Optimization problems involving quantum relative entropies are ubiquitous in quantum information theory. Some prominent examples include the optimization of the fidelity function or the Umegaki relative entropy over a set of states satisfying some specific constraints. In our work, we address this general problem and show that our methods are particularly useful in investigating the additivity properties of a wide class of monotones based on quantum relative entropies. In particular, we develop a very general ansatz-based technique for a wide range of quantum relative entropies that allows us to recast the initial convex optimization problem into a simpler linear one. This method could potentially have many other applications in information theory. We then show that our methods are useful to derive closed-form solutions and new additivity statements of monotones based on quantum relative entropies for several resource theories.
We first consider entanglement theory where we show that the relative entropy of entanglement is additive when at least one of the two states belongs to some specific class. We show that these classes include bipartite pure, maximally correlated, GHZ, Bell diagonal, isotropic, and generalized Dicke states. Previously, additivity was established only if \textit{both} states belong to the same class. Moreover, we extend these results to entanglement monotones based on the \alpha–z Rényi relative entropy. Notably, this family of monotones also includes the generalized robustness of entanglement and the geometric measure of entanglement. Finally, we prove that any monotone based on a quantum relative entropy is not additive for general states.
This result found application in the catalytic transformation of entanglement where to derive fundamental limits, no dimensional or purity assumption can be made on the catalyst state. In this setting, the additivity properties of the monotones based on the \alpha–z Rényi relative entropies are tightly connected with their role in characterizing catalytic state conversions. In particular, different monotones for different values of the parameters have different roles. Therefore, it is crucial to completely characterize the additivity properties of these monotones for all the values of the parameters. In the case of correlated catalysis, i.e. when there are residual correlations between the system and the catalyst at the end of the process, the currently known protocols use catalysts which, for small residual correlations, are typically highly dimensional. Our results allow us to establish fundamental limits on correlated catalytic transformations, which hold for any protocol and do not require any assumption on the structure of the catalyst.
Moreover, we consider the resource theory of magic. The resource theory of magic provides a general resource theory framework to quantify the advantage of quantum computation over classical computation. In this setting, free resources lead to efficiently classically simulable computation, while resource states or `magic states’ provide the quantum advantage over the classical counterpart leading to universal quantum computation. In the resource theory of magic, the set of free states is the set of stabilizer states. All the states that are not stabilizer states are called ‘magic states’.
In this resource theory, very little was known about the additivity of magic monotones based on quantum relative entropies for general mixed states. This is due to the fact that for mixed states, unlike the fidelity when the input state is pure, these monotones are not linear. However, as mentioned above, our methods allow us to recast the original problem for mixed states into a linear one. This. in turn, allows us to provide new results for this general case. We first prove that the stabilizer fidelity is multiplicative for the tensor product of an arbitrary number of single-qubit states. We then show that the relative entropy of magic becomes additive if all the single-qubit states but one belong to a symmetry axis of the stabilizer octahedron. We extend the latter results to include all the \alpha–z Rényi relative entropy of magic. This allows us to identify a continuous set of magic monotones that are additive for single-qubit states and obtain tighter lower bounds for the overhead of probabilistic one-shot magic state distillation. Moreover, we recover some already-known results and provide a complete picture of the additivity properties for single-qubit states for a wide class of monotones based on quantum relative entropies. We also derive a closed-form expression for all single-qubit states for the stabilizer fidelity and the generalized robustness of magic. Finally, we show that all the monotones mentioned above are additive for several standard two and three-qubit states subject to depolarizing noise, for which we give closed-form expressions.
More details of these two projects can be found at Entanglement theory and Resource theory of magic. The first work was presented in TQC23.