Characterizing quantum resourcefulness via group-Fourier decompositions
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913910363258880 |
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| author | Bermejo, Pablo Braccia, Paolo Mele, Antonio Anna Diaz, Nahuel L. Deneris, Andrew E. Larocca, Martin Cerezo, M. |
| author_facet | Bermejo, Pablo Braccia, Paolo Mele, Antonio Anna Diaz, Nahuel L. Deneris, Andrew E. Larocca, Martin Cerezo, M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing quantum resourcefulness via group-Fourier decompositions Bermejo, Pablo Braccia, Paolo Mele, Antonio Anna Diaz, Nahuel L. Deneris, Andrew E. Larocca, Martin Cerezo, M. Quantum Physics Statistical Mechanics Group Theory 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. |
| title | Characterizing quantum resourcefulness via group-Fourier decompositions |
| topic | Quantum Physics Statistical Mechanics Group Theory |
| url | https://arxiv.org/abs/2506.19696 |