Let us consider the multi-resource theory Rmulti introduced in Sec. 3.1, describing a physical task where m constraints and conservation laws are present. We are interested in studying whether the theory is reversible, i.e., whether no resources are lost during an arbitrary cyclic transformation. However, to study this notion of reversibility, one first needs to be able to quantify the amount of resources exchanged in a state transformation. In general, as we have seen in Sec. 1.2 of the background chapter, each resource is quantified by several monotones, and there is not a unique way to assign a value to each state. This reflects the fact that, in resource theories, we can define a partial order between states, rather than a total one.
For single-resource theories, we have shown that if a theory is reversible (in terms of rate of conversion), then there exists a unique quantifier for the resource exchanged during each state transformation, see Thm. 1. However, this result does not apply to multi-resource theories, mainly because defining a rate of conversion for these theories does not seem to be always possible. Indeed, a rate of conversion can be defined only if the theory has a non-empty set of free states, see Def. 15, since the number of copies of the system before and after the transformation are allowed to change. For example, being able to map n copies of ρ into k copies of σ, with n < k, implies that we have the possibility to add k − n copies in a free state to the initial n copies of ρ, and to act globally to produce k copies of σ. In multi-resource theories, the set of free states can be empty, see for example the invariant set structure of the right panel of Fig.3.1, and therefore we cannot define a rate of conversion, nor we can use the results of Thm.1about the uniqueness of the resource quantifier.
For this reason, we start our investigation of reversible multi-resource theories by demanding the following property, which is related to the notion of “seed regularisation” of Ref. [16, Sec. 6],
Definition 30 (Asymptotic equivalence). The multi-resource theory Rmulti satisfies the asymp- totic equivalence property if there exists a set of monotones {fi}mi=1, where each fiis a monotone for the corresponding single-resource theory Ri, such that, for all ρ, σ ∈ S (H), we have that the following two statements are equivalent,
• There exist a sequence of maps {˜εn: S (H⊗n) → S (H⊗n)}n such that lim n→∞ ˜εn(ρ⊗n) − σ⊗n 1 = 0, (3.3)
as well as a sequence of maps performing the reverse process. The maps {˜εn} are defined as ˜ εn(·) = TrA h εn(· ⊗ ηn(A)) i , (3.4)
where A is an ancilla composed by a sub-linear number o(n) of copies of the system, and it is described by an arbitrary state η(A)n ∈ S H⊗o(n), such that fi(η(A)n ) = o(n) for all i = 1, . . . , m. The map εn∈ A
(n+o(n))
multi is an allowed operation of the multi-resource theory.
Here, fi∞ is the regularisation of the monotone fi, and k · k1 is the trace norm, see Def. 7. When a multi-resource theory satisfies the above property, we have that all asymptotic state transformations are regulated by the values of specific monotones (one for each resource), which can be used to quantify the resources. Then, given a theory that satisfies this property, we can study reversibility, since we have a well-defined notion of resources. An example of a multi-resource theory that satisfies asymptotic equivalence is thermodynamics (even in the case in which multiple conserved quantities are present), as shown in Refs. [169,168]. We consider this multi-resource theory in the next chapter.
It is worth noting that, in the above property, we are allowing the agent to act over many copies of the system with more than just the set of allowed operations; we assume the agent to be able to use a small ancillary system, sub-linear in the number of copies of the main system. Roughly speaking, the role of this ancilla is to absorb the fluctuations in the monotones fi∞’s during the asymptotic state transformation. It is important to notice that this ancillary system only contributes to the transformation by exchanging a sub-linear amount of resources. Thus, its contribution per single copy of the system is negligible when n 1, which justifies the use of this additional tool.
Few comments are in order about the meaning of this property. First, the asymptotic equivalence property implies that the state space can be divided into different equivalence classes of states. Each class is labelled by the value of the regularised monotones fi∞’s, and within these classes we can freely move between states in a reversible manner, since we are
not consuming any resource. Secondly, the property only refers to the transformations between states with the same values of these monotones. To study the asymptotic transformations between states with different values of the monotones fi∞’s, we need to introduce the notion of a battery, see the next section. Finally, while the above property allows us to focus on the sole monotones fi∞’s when studying asymptotic state transformations, it alone does not seem to imply the existence of unique resource quantifiers. For example, a priori one might think that other monotones gi’s exist which have constant values over the same equivalence classes singled out by the monotones fi’s, but order these classes in a different way. However, we show in Sec. 3.2.3that, when the monotones satisfy some natural assumptions, they are the unique quantifiers for a theory satisfying asymptotic equivalence.