General state transitions with exact resource morphisms:
a unified resource-theoretic approach

Many problems in information theory and statistics (and quantum generalizations thereof) can be reformulated as to whether a suitable transition between two objects is possible under suitable constraints. For example, noisy channel coding in information theory can be understood as the problem of transforming a number of uses of a noisy channel into less uses of a less noisy (ideally noiseless) channel, under the constraint that no side communication is available between sender and receiver. Another example is provided in mathematical statistics by the task of extracting statistics from a given sample. The exact same intuition becomes paradigmatic in statistical mechanics, most notably in thermodynamics, where a typical problem, for example, is to whether a thermodynamic process satisfying certain constraints (e.g., an isothermal process, an adiabatic process, etc.) exists between two given states.

Whenever the set of allowed transformations is constrained for some reasons, it is natural to consider the possibility of performing a forbidden transformation as a resource DHW (08); HHHH (09); CFS (16); CG (19). The state space of the system (that is, the set of objects that are being transformed) thus ends up being divided into free states (that is, states that can be reached from any other state by means of allowed transformations) and non-free states. The latter are implicitly defined to be the resourceful states in the theory. Notice that the word “state” here is not to be meant in its strict sense of “state of a physical system”, but denotes more generally the objects that are being transformed. These could comprise states proper, like in conventional thermodynamics, but are not limited to these. One could consider as objects, for example, statistical models (as in mathematical statistics), or noisy channels (as in information theory), or entire generalized operational theories for that matter.

In fact, it turns out that it is often easier to study resource theories, in which the first thing being defined is not the set of allowed transformations, but rather the set of free states, while allowed transformations are implicitly defined only later, as all those transformations that map the free set into itself. Even though sometimes less clear operationally, such an approach (that we may call “geometric”, in order to contrast it with the previous “operational” one that puts constraints on allowed operations) has, from a purely mathematical viewpoint, various advantages: for example, resource theories typically become asymptotically reversible only if the set of allowed transformations is taken to be the maximal one (plus epsilon) compatible with the theory BP (08); BG (15).

In this paper we study general resource theories by following the geometric approach, that is, by building the theory on top of a given set of free states $\mathsf{F}$, that we in particular assume to be non-empty, closed, and convex. Accordingly, we allow any transformation that maps $\mathsf{F}$ into itself. For the sake of generality, in this paper we do not even assume any particular rule of composition for free states, so that we essentially work in the single-shot regime, but we still allow the input and the output systems, and their respective free sets, to change as a consequence of the transformation. In this way, even though we cannot say anything about asymptotic rates, we can still discuss in full generality single-shot rates for resource distillation and dilution, for example.

The main aspect that differentiates our work from previous ones is that here we do not focus only on the tasks of resource dilution and distillation. Instead, we consider the more general problem of formulating sufficient (and, in some cases, necessary) conditions for the existence of an $\mathsf{F}$-preserving transformation between an arbitrarily given input–target pair of states. This means that the analysis presented here does not rely on the existence of any privileged maximally resourceful state (like the maximally entangled state in bipartite entanglement theory) and thus applies to quite general resource theories.

Concerning the kind of transformations that we consider in this work, two remarks are in order. First, while we allow for noisy transitions, that is, situations in which the input state is mapped not exactly but “close” to the wanted target, we only allow transitions implemented by operations that are exactly $\mathsf{F}$-preserving. This stands in contrast to other single-shot analyses of general resource theories in which “almost-free” operations are allowed in the finite block-length regime (see, e.g., Ref. BG (15)). Second, we search for conditions that are expressed in terms of resource monotones (a notion to be rigorously introduced in what follows), computed separately for the initial state and for the target state. On the one hand, in this way we are able to “decouple” the initial state from the target state, and to speak of their respective resource’s worth, independently of each other. On the other hand, this means that functions that comprise both states at once, like those that in some situations can be obtained by means of semidefinite linear programs BG (17); GJB⁺ (18), will not be considered in this work. Another aspect of the problem that is not considered in this work is the computational complexity of computing resource monotones.

The paper is structured as follows. After introducing the notation and reviewing some basic notions in Section II, we present our main results and prove them in detail in Section III. Section IV presents some applications of our general analysis to specific cases of physical interest: applications to the resource theories of athermality and asymmetric distinguishability (Section IV.1); bipartite entanglement (Section IV.2); conditions for the existence of maximally resourceful states, together with necessary and sufficient conditions for dilution and distillation which can give, whenever a dimension scaling is provided, optimal dilution and distillation rates (Section IV.3). Section V concludes the paper.

We denote by $\mathsf{D}(\mathbb{C}^{m})$ the set of all $m$-by-$m$ complex density matrices $\rho$, i.e., $\rho\geqslant 0$ and $\operatorname{Tr}\left\{\rho\right\}=1$, which are used here to represent quantum states of $m$-dimensional quantum systems. Within $\mathsf{D}(\mathbb{C}^{m})$, we identify a non-empty closed convex subset $\mathsf{F}$ as the set of “free states”. The closure and the convexity of $\mathsf{F}$ is crucial in various steps of our proofs, for example, when invoking the closure under convex mixtures, or when applying a variant of the minimax theorem that requires convex domain. Here and throughout this work, resource morphisms (or more precisely $\mathsf{F}$-morphisms¹¹1Here we prefer the term “resource morphisms” to the more common “free operations” because it reminds the fact that the foundational concept, in the geometric approach in which we are working, is the free set $\mathsf{F}$, not the set of allowed transformations, which are just defined as all those that map $\mathsf{F}$ into itself.) are defined as completely positive, trace-preserving (CPTP) linear maps $\mathcal{E}:\mathsf{D}(\mathbb{C}^{m})\to\mathsf{D}(\mathbb{C}^{m})$ such that $\mathcal{E}(\mathsf{F})\subseteq\mathsf{F}$. More generally, one may consider CPTP maps that change the dimension of the system, for example, $\mathcal{E}:\mathsf{D}(\mathbb{C}^{m})\to\mathsf{D}(\mathbb{C}^{n})$. Also in this case, whenever the output free set $\mathsf{F}^{\prime}$ is also specified, it is possible to define a notion of resource morphisms by the condition that $\mathcal{E}(\mathsf{F})\subseteq\mathsf{F}^{\prime}$. However in what follows, for the sake of readability, we will restrict ourselves to the case of equal input and output dimensions and $\mathsf{F}=\mathsf{F}^{\prime}$, keeping in mind however that all the results we derive can be straightforwardly extended to the general case. We will go back to the more general setting, with different input and output systems, in Section IV when discussing various applications like the tasks of resource dilution and distillation.

The general structure that we study in this work is the following:

In the above definition, the preorder $\succ_{\epsilon}$ has been introduced with respect to the distance induced by the trace-norm, although it is possible to use any other well-behaved distance measure between density matrices (like the fidelity, for example) without substantially changing the results NC (00); Wil (13). In any case, an important thing to notice in Definition 1 is that, even though errors are allowed in the state transformation, we always require the constraint $\mathcal{E}(\mathsf{F})\subseteq\mathsf{F}$ to be strictly satisfied.

The resourcefulness preorder naturally lead us to define a maximally resourceful element as follows:

Given a general resource theory, an important question is to whether the theory possesses maximally resourceful elements or not. In Section IV.3 we will consider sufficient conditions for their existence. However, we recall that our main results do not rely in any way on the existence of maximally resourceful elements.

In this section, we state and prove the main results of this paper. Firstly we derive, for any finite-dimensional resource theory in which the set of free states is non-empty closed and convex, sufficient conditions for the existence of a resource morphism between any two states, given in terms of resource monotones. Such conditions are formulated so to allow, in general, non-zero errors in the state transition, while the operation implementing the transition is an exact resource morphism.

Condition (17) is stronger than condition (16), in the sense that if the former is satisfied, the latter is also satisfied. This is because, by multiplying both sides by $2^{D_{\max,\mathsf{F}}(\tilde{\sigma}\|\tilde{\sigma}_{+})}-1>0$ (see preceding remark), condition (16) becomes

and this, if $\max_{\omega\in\mathsf{F}}{D}_{h}^{\epsilon_{1}}(\rho\|\omega)=\min_{\omega\in\mathsf{F}}{D}_{h}^{\epsilon_{1}}(\rho\|\omega)$, becomes equivalent to

which is always trivially satisfied, due to the non-negativity of the hypothesis testing relative entropy. In other words, we have shown the following:

Corollary 1, for rank-one $\rho$ and $\epsilon_{1}=0$, recovers Theorem 2 in Ref. LBT (19).

A less general, but simpler, statement stemming from Theorem 1 is the following:

A merit of Corollary 2 is to provide a simple compact sufficient condition, free of supplementary caveat like condition (16), which is difficult to interpret operationally. However, the right-hand side of (36) is not yet a valid resource monotone. The following result fills the gap.

Theorem 2, when $\epsilon_{2}=0$ and $\sigma$ is rank-one, recovers Theorem 5 in Ref. LBT (19) (see also Corollary 17 of RBTL (19)).

Theorem 1 and Theorem 2 are independent of each other. This is because, on the one hand, it is possible that $\mathfrak{D}^{\epsilon_{1}}_{h}(\rho)\geqslant\mathfrak{D}_{\max}^{\epsilon_{2}}(\sigma)$ even though $\mathfrak{D}^{\epsilon_{1}}_{h}(\rho)\centernot{\geqslant}\mathfrak{D}_{\max,\mathsf{F}}^{\epsilon_{2}}(\sigma)$, so that Theorem 2 would be inconclusive. On the other hand, it is possible that $\mathfrak{D}^{\epsilon_{1}}_{h}(\rho)\geqslant\mathfrak{D}_{\max,\mathsf{F}}^{\epsilon_{2}}(\sigma)$ even though neither condition (16) nor (17) hold, so that Theorem 1 would be inconclusive. In other words, Theorem 1 and Theorem 2 in general apply to two different regimes and are logically independent of each other. Nonetheless, since $\tilde{\sigma}\in\mathsf{B}^{\epsilon_{2}}(\sigma)$ and $\tilde{\sigma}_{+}\in\mathsf{F}$, we see that $D_{\max,\mathsf{F}}(\tilde{\sigma}\|\tilde{\sigma}_{+})\geqslant\mathfrak{D}_{\max,\mathsf{F}}^{\epsilon_{2}}(\sigma)$. This implies that Corollary 2 above can be as well derived as a consequence of Theorem 2.