Characterizing quantum resourcefulness via group-Fourier decompositions

Abstract: In this work we present a general framework for studying the resourcefulness in pure states for quantum resource theories (QRTs) whose free operations arise from the unitary representation of a group. We argue that the group Fourier decompositions (GFDs) of a state, i.e., its projection onto the irreducible representations (irreps) of the Hilbert space, operator space, and tensor products thereof, constitute fingerprints of resourcefulness and complexity. By focusing on the norm of the irrep projections, dubbed GFD purities, we find that low-resource states live in the small dimensional irreps of operator space, whereas high-resource states have support in more, and higher dimensional ones. Such behavior not only resembles that appearing in classical harmonic analysis, but is also universal across the QRTs of entanglement, fermionic Gaussianity, spin coherence, and Clifford stabilizerness. To finish, we show that GFD purities carry operational meaning as they lead to resourcefulness witnesses as well as to notions of state compressibility.

• The paper introduces a framework using group Fourier decomposition (GFD) purities to universally quantify quantum resourcefulness in states for resource theories with unitary group representations.
• GFD purities serve as resource witnesses with operational meaning, demonstrating that free states are maximally compressible based on their smallest irreducible representation components.
• Empirical analysis across entanglement, Gaussianity, spin coherence, and stabilizerness shows resourceful states distribute purity in higher-dimensional irreps, while free states concentrate it in smaller ones.

Characterizing Quantum Resourcefulness via Group-Fourier Decompositions

This work introduces a comprehensive framework for quantifying quantum resourcefulness in pure states within quantum resource theories (QRTs) where the free operations are described by unitary group representations. The central thesis is that the group Fourier decomposition (GFD) of a quantum state—its projection onto the irreducible representations (irreps) of the relevant group—serves as a universal fingerprint of resourcefulness and complexity. The authors formalize this intuition by defining GFD purities: the norms of a state's projections onto each irrep, which collectively form a probability distribution for normalized states.

Framework and Theoretical Foundations

The framework is applicable to any QRT where free operations are given by a group representation $R: G \to \mathrm{GL}(H)$, with $H$ a finite-dimensional Hilbert space. The set of free pure states is the orbit of a reference state under $R(G)$. The key mathematical structure is the decomposition of the relevant vector space (e.g., $H$, $L(H)$, $H^{\otimes 2}$) into irreps of $G$, as guaranteed by Maschke's theorem. For a state $v \in V$, its GFD is the set of orthogonal projections $v_{j,\alpha}$ onto each irrep $V_{\alpha,j}$, and the associated GFD purities $P_{j,\alpha}(v) = \|v_{j,\alpha}\|^2$.

The GFD purities are invariant under the action of the free group, and for normalized states, their sum is unity. This invariance ensures that GFD purity profiles are well-defined resource monotones within the QRT.

Operational Significance

The authors establish that GFD purities have direct operational meaning:

• Resource Witnesses: In Lie group-based QRTs, the purity in the irrep corresponding to the complexified Lie algebra is maximized if and only if the state is free. This generalizes to discrete groups, as shown for Clifford stabilizerness, where the stabilizer entropy is proportional to a GFD purity.
• State Compressibility: For free states, the information in the smallest non-trivial irrep suffices to reconstruct the state. This is formalized via a MaxEnt principle: if a pure state matches a free state in all expectation values over the smallest irrep, the states are identical. Thus, free states are maximally compressible in the GFD basis.

Empirical Analysis Across QRTs

The framework is instantiated for several QRTs, revealing universal and distinctive behaviors:

1. Bipartite and Multipartite Entanglement

• Two-Qubit Case: The operator space $L(H)$ decomposes into four irreps. Product (free) states concentrate purity in lower-dimensional irreps, while maximally entangled states shift purity to the largest irrep. The purity in certain irreps serves as an entanglement witness, being proportional to the reduced state purity.
• $n$-Qubit Case: The number of irreps grows exponentially. Product states again concentrate purity in small irreps, while highly entangled states (e.g., GHZ, W, Haar random) distribute purity among higher-dimensional irreps. Absolutely maximally entangled states maximize purity in the largest irrep.

2. Fermionic Gaussianity

• The operator space decomposes according to the number of Majorana operators. Gaussian (free) states have purity concentrated in small irreps; non-Gaussian states (e.g., GHZ, high-extent states) shift purity to larger irreps, often exhibiting a zig-zag pattern. The purity in the irrep corresponding to the Lie algebra is a non-Gaussianity witness, but there is no monotonic relationship between extent and this purity.

3. Spin Coherence

• For spin-$s$ systems, the operator space decomposes into $2s+1$ irreps. Free states maximize purity in the smallest non-trivial irrep, while resourceful states (e.g., GHZ, $|s,0\rangle$) shift purity to higher-dimensional irreps. The purity in the spin-1 irrep is a coherence witness.

4. Clifford Stabilizerness

• The Clifford group induces a decomposition of $L(H^{\otimes 2})$ into 14 irreps. All GFD purities are linearly dependent on the purity in a single irrep, which is proportional to the stabilizer entropy. This indicates a simpler resource structure compared to Lie group-based QRTs.

Implications and Comparative Analysis

A key finding is the universality of the relationship between resourcefulness and support in higher-dimensional irreps: free states are compressible and have support in small irreps, while resourceful states are distributed among larger irreps. This mirrors classical harmonic analysis, where complex signals require higher-frequency components.

However, the relationship between GFD purities and other resource measures (e.g., extent) is not universal. In stabilizerness, small purity implies large extent, but in fermionic Gaussianity, states with large extent can have significant purity in small irreps. This suggests that different QRTs capture distinct aspects of quantum complexity, with implications for classical simulability: states with large extent but high small-irrep purity may be efficiently simulable by Lie-algebraic methods but not by standard techniques.

Future Directions

The framework opens several avenues for further research:

• Extension to mixed states and higher tensor powers.
• Exploration of GFD purities for other QRTs and non-unitary operations.
• Investigation of linear combinations of GFD purities as resource monotones.
• Application to quantum machine learning, where trainability may correlate with small-irrep support.
• Analysis of redundancy in GFD purities and the development of more efficient state representations.

Numerical Results and Claims

The paper provides explicit analytical and numerical results for GFD purities across QRTs, including:

• Exact formulas for GFD purities in entanglement, Gaussianity, and spin coherence.
• Demonstration that free states are fully characterized by their small-irrep purities.
• Identification of zig-zag purity profiles for highly structured resourceful states (e.g., GHZ).
• Quantitative comparison of GFD purities for Haar random, free, and maximally resourceful states.

Conclusion

This work establishes group-Fourier decompositions as a unifying and operationally meaningful framework for characterizing quantum resourcefulness. By connecting representation theory, harmonic analysis, and quantum information, it provides both a theoretical foundation and practical tools for analyzing and compressing quantum states in diverse QRTs. The identification of universal behaviors and resource witnesses via GFD purities has significant implications for quantum computation, simulation, and machine learning, and suggests new directions for the study of quantum complexity and resource theories.