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Suppose that, in our theory, the agent has access to the energy and information batteries introduced in the previous section. We are now interested in studying the interconversion of these two resources, which can be realised only if an additional system, the bank, is introduced, see Ch. 3 for further details. In particular, we need to identify the set of states which can be used to describe the bank. However, it is worth noting that in the current formulation of the multi-resource theory, there seems to be some arbitrariness in the definition of this set, which is composed of those states with the minimum possible values of both resources, see Eq. (3.28) in the previous chapter. Indeed, the operations the agent can use always preserve energy and information, so that storing and extracting any one of them seem to be equally important. For example, we have already noticed that, in this theory, entropy and information are equivalent, although they actually quantify the same resource in opposite way.

In order to unambiguously define the bank system, we now consider a multi-resource theory whose allowed operations are a superset of Athermo. This theory describes the situation in which the agent is acting on a closed system, and has a coarse-grained control over the unitary operations they can perform. Thus, the operations they implement are described by mixtures of unitary operations, a strict subset of the unital maps. Furthermore, we assume that the system can exchange energy (but no other quantities) with an ancillary system which acts as an energy sink, so that the operations the agent performs cannot increase the energy of the system. As a result, the allowed operations we consider in this section are given by the intersection between the set of mixtures of unitary operations, and the set of average-energy- not-increasing maps, introduced in Sec.3.2.3of the previous chapter. Within this new resource theory, energy and information are the resources required to perform state transformations, since they never increase under the allowed operations. Clearly, this resource theory satisfies the asymptotic equivalence property, since the class of allowed operations is larger than the

one of energy-preserving unitary operations. Then the bank states are the ones with minimum values of information and energy, and it is easy to show, for example by using the method of Lagrange multipliers, that this set is given by the Gibbs state of the system Hamiltonian at a positive temperature, Fbank=  τβ ∈ S (H) | τβ = e−β H Z , ∀ β ∈ R +  . (4.54)

The inverse temperature β is a continuous label for this set, and for each fix value of this parameter we obtain a different bank, with a different exchange rate.

Following the procedure in Sec.3.3.1 of the previous chapter, we can now define the bank monotone associated with the state τβ. We can use the fact that the bank monotone is repre- sented, in the energy-information diagram, as a tangent to the state space in the point associ- ated with τβ, see Fig.4.3. We recall that the von Neumann entropy of a thermal state can be expressed as S(τβ) = β E(τβ) + log Z. Using this information, is is easy to show that

dI dβ = β h∆ 2_Hi β, (4.55a) dE dβ = −h∆ 2_Hi β, (4.55b)

where h∆2Hiβ = TrH2τβ − (Tr [H τβ])2 is the variance of the energy over the system. The linear coefficient of the tangent line we are interested in is then given by

dI dE = dI dβ  dE dβ −1 = −β. (4.56)

The absolute value of this coefficient gives the exchange rate at which the agent can inter-convert the resources. Furthermore, with this linear coefficient we can define the bank monotone,

f_bankβ (ρ) = (E(ρ) − E(τβ)) + β−1(I(ρ) − I(τβ)) = F (ρ) − F (τβ), (4.57) where the last equality follows from the definition of Helmholtz free energy.

Resource interconversion is obtained, in this theory, through the following transformation, τ_β⊗n⊗ ωE(k) ⊗ ωI(h)

asympt

←−−−→ ˜τ_β⊗n⊗ ωE(k0) ⊗ ωI(h0), (4.58) where ∆k = k − k0 and ∆h = h − h0 are finite, while we send the number of copies of the bank state, n, to infinity, so that

  f β bank(τβ) − f β bank(˜τβ)  

Figure 4.3: The state space S (H) is represented in the energy-information diagram (blue region), together with the line (in red) tangent to this convex region in the point associated with the bank state τβ. The tangent line represent the set of states with bank monotone equal to zero. The bank monotone for the thermodynamic theory is proportional to the Helmholtz free energy, see Eq. (4.57), which is indeed a linear combination of the two monotones we are considering, energy and information. The absolute value of the linear coefficient of the line is the exchange rate at which the agent can inter-convert the resources, and from the figure it is clear that this rate changes when we change the state of the bank to be at a different temperature – since the tangent line in the new point as a different slope.

From the transformation of Eq. (4.58), we find that the interconversion relation of this theory is given by

∆WE = −β−1∆WI. (4.59)

This equation regulates the amounts of resources that the agent can exchange using a bank. As expected, the agent needs to provide one resource in order to extract the other, and the exchange rate is given by the inverse temperature, i.e., by the linear coefficient of the tangent line in Fig. 4.3. As already mentioned, examples of resource interconversion are Landauer’s erasure, where energy is traded with a thermal reservoir in order to gain neg-entropy, which in turn is used to erase the state of an unknown bit, and the Maxwell demon, where information

is traded with the reservoir in order to extract energy.

Let us now consider the relative entropy distance from the bank state. It is easy to show that Cor. 2 is satisfied in the multi-resource theory we are studying, and therefore the bank monotone coincides (modulo a multiplicative constant) with the relative entropy distance from τβ, which is

Eτβ(ρ) = D(ρ k τβ) = β E(ρ) − β

−1_{S(ρ) + β}−1 _{log Z
= β f}β

bank(ρ). (4.60)

This monotone is a measure of athermality for states, meaning that conditionX1 introduced in the previous chapter corresponds, in this multi-resource theory, to the demand that the bank system only changes its athermality by an infinitesimal amount.

We conclude the section with the derivation of the First Law of Thermodynamics within the multi-resource theory we are considering. Recall that, as shown in Cor. 3, an agent who has access to bank and batteries can modify the state of the main system if the amounts of resources exchanged satisfy a single relation, which is the first law for general multi-resource theories, see Eq. (3.49). This relation, for our theory, is given by

∆WE + β−1∆WI = F (ρ) − F (σ), (4.61) which tell us that an asymptotic transformation mapping ρ into σ can be achieved if the weighted sum of the resources ∆WE and ∆WI exchanged with the batteries is equal to the athermality change in the system. Notice that the weight in the lhs is given by the inverse of the exchange rate, see Eq. (4.59), and it is proportional to the temperature of the thermal reservoir (the bank) which the agent has access to.