A short story of quantum and information thermodynamics

The author gives here a very nice introduction to quantum thermodynamics. The first two parts give good reminders on classical thermodynamics, both from a macroscopic and a stochastic points of view. It also introduces some details about information theory and technical methods essential for quantum thermodynamics such as "two-points trajectories".

The last part introduces quantum thermodynamics, and deals with an interesting result about quantum measurement and the use of information as a thermodynamic resource.

I found this article very interesting, clear, and built with many examples / intuitions which allow to develop new intuitions about quantum thermodynamics. I highly recommend this article.

I noted some mistakes, and some sentences that, I think, could be improved. I hope it will allow to improve an overall good paper.

1- Page 2, paragraph 1; I would replace "and from the first law of thermodynamics, there is nothing
wrong in converting one into another." by "and according to the first law of thermodynamics, it is possible to convert one into another.";

2- Page 2, paragraph 2; I would replace "caveats" by "constraints";

3- Page 2, paragraph 2; I would relace "turned out to become" by "turned out to be";

4- Page 2, paragraph 2; the sentence "This is how an initially
applied area of physics turned out to deliver fundamental concepts like entropy and time
arrow." might need to me reformulated, for example in two parts, to make it clearer and increase its fluidity;

5- Page 2, paragraph 3; I would replace "circulated" by "flows";

6- Page 2, paragraph 3; for the last sentence, equality is reached "if" or "only if " $T_c = T_h$ ?;

7- Page 2, paragraph 3; I would replace "exchanged reversibly" by "reversibly exchanged" or " exchanged in a reversible case";

8- Page 2, paragraph 4; in the last sentence, I would replace "motivation of quantum thermodynamics" by "motivation for developping a quantum thermodynamics";

9- Page 2, paragraph 5; I would replace "the gas while it is coupled " by "the gas while being coupled";

10- Page 2, paragraph 5; I would replace "settings of the engine" by "engine settings";

11- Page 2, paragraph 5; I would replace "particle of the gas" by "gas particles";

12- Page 2, paragraph 5 (end on page 3); I think that the last sentence can be clarified by replacing "hence the need for at least two baths at two dierent temperatures." by "meaning that two baths at two different tempratured are required for work extraction.";

13- Page 3, paragraph 1; I would replace "Such mechanism blatantly violates " by "A so called Maxwell demon applying such mechanism would obviously violates", in order to introduce the "Maxwell demon" which is mentionned below without being introduced before;

14- Page 3, paragraph 2; I think that the sentence "To understand how information and thermodynamics
are related, Carnot engines are enlightening - however, instead of considering a gas made
of a large number of particles, one can consider a 'single-particle gas' rather, that is either
positioned on the left or on the right of the chamber." should be splitted in two parts for clarity.

15- Page 3, paragraph 3; I think that replacing "in information theory a RESET operation:" by " a RESET operation in information theory" would be clearer;

16- Page 3, paragraph 3; if "$W_0$" expression's can be directly seen on Eq. 1, I think it should be written here "that can be lower than $W_0 =$ ... - the bound being reached";

17- Page 3, last paragraph; I would replace "that deals with small enough working substances such that fluctuations become predominant" by "that deals with small enough working substances for fluctuations to become predominant ";

18- Page 4, paragraph 4; I would replace "$\gamma$ is more probable forward than backward" by "the forward trajectory $\gamma$ is more probable than its corresponding backward trajectory $\tilde{\gamma}$", because $\gamma$'s probability is not always defined for a backward process, and is not the one interesting here. For example, if we consider a two Qbits system, with a trajectory $\gamma: (1,0) \rightarrow (1,1)$, its probability has to be compared with $\tilde{\gamma}$ probability where $\tilde{\gamma}: (1,0) \leftarrow (1,1)$ and not with $\gamma: (1,1) \leftarrow (1,0)$ which I believe would be a backward $\gamma$;

13- Page 4, paragraph 5; I would replace "exponential" by "exponential function";

14- Page 4, paragraph 5; I would replace "saturated" by "reached", which in my opinion fits better;

15- Page 5, paragraph 3; I would replace "build many useful intuitions" by "develop many useful intuitions";

16- Page 5, paragraph 3; I would replace "probability of jump" by "probability of jumping" or "probability of a jump";

17- Page 5, paragraph 3; for $D_\infty$ computation, I don't understand how we get to this result, and what are the $p(\sigma_i)$, since only $p_0$, $ p_1$ and $p_\infty$ were previously defined. There is probably a simple argument to come to that result, but I think that some details might help the reader here;

18- Page 5, paragraph 4; I would replace "We get Eq.4 becomes" by " Eq.4 then becomes";

19- Page 5, paragraph 4; I think that precisions on $Q[\gamma]$ expression could make the result clearer;

20- Page 6, paragraph 2; I would replace "One focuses on" by "One can focus on";

21- Page 6, paragraph 2; I would replace "knowledge on the system" by "knowledge about the system";

22- Page 6, paragraph 2; I would replace "An ideal feedback is perfectly defined by the memory state" by "A feedback is said to be ideal if it is perfectly defined by the memory state", to make it clearer that it is how we define an ideal feedback (which I believe is this sentence purpose);

23- Page 6, paragraph 3; I would replace "The same kind of argument as developped for the IFT" by "The same kind of argument developped for the IFT" or "The same kind of argument as those developped for the IFT";

24- Page 6, paragraph 3; I would replace "rigorous path to exorcize" by "rigorous way to exorcize";

25- Page 6, paragraph 3; I would replace "run reversibly" by "reversibly run";

26- Page 6, paragraph 4; I would replace "engine" by "heat engine", because I think that macroscopic thermodynamics firstly focused on heat engines;

27-Page 7, paragraph 3; I would replace "What are irreversibility, work and heat in the quantum realm? is one of the most important questions." by "One of the most important questions is to know what are irreversibility, work and heat in the quantum realm ", or put quotation mark;

27-Page 7, paragraph 3; I would replace "coherence" by "coherences", since "correlations" are plural, for consistency;

28- Page 7, paragraph 4; I would replace "driven quantum systems interacting with one or several baths" by something like "quantum systems interacting with a driven force and one or several baths" to emphasise the fact that there is a driving system on one side, and baths on the other;

29- Page 7, paragraph 4; I would replace "the action of the drive" by "drive's action";

30- Page 7, paragraph 4; I would replace "Hence, the drive can exchange energy with the system," by "Hence, the drive can unitarily exchange energy with the system,", in order to insist "unitary / non-unitary" dichotomy between drive and baths;

31- Page 7, paragraph 5; I would replace "Quantum measurement pertubs" by "Quantum measurement perturbs the system", in order to make the entence smoother (it is true that quantum mechanics might be disturbing, but here we mainly focus on the system);

32- Page 7, paragraph 6; I would replace "Knowing the outcomes of the measurement" by "On the other hand, knowing the measurement outcomes" to fit better with the "on the one hand" structure from the previous sentence;

33- Page 8; paragraph 1; I would replace "the trajectory of pure quantum states" by "the trajectories of pure quantum states", since we have several quantum states, I also expect several trajectories;

34- Page 8, paragraph 1; I would replace "that provide" by "which provided", for temporal consistency;

35- Page 8, paragraph 2; I would replace "the expectation value of the Hamiltonian" by "Hamiltonian expectation value"

36- Page 8, paragraph 2; I would replace "on the other hand, being identified with the sudden energy changes during the quantum jumps." by "or can be indentified with the sudden energy changes during the quantum jumps" or "which corresponds to the sudden energy changes during the quantum jumps" depending on what you meant;

37- Page 8, paragraph 3; I would replace "of the observable $\hat{M}$ " by " of an observable $\hat{M}$", because at this point it can by any observable;

38- Page 8, paragraph 3; I would replace "a unitary evolution defined by the operator $\hat{U}$" by "an unitary evolution $\hat{U}$", since we did not say that $\hat{M}$ is an operator, it is not needed for $\hat{U}$ (and it is "an unitary" I guess);

39- Page 8, paragraph 3; I would replace "probability $p_k$ by "probabilities $p_k$"; for consistency, since we said "pure states" right before;

40- Page 8/9 paragraph 1; I would replace "the backward protocol [...] and performing a final measurement" by a sentence like "Reciprocally, the bacwkard protocol is defined as follows. Observable $\hat{M}$ is measured on the system, in state $\rho$. Then, the system follows the backward evolution $\hat{U}^\dagger$ after which a final measurement is performed". Defining the backward evolution is a key point here, and I think that a smoother explanation with two sentences might be more effective;

41- Page 9, paragraph 1; I would replace " 'how much irrversible' a measurement is" by " 'how reversible' a measurement is" or "how much urreversibility a measurement creates", since "irreversible" is not a quantity;

42- Page 10, line 4; a reference is missing "[?]";

43- Page 10, line 5; I would replace the coma after $\ket{1}$ by "and" and add a coma after $\omega_0$;

44- Page 10, paragraph 1; I would replace "slope of the Rabi" by "slope for the Rabi";

45- Page 10, paragraph 2; I would replace "extract work cyclically" by "cyclically extract work";

46- Page 10, paragraph 4; I would replace "is its yield" by "is its yield $\eta$", since this notation is used in the following ;

47- Page 11, paragraph 3; I would replace "why not ? " by " – why not ? –" for clarity;

48- Page 11, paragraph 4; I would replace "being more specific" by "to be more specific";

49- Page 11, paragraph 4; I would replace "Th quantum state of the register" by "The register quantum state";

50- Page 11, paragraph 4; I would replace "the agent performing the computation records the outcome, and then reverses" by "the agent performing the computation can record the outcome, and then reverse", since all this process is hypothetical;

51- Page 11, paragraph 4; I would replace "the register to its initial state" by "the register into its initial state";

52- Page 11, paragraph 5; I would replace "to run it is to let the qubit interact resonantly" by "to run it is reached when we let the Qbit resonantly interact " for clarity and consistency;

53- Page 12, paragraph 1; I would replace "at least 1000 photons in the field to keep" by "at least 1000 photons are needed in the field to keep";

54- Figure 2; a point is missing after "(Rabi oscillations)."

55- Figure 2; I would replace "Maxwell?s demon" by "Maxwell's demon", typing error I guess;

56- Figure 2; I would replace "while it is periodically measured" by "while being periodically measured";