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

TL;DR

The paper develops and tests a unified framework to analyze how free states from one quantum resource theory can serve as resourceful states under other theories, spanning eight QRTs and employing group-Fourier purity-type witnesses. It combines analytical results (exact averages for Haar, Gaussian, and stabilizer ensembles) with large-scale numerical studies (datasets of free states and Haar random states across $n=3$ to $8$ qubits) to map cross-QRT resourcefulness and inter-theory relationships. A key theoretical finding is the uniform-entanglement inequality $\Lambda_{\mathrm{uent}} \le \Lambda_{\mathrm{ent}}$, with equality conditions tied to site-invariant Pauli expectations; the work also characterizes discrete witness values for stabilizer states and derives asymptotics for several witnesses across different state ensembles. Overall, the results show that “resource” is highly relative to the chosen QRT, and even simple free states can be highly resourceful when evaluated under other QRTs, highlighting rich structure and guiding future rigorous proofs and widened witness sets.

Abstract

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.

Figures (4)

• Figure 1: Graphic representation of our results. Our work showcases the fact that the resourcefulness, or the "quantumness" of a given quantum state $\ket{\psi}$ is relative to the QRT through which it is analyzed. For instance, a resource-free state of QRT 1, could have widely varying resourcefulness when examined through QRTs 2 and 3.
• Figure 2: Average resourcefulness. Violin plot of the average resourcefulness across all considered QRTs grouped by each family of the dataset. Dashed lines, represent the median and colors the system size, as indicated.
• Figure 3: Pair plot of QRT vs QRT witnesses. We computed the witnesses for the states in the datasets on $n=4$ (lower-left half) and $n=8$ qubits (top-right half). These plots serve as a visual aid to look for any relationships between resource witnesses. As indicated by some example figures (with a thick colored edge and a number), the plots are reflected with respect to the diagonal. For instance, one can see how the relation between $\Lambda_{{\rm stab}}(\ket{\psi})$ and $\Lambda_{{\rm ferm}}(\ket{\psi})$ changes with system size by comparing the bottom left plot, and the top-right plot.
• Figure 4: Principal component plot for the (a) $n=4$ and (b) $n=8$ datasets. We perform PCA analysis on the dataset (excluding $\Lambda_{\rm imag}$), allowing us to project each data vector $\boldsymbol{v}(\ket{\psi})$ onto the subspace spanned by the first two principal components. The black arrows represent the loading vectors for each QRT's witness.

Theorems & Definitions (16)

• Proposition 1
• Proposition 2
• Corollary 1
• Corollary 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Theorem 1
• Corollary 3