To introduce our approach to the problem of quantifying work, I will use a language that might be unfamiliar from the usual physics literature. It is inspired from computer science
and tries to describe a problem as an interaction between two agents, one of which, called Merlin, is very powerful and the other one, called Arthur, is limited in his capabilities. I want to stress, however, that the language captures the usual thermodynamic setting and bounds the very same quantities usually under consideration in thermodynamics. This will become clear in the later sections.
Consider the two players, Arthur and Merlin living in a world of inverse temperature β. They both agree upon the fact that the environment has inverse temperature β. They also agree that this fact implies the existence of a set of "(catalytic) free transitions", i.e., a set of transitions on physical systems that can be implemented without additional resources. This should be understood in the sense of chapter 2, but in fact all our results will be a function of the choice of model for the free transitions and therefore other models than thermal operations can be considered as well (in section 14.3 I discuss in detail which are the minimal properties of the free operations that are explicitly needed to derive our results). For the purpose of this chapter, let us introduce the following notation. We will denote by
F (p)the set of objects that can be reached from p by any free transition. Similarly, we will denote byFC(p)the set of objects that can be reached by a catalytic free transition.
In mathematical terms, this means thatq∈ FC(p)whenever there is a catalystc such that
q⊗c∈ F (p⊗c). Here, we introduced the notation
(ρ, H) ⊗ (σ, K):= (ρ⊗σ, H⊗1+1⊗K). (4.3)
Merlin now claims to own a machine which can perform thermodynamically useful tasks and trades quantum systems for a living. Arthur is one of his customers whose system takes the role of the work-storage device.
We then imagine that Arthur hands to Merlin his system described by p= (ρ, H) ∈ P
and leaves again. Later, Arthur comes back and Merlin returns the system described by a different objectq ∈ P (along with the classical description of the state). The new state might be more or less useful for Arthur. In general Arthur will have to pay some money to Merlin (or vice-versa if Merlin made the system less useful for Arthur). The two thus have to find a fair agreement on how they value such state-transitions. We will denote this price by
W (p→q, β) ∈R. (4.4)
Note thatW is a function that associates a real number only to a transition between valid work-storage devices, that is objects inP. No price is associated if Merlin does not return a valid work-storage device. We will make the convention thatW ≥0 if Arthur has to pay money to Merlin and will refer toW as a work quantifier.
Importantly, Arthur does not know how exactly Merlin performed the transition and it is completely unimportant to him as he is only interested in what happens to his system. This is in analogy to the case of mechanical work or cooling: One can check whether (and how much) work a machine performs without knowing the internal details of the machine by simply looking at how the machine acts on auxiliary systems. It is important to understand that this is not in contradiction with the fact that thermodynamic work is a path-dependent quantity from the viewpoint of the machine, as is explained in Fig. 4.2.
As stated above, Arthur and Merlin have to agree on a "fair" agreement on the work- quantifierW. Such an agreement is of course subjective in principle and hence has to be assumed. The rest of this chapter will be devoted to establishing natural and mathematically precise Axioms that any valid work-quantifier should fulfill and derive from these Axioms a series of properties for any such work-quantifier. Before we come to the precise Axioms, however, we have to introduce a few more important concepts.
4.3.1
Work of transition, work cost and work value
The work-quantifierW evaluates transitions on Arthur’s work-storage device. It is easy to confuse this quantity with other quantities in thermodynamics. After having definedW
one is often interested in the optimal transition (as measured byW) that one can imple- ment on a work-storage device given some non-equilibrium resourcep(i)M. In other words, suppose that Merlin has, inside of his lab, a system described by the objectp(i)M but no other
A Q UA N T U M O F T H E R M O DY N A M I C S 47 Work-storage device t mgh(t) W = mg∆h Machine V p W = mg∆h Fuel initially Catalyst/Machine Fuel finally Work-storage device Mass m ∆h
Figure 4.2: Left: Phenomenological analogy of the setting in the case of a machine that burns fuel to perform mechanical work. The catalyst corresponds to the machine that returns to its initial state, using up burning fuel to lift a weight. The burning fuel corresponds to Merlin’s non- equilibrium system and the lifted weight corresponds to Arthur’s work-storage device. Right: Work from two points of view. Path-dependent work obtained by looking at the time-dependent thermodynamic state of the thermal machine at the top and operational path-independent work obtained by looking at the weight (work-storage device) at the bottom. All processes happen at some background-temperatureT. The work of transition Wtransof the fuel corresponds to the maximal height that the weight can be lifted by arbitrary machines leaving the fuel in the corresponding final state and operating at background-temperatureT. (Figure from Ref. [2].)
resources. Then it is interesting to know how much money he can earn if a customer comes with a work-storage device. This corresponds to what is often called the "extractable work" or "work value" in the literature [47, 49, 50, 70, 150]. Similarly, we could not only fix the initial, but also the final objectp( f )M and ask how much Merlin can in principle earn from a transition between the two states. We will call this quantity the work of transition [47].
Definition 4.1 (Work of transition). Given a work-quantifierW, an inverse temperature β, a class of work-storage devicesP, and initial and final objectsp(i)M andp( f )M , thework of transition is defined as
Wtrans(p(i)M →p( f )M, β):= sup p(i)A,p(fA)∈P; p(fM)⊗p(f)A ∈FC(p(i)M⊗p(i)A)
W (p(i)A →p( f )A , β). (4.5)
According to our convention, if the work of transition is positive, then Merlin receives money from Arthur. It is important to realize that the work of transition depends on all possible catalytic free transitions with all possible work-storage devices that can be im- plemented in such a way that the transition on Merlins non-equilibrium resource matches the given one. In contrast toW, the work of transitionWtrans is not defined on the work-
storage device, but on the transition on Merlin’s system. Its input objects thus do not have to be inP. In fact the work (as measured byW on the work-storage device) associated to some process can never be deduced from the initial and final state onM alone. Thus it is impossible to defineW as a function on transitions p(i)M → p( f )M. One either needs to specify the precise process including the work-storage device or, if one has only access to M, consider the optimal possible value as in (4.5). It is precisely in this sense that work, as a function of transitions on M, is a path dependent quantity when evaluated in transitions onM, and a path-independent quantity when evaluated in transitions on A. This is also the case in phenomenological thermodynamics: work can be specified by knowing only the initial and final height of the lifted weight, however it is path-dependent as function of the state of the machine (see Fig. 4.2).
After having defined the work of transition, we can also define the work value and the work costas the maximum possible value that Merlin can obtain frompMand the minimum
possible value that he has to pay to be able to createpM, respectively:
Wvalue(pM, β):=Wtrans(pM→ωβ, β), (4.6)
wherewβdenotes an object representing a system in thermal equilibrium. While both these
quantities are important in thermodynamics, it is clear that they can only be defined once a work-quantifierW has been established, which is the primary concern in this chapter. We will however see, that our Axioms implyWvalue ≤ Wcost, which can be seen as an
expression of the second law of thermodynamics.
Finally, note that in (4.5), it is assumed that the work-storage device and the system of Merlin are uncorrelated both in the beginning and the end of the transition. This is in fact important and in section 4.7 we will discuss in detail the role of correlations betweenM andA.