Group Fourier filtering of quantum resources in quantum phase space

TL;DR

This work develops a representation-theoretic bridge between quantum resource theories and quantum phase spaces by analyzing group Fourier structure under a compact Lie group $\mathbb{G}$. It derives the master irrep spectrum $\tau_{\lambda}=\mathcal{P}_{\lambda}(\ket{\mathrm{hw}}\!\bra{\mathrm{hw}})/d_{\lambda}$ and shows that phase-space representations $F_{\rho}(\Omega,s)$ are obtained from operator-space purities via a group-Fourier filter with $\widetilde{\mathcal{P}}_{\lambda}(F_{\rho}(\Omega,s))=\tau_{\lambda}^{-s}\mathcal{P}_{\lambda}(\rho)$, so the Cahill–Glauber parameter $s$ tunes from low-pass ($s=-1$) to high-pass ($s=1$) behavior and interpolates between free-adapted and resource-sensitive representations. The paper reveals an $s$-duality linking the phase-space spectra of free states and Haar-random highly resourceful states through a shift $s\to s+1$, and it introduces a twisted-product structure on phase space with couplings controlled by $\tau_{\lambda}$. These results are illustrated via spin-coherence, multipartite entanglement, and fermionic Gaussianity, and extended beyond SW kernels to general covariant QPS mappings. Collectively, this provides a signal-processing viewpoint for diagnosing, filtering, and visualizing quantum resources across QRTs, with potential implications for resource witnesses and phase-space-based quantum algorithms.

Abstract

Recently, it has been shown that group Fourier analysis of quantum states, i.e., decomposing them into the irreducible representations (irreps) of a symmetry group, enables new ways to characterize their resourcefulness. Given that quantum phase spaces (QPSs) provide an alternative description of quantum systems, and thus of the group's representation, one may wonder how such harmonic analysis changes. In this work we show that for general compact Lie-group quantum resource theories (QRTs), the entire family of Stratonovich-Weyl quantum phase space representations-characterized by the Cahill-Glauber parameter $s$-has a clear resource-theoretic and signal-processing meaning. Specifically, changing $s$ implements a group Fourier filter that can be continuously tuned to favor low-dimensional irreps where free states have most of their support ($s=-1$), leave the spectrum unchanged ($s=0$), or highlight resourceful, high-dimensional irreps ($s=1$). As such, distinct QPSs constitute veritable group Fourier filters for resources. Moreover, we show that the norms of the QRT's free state Fourier components completely characterize all QPSs. Finally, we uncover an $s$-duality relating the phase space spectra of free states and typical (Haar-random) highly resourceful states through a shift in $s$. Overall, our results provide a new interpretation of QPSs and promote them to a signal-processing framework for diagnosing, filtering, and visualizing quantum resources.

Figures (3)

• Figure 1: Schematic representation of our main results. We show that the SW kernels $\Delta(\Omega,s)$ serve as a signal-processing-like Fourier filter for the irrep components of quantum operators, with the Cahill--Glauber parameter $s$ interpolating between low-pass ($s=-1$) and high-pass ($s=1$) actions. Moreover, all the properties of the kernel are fully determined by the QRT's free states, further unraveling the deep connection between QPSs and QRTs. The illustrated QPS corresponds to the Robinson projection of the sphere $S^2$ for a Greenberger–Horne–Zeilinger (GHZ) state in the spin coherence QRT. See below for more details.
• Figure 2: SU(2) spin-coherence QPS for four states ($\ket{S,S}$, $\ket{S,0}$, GHZ, and a Haar-random pure state) and for $S=5$. (a, top) GFD purities $\mathcal{P}_\lambda(\rho)$ in operator space $\mathcal{L}(\mathcal{H})$. By mapping to the QPSs via SW kernels with $s=-1,0,1$, the components in each irrep get modulated by $\tau_\lambda^{-s}$ (see Eq. \ref{['eq:tau-su2']}). We show said modulation for $s=\pm1$ (as the $s=0$ case is trivial). (a, middle): Corresponding phase space purities $\widetilde{\mathcal{P}}_\lambda(\rho,s)=\tau_\lambda^{-s} \mathcal{P}_\lambda(\rho)$ for $s=-1,0,1$, showing how the kernel reweights irreps. (b) Robinson projection of phase space functions $F_\rho(\Omega,s)$ on $\mathcal{X} \simeq S^2 = \mathrm{SU}(2)/\mathrm{U}(1)$ for $s=-1,0,1$, illustrating how changing $s$ (i.e. the kernel $\Delta(\Omega,s)$) acts as a low-, neutral-, or high-pass filter and yields distinct phase space representations of the same state.
• Figure 3: Coefficient $\tau_\lambda^{-s}$ for the QRTs of multipartite entanglement and fermionic Gaussianity on $n=10$ qubits. (a) For the QRT of multipartite entanglement we show $\tau_\lambda^{-s}$ versus the Hamming weight $w(\lambda)$ of the irrep label $\lambda\in\{0,1\}^{\otimes n}$. (b) For the fermionic Gaussinity we plot $\tau_\lambda^{-s}$ for $\lambda=0,2,4,\ldots,n$ (as $\lambda=n+1,\ldots,2n$ are the same due to the binomial coefficient symmetry). In both cases the top row corresponds to $s=1$, while the bottom one to $s=-1$.

Theorems & Definitions (20)

• Definition 1: Irrep decomposition of $\mathcal{L}(\mathcal{H})$
• Definition 2: adjoint representation of $\mathfrak{g}$ in $V_\lambda$
• Definition 3: Adjoint representation of $\mathbb{G}$ in $V_\lambda$
• Definition 4: GFD and purities of operators in $\mathcal{L}(\mathcal{H})$
• Definition 5: Free operations and states in Lie-algebraic QRTs
• Definition 6: Quantum phase space
• Definition 7: Properties of phase space functions
• Definition 8: Properties of the SW kernels
• Theorem 1: Irrep decomposition of $L^2(\mathcal{X})$
• Definition 9: GFD and purities of operators in $L^2(\mathcal{X})$