Towards the simulation of higher-order quantum resources: a general type-theoretic approach
Samuel B. Steakley, Elia Zanoni, Carlo Maria Scandolo
TL;DR
This work addresses the challenge of formulating a uniform theory for higher-order quantum resources, spanning all levels of the resource hierarchy. It introduces a type-theoretic framework with a family of spaces $\textsf{L}(x)$ for each type $x$, a generalized parallel product $\boxtimes$ that extends the tensor product to arbitrary orders, and a family of convex cones $\mathsf{K}(x)$ that generalize complete positivity to all orders. The authors show how $\mathsf{K}(x)$ forms convex cones isomorphic to positive semidefinite cones via a Choi-like correspondence, and they establish foundational properties such as closure under composition and parallel product. The resulting framework enables a unified analysis of static, dynamical, and higher-order quantum resources, with applications to indefinite causal order and quantum resource theories, and it provides tools for extending convex optimization techniques to higher-order maps. Overall, the paper lays down a scalable mathematical foundation for studying, interconverting, and optimally exploiting higher-order quantum protocols across all orders.
Abstract
Quantum resources exist in a hierarchy of multiple levels. At order zero, quantum states are transformed by linear maps (channels, or gates) in order to perform computations or simulate other states. At order one, gates and channels are transformed by linear maps (superchannels) in order to simulate other gates. To develop a full hierarchy of quantum resources, beyond those first two orders, and to account for the fact that quantum protocols can interconvert resources of different orders, we need a theoretical framework that addresses all orders in a uniform manner. We introduce a framework based on a system of types, which label the different kinds of objects that are present at different orders. We equip the framework with a parallel product operation that modifies and generalizes the tensor product so as to be operationally meaningful for maps of distinct and arbitrary orders. Finally, we introduce a family of convex cones that generalize the notion of complete positivity to all orders, with the aim of characterizing the objects that are physically admissible, facilitating an operational treatment of quantum objects at any order.