GREATLY, I would have expected the opposite! Is it new, or is it already known? How can it be explained? Answer: We have added some commenting paragraphs after key derivations, in particular in the section V.A. Regarding the question on the Log Z/beta term, this is expected because in the last equation that the Referee has mentioned we consider the extractable work in contact with a thermal environment; this quantity is always greater than the ergotropy (see for example Nat. Commun. 12, 918 (2021)) and we proved that this hierarchy survives in the presence of dephasing noise.

Question: The manuscript defines quantities that they call "Quantum Work Capacitances" (the title of the paper) which are only defined in the limit of taking the number of systems to infinity. This is not the usual macroscopic limit, because one still need perfect control of each individual system (to perform an arbitrary unitary rotation on each system individually). The authors give no physical motivation for WHY this limit is interesting.

(a) WHY define the "work capacitance" as the infinite n limit? (b) Why not define it for n systems?

As a physicist, I know we will never be able to control an infinite number of systems perfectly, but I know that experiments exist with good individual control of tens of qubits (Google's Sycamore quantum computer, etc). So what can one learn about the physics with n=20 or n=30 from the infinite n "quantum work capacitances" presented here. Answer: The evaluation of the asymptotic (n to infinity limit) to describe the properties of many copies of quantum object is a formal procedure, and its physical meaning is analogous to the thermodynamic limit, or to the capacities of quantum channels, that are similarly defined in terms of (n to infinity) copies. Our figures of merit provide an upper bound on the capabilities of quantum batteries under a certain model of noise, and while in any physical implementation the number of copies will be inevitably finite, this is only one of many constraints that will prevent the perfect attainability of these upper bounds.

Question: Why is it "lim" in Eq. (5) and "lim sup" in Eq (25)? Answer: Using the limsup ensures that the work capacitance is well-defined even in the cases when the succession defined by the righthand-side of (25) may not be converging to a limit. As we wrote in the text after equation (5), for the definition of total ergotropy (5) this is not a problem, as one can easily show that this succession always converges.

Question: Why does Eq. (10) have the second term "Log Z"? What is its physical meaning. In normal thermodynamics, the capacity to do work is the free energy F, indeed "free energy" is DEFINED as the capacity of a system to do work when coupled to a thermal bath. Thus the Log Z term here requires explanation. Answer: The term (-1/beta)log Z expresses the free energy of a system in thermal equilibrium, both in classical and in quantum thermodynamics. The maximum amount of work that can be extracted from a thermal bath is the difference between the free energy of the system and the free energy at thermal equilibrium, giving rise to the term in question. We have added an explanation in the revised version after this equation.

Question: Finally the manuscript is hard to read, with poorly chosen or poorly defined notation, and strange choices for acronyms. Also some sections contain multiple typos. These are specific points that are probably easy to fix, and I list them below. Answer: We thank the Referee for signaling those issues, and we have addressed them (including those not explicitly cited here) in the revised version.

Reply to Referee 2 We thank Referee 2 for their positive feedback on our work. The referee's main concerns regard the clarity of presentation, which we have tried to improve in our revised version. In particular: - The introduction highlights how our work opens a new avenue in quantum batteries by considering noise resilience and long-term efficiency. The definition and meaning of work capacitances is expanded. - Explanations are added for how the asymptotic limits capture useful properties like power and efficiency scaling. - The measures are related to existing literature and their novelty and necessity discussed. - Notation is streamlined and confusing acronyms eliminated.

Additionally, in response to the referee's specific comments:

Question: In Eq. (10) it seems like free energy of a specific state (first term) and the global free energy (second term) is added together. Why? Doesn’t the contribution of the specific state already included in the global free energy? The same goes for Eq. (6). Can authors explain the logic behind those formulas? Answer: The total ergotropy is defined as the difference in energy between the initial state and the state with minimum energy that can be reached with isoentropic transformations, i.e. the Gibbs state. The work extraction functional $\mathcal{W}_\beta$ in presence of a thermal bath at inverse temperature beta can be expressed as the difference between the free energy of the initial state and the free energy of the state once brought in equilibrium with the heath bath. Both formulas express the maximum work that can be extracted from a stem as the difference, in an energy state function, between the initial state of the system and the optimal state reachable through the allowed operations.

Question: I find the manuscript to be too technical. Many concepts are introduced but not justified properly. Why are the proposed measures (capacitances and MAWER) defined in those ways? Are the definitions general/universal or system/condition specific? What are the physical justifications? In addition to clarify these points, as a general comment, the authors should consider providing a more accessible introduction to these concepts to make the paper more accessible to a broader audience. Although authors have already used the Appendix to reduce the mathematical load, I still think they could use even more. Making a concise paper focusing on the physics of the problem and putting detailed derivations, even the properties to Appendix could increase the focus and clarity much more than its current form. Answer: We agree with the Referee that the lack of physical intuition was a problem of the manuscript. In our revised version, we have improved the introduction and written several new paragraphs to try to provide a physical meaning that conceptually justifies the results.

Question: Authors provide new measures and approaches but there is no proper comparison given with the existing methods. They mention other works on this as references, however a more detailed comparison should be given. How are they related to the existing methods? Why we need new ones? How the proposed approaches are different? How are they better or worse in various ways? Pros\&cons etc. Too many new concepts are introduced in the paper. Each of them needs to be justified properly, as this is the main novelty of the work. Answer: While ours is to our knowledge the first paper to formulate and address the general problem, several previous papers had proposed specific physical models of quantum batteries in a noisy environment. We have added to the introduction a more detailed comparison with the previous literature.

Question: I find the term “work capacitance” to be a bad choice. Capacitance has a well-established meaning in physics, and I do not think what is proposed in the manuscript relates to that. This discrepancy may lead to confusion and misinterpretation among readers. “Work capacity” might be a better term for the proposed measures. Answer: We find the Referee's concerns well motivated, but we would like to stress that we have defined these quantities for the first time: while the word "capacitance" alone has a different meaning, "work capacitance" is so far unused. Our hope is that they will be used to certify the quality of quantum batteries, which is a research field at an infant stage, so we believe that using the word "capacitance" may not be confusing. We have also used this terminology in two following articles (Phys. Rev. Lett. 131, 060402 (2023) and arXiv:2305.16803), so we would like to keep consistency in all of the definitions. Moreover, the term "thermodynamic capacity" has already be used in a landmark work by Faist, Berta and Brandao (Phys. Rev. Lett. 122, 200601 (2019)), so we have thought that using the term "work capacity" may result ambiguous.

Question: I have a similar comment about the title. It is generally suggested to avoid using newly introduced terms in the title as it does not connote anything in people’s minds. The title in its current form says not much about the content of the manuscript. People will hesitate to read something they don’t understand right away. I suggest authors to reconsider the title. Defining new figures of merit could be mentioned in the title. Answer: We thank the Referee for this suggestion. We added a sentence to clarify in a straightforward manner the concept that we are studying. Also, we believe that this new term can be used in the title if properly described.

Question: A qubit system is chosen for the application. However, the limitations of the measures are not properly discussed. Can they be applied to other quantum systems? Are the results general? If yes, it should be explicitly shown and emphasized, which would increase the impact of the work. Answer: We have added some remarks at the beginning of section V to clarify that, while we are focusing on battery models with wubit cells in our examples, our formalism can be applied irrespectively of the dimension of the cell quantum system. In other works, we have already used the conceptual tool developed in this paper to analyze systems which are not qubits.

- We have modified the title of the manuscript.
- We have modified the abstract
- We have significantly rewritten the introduction following the Referees' suggestions.
- We have added several new comparisons of our work with the existing literature.
- We have clarified numerous expressions by adding physical meaning to them.
- We have corrected the typos and we have streamlined the notation.

For more specific points please see the resubmission letter.