Analyzing the free states of one quantum resource theory as resource states of another

In the context of quantum resource theories (QRTs), free states are defined as those which can be obtained at no cost under a certain restricted set of conditions. However, when taking a free state from one QRT and evaluating it through the optics of another QRT, it might well turn out that the state is now extremely resourceful. Such realization has recently prompted numerous works characterizing states across several QRTs. In this work we contribute to this body of knowledge by analyzing the resourcefulness in free states for–and across witnesses of–the QRTs of multipartite entanglement, fermionic non-Gaussianity, imaginarity, realness, spin coherence, Clifford non-stabilizerness, $S_n$-equivariance and non-uniform entanglement. We provide rigorous theoretical results as well as present numerical studies that showcase the rich and complex behavior that arises in this type of cross-examination.

💡 Research Summary

This paper investigates a fundamental question in quantum resource theories (QRTs): how “free” states in one resource theory behave when examined through the lens of another. The authors develop a unified framework that treats the free operations of a QRT as a unitary representation of a group (G) and defines the set of pure free states as the orbit of a reference state (|\psi_{\text{ref}}\rangle) under (G). While the reference state is often chosen as the highest‑weight state of the Lie algebra associated with (G), the authors deliberately explore the freedom in this choice, showing that different reference states lead to distinct families of free states even for the same free operations.

To quantify resourcefulness across different QRTs, the paper adopts a family of “purity‑type witnesses” derived from group‑Fourier analysis. Each witness takes the form
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