Pith. sign in

REVIEW 1 major objections 5 minor 76 references

A Nonstabilizerness Resource Law for Universal Quantum State Purification

T0 review · 1 major / 5 minor · reviewed 2026-07-10 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Magic cost of quantum purification is exactly linear in fidelity gain

desk verdict Letter on arXiv:2607.08626 read the letter →

arxiv 2607.08626 v1 pith:3LML5SX6 submitted 2026-07-09 quant-ph

Keming He , Enji Xiong , Xin Wang This is my paper · ORCID
classification quant-ph PACS 03.67.Ac03.67.Pp
keywords quantumstatepurificationmagicresourcetheorynonstabilizernessmanarobustnessofdepolarizingnoisesemidefiniteprogrammingstabilizerstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks a sharp quantitative question: if you want to clean up noisy quantum states using a purification protocol that works for any unknown input state, how much nonstabilizerness (magic) must the successful branch of that protocol carry? Prior work showed that zero-magic operations cannot improve fidelity at all in the universal setting. This paper goes beyond that no-go result by proving that, for two-copy purification under depolarizing noise, the minimum magic required is governed by an exact linear law in the fidelity improvement over the noisy copy. In odd prime-power dimensions, the exponentiated mana (a discrete-Wigner-function-based magic measure) of the successful operation equals one plus a constant times the fidelity gain. In multi-qubit systems, the Choi-state robustness of magic is sandwiched between two linear functions of the same fidelity gain, and the two bounds coincide for a single qubit, giving an exact law there too. The authors construct an explicit two-copy probabilistic map, built from weighted symmetric and antisymmetric projections, that achieves any prescribed success probability and target fidelity while saturating these bounds. The construction reveals a clean separation: the fidelity gain is controlled by the ratio of antisymmetric to symmetric branch weights, while the success probability is controlled by an overall scale factor, so the magic cost depends only on the fidelity gain and not on the success probability. The proofs combine semidefinite programming duality for the lower bounds with explicit stabilizer decompositions for the upper bounds, exploiting the structure of tripartite stabilizer states and partially transposed permutation algebras.

What carries the argument

The central objects are: (1) a completely positive trace-non-increasing map representing the successful branch of a probabilistic purification protocol, whose Choi state is the object whose magic is quantified; (2) the mana, defined as the logarithm of the maximum column-sum norm of the channel's discrete Wigner function, used in odd dimensions; (3) the robustness of magic, defined as the minimum L1 norm over all stabilizer-state decompositions of the Choi state, used for multi-qubit systems; (4) an explicit two-copy purification map built from weighted symmetric and antisymmetric projections, parametrized so that the ratio t = mu_2/mu_1 fixes the conditional fidelity and the scale s = 2(mu_

What would settle it

A counterexample would be a two-copy universal purification protocol achieving a nonzero fidelity gain whose successful branch has magic strictly below the linear law's prediction, or a protocol whose magic cost depends on the success probability in a way not captured by the fidelity gain alone.

Watch

Extended reading notes

Core claim

For two-to-one universal probabilistic purification under depolarizing noise, the minimum magic of the successful operation is exactly linear in the fidelity gain. In odd dimensions, exponentiated mana equals one plus a dimension- and noise-dependent constant times the fidelity gain, with matching primal and dual SDP bounds. In multi-qubit systems, Choi-state robustness of magic is bounded above and below by linear functions of the same gain, collapsing to an exact law for a single qubit. The resource cost is set entirely by the fidelity gain, not by the success probability, because an explicit two-copy map separates these two parameters cleanly.

Load-bearing premise

The exact linear laws and tight bounds are proven only for the two-copy setting under depolarizing noise. The symmetry algebra that makes the proofs tractable becomes substantially richer at higher copy numbers, and the stabilizer decompositions needed for the upper bounds grow super-exponentially, so the resource law is structurally tied to the tractability of the two-copy case.

Editorial extensions

If this is right

  • The exact linear law for two-copy purification provides a concrete benchmark: any experimental two-copy universal purification protocol that achieves a fidelity gain must consume magic proportional to that gain, with the proportionality constant determined by dimension and noise level.
  • The separation between fidelity gain (controlled by branch ratio) and success probability (controlled by scale) means that postselection can trade rate for fidelity but cannot reduce the per-success magic cost, making this a resource-theoretic analogue of rate-fidelity tradeoffs.
  • The framework extends in principle to more copies and other noise models via the same SDP formulation, though the authors note that the symmetry algebras and stabilizer decompositions become substantially harder at higher copy numbers.
  • The connection between magic and purification performance suggests that magic state distillation and state purification are not independent resource tasks but are linked through a common quantitative law, potentially unifying error mitigation and fault-tolerant computation resource accounting.
  • The Clifford-twirling reduction used for the lower bound shows that the worst-case magic cost can be assessed on Clifford-invariant Choi states, which may simplify experimental verification of the bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the linear resource law extends to higher copy numbers (which the paper does not prove but the framework allows), one could derive an asymptotic magic-per-fidelity-gain rate for large-scale purification, analogous to distillation rates in entanglement theory.
  • The exact single-qubit law and the odd-dimensional exact law having different magic measures (robustness vs. mana) raises the question of whether a unified magic measure exists for which the law is exact across all dimensions, or whether the dimension-dependent choice of measure is fundamental.
  • The explicit two-copy map achieving the bounds is implementable via a swap test plus an acceptance step, suggesting that near-term experiments could verify the linear law in the single-qubit case with existing hardware.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The manuscript studies the magic (nonstabilizerness) resource cost of universal quantum state purification under depolarizing noise. Given two noisy copies of an unknown pure state, the authors ask: for a prescribed success probability $p$ and target fidelity $f$, what is the minimum magic required in the successful branch of a probabilistic purification protocol? They define two resource quantifiers—the mana of purification (for odd-dimensional qudits) and the robustness of purification (for multi-qubit systems)—and formulate both as semidefinite programs (Appendix B). The main results are: (1) Theorem 1, an exact linear law stating that the exponentiated mana of two-copy universal purification equals $1 + K_M(f - f_0)$, where $f_0$ is the single-copy fidelity; and (2) Theorem 2, two-sided linear bounds on the Choi-state robustness for multi-qubit systems, with the claim that the bounds coincide for a single qubit ($d=2$), yielding an exact law. An explicit purification map (Eq. 5) achieving the tradeoff is constructed, and Corollary 3 recovers a prior no-go result as the zero-magic boundary. The proofs use SDP duality, Clifford twirling, and the tripartite stabilizer normal form.

Significance. The paper addresses a well-motivated question at the intersection of resource theory of magic and quantum error mitigation. The framing of purification as a resource-pricing problem—fixing $(p, f)$ and minimizing magic—is natural and operationally meaningful. The SDP formulations in Appendix B are carefully derived and provide a reproducible framework. The explicit construction of the two-copy purification map (Eq. 5) with transparent parameterization in terms of the ratio $t = f_2/f_1$ and scale $s$ is a concrete strength. The connection between the accepted branch's magic and the fidelity gain, separated from the success probability, is a clean conceptual contribution. However, the significance of the headline claim of an exact single-qubit robustness law is undermined by an apparent algebraic inconsistency in the constants (see Major Comment 1).

major comments (1)
  1. Appendix D, Eq. (S2) and the concluding statement of the proof (Eq. S110): The claim that the upper and lower robustness bounds coincide for $d=2$ appears to be algebraically inconsistent with the derived formulas. The lower-bound slope is $K_R^l = (d-2)/2 + 1/(d-1) + (2f - f^2)/(f_0 f(1-f))$, which for $d=2$ gives $K_R^l = 1 + (2f-f^2)/(f_0 f(1-f))$. The upper-bound slope is $K_R^u = (2d-3 + 2f - f^2)/(f_0 f(1-f))$, which for $d=2$ gives $K_R^u = (1 + 2f - f^2)/(f_0 f(1-f))$. These are not equal: $K_R^l$ has a standalone constant 1, while $K_R^u$ has its constant 1 divided by $f_0 f(1-f)$. For $f=0.5$, $f_0 = 1 - f/2 = 0.75$: $K_R^l = 1 + 1.75/0.1875 = 10.33$, while $K_R^u = 2.75/0.1875 = 14.67$. The text's argument (Eq. S110) that $2d-3=1$ and $(d-2)/2+1/(d-1)=1$ for $d=2$ does not address the differing placement of the constant 1 relative to the denominator $f_0 f(1-f)$. This affects:
minor comments (5)
  1. Figure 2: The caption mentions 'exponentiated mana and robustness bounds' but the axes labels and panel descriptions are not fully legible in the provided text. Ensure axis labels clearly distinguish $2^{M_{D_¥}} - 1$ from $R_{D_¥}$ and specify the value of $d$ for each curve.
  2. Eq. (S99) in Appendix D: The intermediate algebraic step from the $S$-basis expression to the factored form $p[1 + K_R^l(f-f_0)] + c_+ s_+ + c_- s_-$ is nontrivial. Providing a few additional lines of simplification, or a symbolic verification script, would help readers verify the lower-bound derivation independently.
  3. The manuscript uses $f$ for both the depolarizing error parameter and the target fidelity $f$ in some contexts (e.g., $f_0 = 1 - (d-1)/d ¥ f$). While the notation is technically unambiguous, it could cause confusion; consider using a different symbol for the error parameter.
  4. Reference [31] is cited as a prior no-go result by the same authors. The relationship between the present quantitative result and [31] is clear, but a brief sentence in the introduction explicitly stating that the present work supersedes/complements [31] would help readers.
  5. The paper notes that extending beyond $n=2$ copies faces super-exponential complexity in stabilizer decompositions. A brief discussion of whether numerical SDP solutions for $n=3$ or $n=4$ are feasible (even if analytic laws are not) would add value.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; derivation is self-contained via SDP duality with independently derived constants. The skeptic's algebraic objection targets correctness of the d=2 simplification, not circularity.

full rationale

The paper's central results (Theorems 1 and 2) are derived through semidefinite programming duality: explicit primal constructions (Eq. S5/S6) give upper bounds, and explicit dual feasible solutions (Eq. S25 for mana; the witness W in Eq. S57 for robustness) give matching lower bounds. The constants K_M (Eq. S2) and K_R^l, K_R^u (Eq. S2/S109) are derived analytically from the depolarizing parameter δ and dimension d, not fitted to data or defined in terms of the target quantity. The prior no-go result [31] by overlapping authors is cited as motivation ('These results identify the zero-magic boundary of the task, but leave open the resource law beyond this boundary'), not as a load-bearing premise for the quantitative laws. The skeptic's claim that K_R^l ≠ K_R^u for d=2 is a correctness concern about algebraic simplification (whether the standalone 1 in K_R^l equals 1/[λ₀δ(1-δ)] in K_R^u), not a circularity issue: the formulas are derived from independent SDP analysis, and if the d=2 equality fails, that would be an algebraic error, not a self-definitional reduction. No step in the derivation chain reduces to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies on standard mathematical results from quantum information theory (Hudson theorem, Clifford group design properties, stabilizer normal forms) and a standard noise model (depolarizing). No new physical entities or ad-hoc parameters are introduced; the constants in the linear laws are derived from the dimension d and noise parameter δ.

assumptions (4)
  • standard math Discrete Hudson theorem: a pure state in odd dimension is a stabilizer state iff its Wigner function is non-negative.
    Used to justify mana as a magic witness in odd dimensions (Preliminaries).
  • standard math Multi-qubit Clifford group is a unitary 3-design.
    Used in Appendix D to justify Clifford twirling reducing to the partially transposed permutation algebra.
  • standard math Tripartite stabilizer normal form: any tripartite qubit stabilizer state is locally Clifford equivalent to a tensor product of GHZ states, Bell pairs, and single-qubit stabilizer states.
    Used in Lemma S7 (Appendix D) to bound the robustness witness on stabilizer states.
  • domain assumption Depolarizing noise model: each copy undergoes the depolarizing channel D_δ.
    The exact linear laws are derived specifically for depolarizing noise; extension to general noise is stated as future work.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Nonstabilizerness Resource Law for Universal Quantum State Purification." pith.science (2026). https://pith.science/paper/3LML5SX6

@misc{pith2026260708626,
  author       = {Pith},
  title        = {Pith review of: A Nonstabilizerness Resource Law for Universal Quantum State Purification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LML5SX6}},
  note         = {Machine review of arXiv:2607.08626}
}
read the original abstract

Quantum state purification aims to recover higher-fidelity quantum states from multiple noisy copies and is a fundamental primitive for quantum information processing. Magic resources enable operations beyond classically simulable dynamics and are central to universal fault-tolerant quantum computation. Recent no-go results show that classically simulable operations cannot achieve a nontrivial universal fidelity gain. This motivates a quantitative theory of the magic required for purification at prescribed success probability and target fidelity. For universal purification with two input copies, we prove an exact linear mana law in odd dimensions and a two-sided linear robustness law for multi-qubit systems, which becomes exact for a single qubit. We also identify an explicit successful purification map that makes the tradeoff transparent. These results establish universal purification as a task obeying a quantitative magic-fidelity law and link magic resources to error mitigation and fault-tolerant quantum information processing.

Figures

Figures reproduced from arXiv: 2607.08626 by the authors.

Figure 1
Figure 1. Schematic illustration of the role of magic in universal [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Resource laws for two-copy universal purification at [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages

  1. [31]

    K. He, C. Zhu, H. Yao, J. Liu, Y . Li, and X. Wang, No- go theorems for universal quantum state purification via classically simulable operations, Physical Review Let- ters136, 090204 (2026)

    work page 2026

  2. [1]

    Barenco, A

    A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, Stabilization of quan- tum computations by symmetrization, SIAM Journal on Computing26, 1541 (1997)

    work page 1997

  3. [2]

    J. I. Cirac, A. Ekert, and C. Macchiavello, Optimal pu- rification of single qubits, Physical Review Letters82, 4344 (1999)

    work page 1999

  4. [3]

    Keyl and R

    M. Keyl and R. F. Werner, The rate of optimal purifi- cation procedures, inAnnales Henri Poincare, V ol. 2 (Springer, 2001) pp. 1–26

    work page 2001

  5. [4]

    Fiurášek, Optimal probabilistic cloning and purifica- tion of quantum states, Physical Review A—Atomic, Molecular, and Optical Physics70, 032308 (2004)

    J. Fiurášek, Optimal probabilistic cloning and purifica- tion of quantum states, Physical Review A—Atomic, Molecular, and Optical Physics70, 032308 (2004)

    work page 2004

  6. [5]

    Fu,Quantum state purification, Master’s thesis, Uni- versity of Waterloo (2016)

    H. Fu,Quantum state purification, Master’s thesis, Uni- versity of Waterloo (2016)

    work page 2016

  7. [6]

    Yao, Y .-A

    H. Yao, Y .-A. Chen, E. Huang, K. Chen, H. Fu, and X. Wang, Protocols and trade-offs of quantum state pu- rification, Quantum Science and Technology10, 35020 (2025)

    work page 2025

  8. [7]

    Z. Li, H. Fu, T. Isogawa, C. Silva, and I. Chuang, Optimal quantum purity amplification, arXiv preprint arXiv:2409.18167 (2024)

    work page arXiv 2024

Show all 76 references
  1. [8]

    Z. Li, E. Theil, A. W. Harrow, and I. Chuang, Quantum purity amplification for arbitrary eigenstates and multi- ple outputs, arXiv preprint arXiv:2605.21570 (2026)

    2026 arXiv

  2. [9]

    C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy chan- nels, Physical Review Letters76, 722 (1996)

    1996

  3. [10]

    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Physical Review A54, 3824 (1996)

    1996

  4. [11]

    Deutsch, A

    D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy am- plification and the security of quantum cryptography over noisy channels, Physical Review Letters77, 2818 (1996)

    1996

  5. [12]

    Dür and H

    W. Dür and H. J. Briegel, Entanglement purification and quantum error correction, Reports on Progress in Physics70, 1381 (2007)

    2007

  6. [13]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of modern physics81, 865 (2009)

    2009

  7. [14]

    Zhao, Y .-A

    B. Zhao, Y .-A. Chen, X. Zhao, C. Zhu, G. Chiribella, and X. Wang, Power and limitations of distributed quan- tum state purification, Physical Review Letters136, 090203 (2026)

    2026

  8. [15]

    Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

    D. Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

    1997

  9. [16]

    Gottesman and I

    D. Gottesman and I. L. Chuang, Demonstrating the vi- ability of universal quantum computation using tele- portation and single-qubit operations, Nature402, 390 (1999)

    1999

  10. [17]

    X. Zhou, D. W. Leung, and I. L. Chuang, Methodology for quantum logic gate construction, Physical Review A 62, 052316 (2000)

    2000

  11. [18]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Physi- cal Review A—Atomic, Molecular, and Optical Physics 71, 022316 (2005)

    2005

  12. [19]

    Bravyi and D

    S. Bravyi and D. Gosset, Improved Classical Simula- tion of Quantum Circuits Dominated by Clifford Gates, 6 Physical Review Letters116, 250501 (2016)

    2016

  13. [20]

    Qassim, H

    H. Qassim, H. Pashayan, and D. Gosset, Improved up- per bounds on the stabilizer rank of magic states, Quan- tum5, 606 (2021)

    2021

  14. [21]

    R. S. Gupta, N. Sundaresan, T. Alexander, C. J. Wood, S. T. Merkel, M. B. Healy, M. Hillenbrand, T. Jochym- O’Connor, J. R. Wootton, T. J. Yoder,et al., Encoding a magic state with beyond break-even fidelity, Nature 625, 259 (2024)

    2024

  15. [22]

    Sales Rodriguez, J

    P. Sales Rodriguez, J. M. Robinson, P. N. Jepsen, Z. He, C. Duckering, C. Zhao, K.-H. Wu, J. Campo, K. Bag- nall, M. Kwon,et al., Experimental demonstration of logical magic state distillation, Nature645, 620 (2025)

    2025

  16. [23]

    Rosenfeld, C

    E. Rosenfeld, C. Gidney, G. Roberts, A. Morvan, N. Lacroix, D. Kafri, J. Marshall, M. Li, V . Sivak, D. Abanin,et al., Magic state cultivation on a superconducting quantum processor, arXiv preprint arXiv:2512.13908 (2025)

    2025

  17. [24]

    Bluvstein, A

    D. Bluvstein, A. A. Geim, S. H. Li, S. J. Evered, J. P. Bonilla Ataides, G. Baranes, A. Gu, T. Manovitz, M. Xu, M. Kalinowski,et al., A fault-tolerant neutral- atom architecture for universal quantum computation, Nature649, 39 (2026)

    2026

  18. [25]

    X. Wang, M. M. Wilde, and Y . Su, Quantifying the magic of quantum channels, New Journal of Physics21, 103002 (2019)

    2019

  19. [26]

    J. R. Seddon and E. T. Campbell, Quantifying magic for multi-qubit operations, Proceedings of the Royal Soci- ety A475, 20190251 (2019)

    2019

  20. [27]

    Howard and E

    M. Howard and E. Campbell, Application of a resource theory for magic states to fault-tolerant quantum com- puting, Physical Review Letters118, 090501 (2017)

    2017

  21. [28]

    Veitch, S

    V . Veitch, S. H. Mousavian, D. Gottesman, and J. Emer- son, The resource theory of stabilizer quantum compu- tation, New Journal of Physics16, 013009 (2014)

    2014

  22. [29]

    Y .-A. Chen, G. Gour, X. Wang, L. Zhang, and C. Zhu, Physical implementability for reversible magic state manipulation, Communications Physics8, 4 (2025)

    2025

  23. [30]

    J. R. Seddon, B. Regula, H. Pashayan, Y . Ouyang, and E. T. Campbell, Quantifying quantum speedups: Im- proved classical simulation from tighter magic mono- tones, PRX Quantum2, 010345 (2021)

    2021

  24. [32]

    Gross, Hudson’s theorem for finite-dimensional quantum systems, Journal of mathematical physics47 (2006)

    D. Gross, Hudson’s theorem for finite-dimensional quantum systems, Journal of mathematical physics47 (2006)

    2006

  25. [33]

    Delfosse, P

    N. Delfosse, P. Allard Guerin, J. Bian, and R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits, Physical Review X5, 021003 (2015)

    2015

  26. [34]

    Raussendorf, D

    R. Raussendorf, D. E. Browne, N. Delfosse, C. Okay, and J. Bermejo-Vega, Contextuality and wigner- function negativity in qubit quantum computation, Physical Review A95, 052334 (2017)

    2017

  27. [35]

    Raussendorf, J

    R. Raussendorf, J. Bermejo-Vega, E. Tyhurst, C. Okay, and M. Zurel, Phase-space-simulation method for quan- tum computation with magic states on qubits, Physical Review A101, 012350 (2020)

    2020

  28. [36]

    Vidal, Entanglement monotones, Journal of Modern Optics47, 355 (2000)

    G. Vidal, Entanglement monotones, Journal of Modern Optics47, 355 (2000)

    2000

  29. [37]

    Regula, Probabilistic transformations of quantum re- sources, Physical Review Letters128, 110505 (2022)

    B. Regula, Probabilistic transformations of quantum re- sources, Physical Review Letters128, 110505 (2022)

    2022

  30. [38]

    K. Fang, X. Wang, L. Lami, B. Regula, and G. Adesso, Probabilistic distillation of quantum coherence, arXiv preprint arXiv:1804.09500 (2018)

    2018 arXiv

  31. [39]

    Takagi and B

    R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational charac- terization via discrimination tasks, Physical Review X 9, 031053 (2019)

    2019

  32. [40]

    Bravyi, D

    S. Bravyi, D. Fattal, and D. Gottesman, Ghz extraction yield for multipartite stabilizer states, Journal of Math- ematical Physics47(2006)

    2006

  33. [41]

    S. Y . Looi and R. B. Griffiths, Tripartite entanglement in qudit stabilizer states and application in quantum er- ror correction, Physical Review A—Atomic, Molecular, and Optical Physics84, 052306 (2011)

    2011

  34. [42]

    Nezami and M

    S. Nezami and M. Walter, Multipartite entanglement in stabilizer tensor networks, Physical Review Letters125, 241602 (2020)

    2020

  35. [43]

    Zhu, Multiqubit clifford groups are unitary 3- designs, Physical Review A96, 062336 (2017)

    H. Zhu, Multiqubit clifford groups are unitary 3- designs, Physical Review A96, 062336 (2017)

    2017

  36. [44]

    Gross, S

    D. Gross, S. Nezami, and M. Walter, Schur–weyl du- ality for the clifford group with applications: Property testing, a robust hudson theorem, and de finetti repre- sentations, Communications in Mathematical Physics 385, 1325 (2021)

    2021

  37. [45]

    Bittel, J

    L. Bittel, J. Eisert, L. Leone, A. A. Mele, and S. F. Oliviero, A complete theory of the clifford commutant, arXiv preprint arXiv:2504.12263 (2025)

    2025 arXiv

  38. [46]

    G.-C. Li, L. Chen, S.-Q. Zhang, X.-S. Hong, H. Xu, Y . Liu, Y . Zhou, G. Chen, C.-F. Li, G.-C. Guo,et al., Invested and potential magic resources in measurement- based quantum computation, Physical Review Letters 135, 160203 (2025)

    2025

  39. [47]

    Bu and A

    K. Bu and A. Jaffe, Magic resource can enhance the quantum capacity of channels, Physical Review Letters 134, 050202 (2025)

    2025

  40. [48]

    C. Zhu, Z. Liu, C. Zhu, and X. Wang, Limitations of classically simulable measurements for quantum state discrimination, Physical Review Letters133, 010202 (2024)

    2024

  41. [49]

    H. Kwon, Nonstabilizerness without magic: Classically simulatable quantum states that are indistinguishable by classically simulatable quantum circuits, arXiv preprint arXiv:2509.25790 (2025)

    2025

  42. [50]

    G. Zhen, X. Yang, C. Zhu, R. Chen, and X. Wang, Invariant measures and weak-magic-injection asymp- totics in random monitored quantum circuits, arXiv preprint arXiv:2606.13470 (2026)

    2026 arXiv

  43. [51]

    Hernández-Yanes, P

    T. Hernández-Yanes, P. Sierant, J. Zakrzewski, and M. Płodzie´n, Nonstabilizerness in quantum-enhanced metrological protocols, Physical Review A113, 012416 (2026)

    2026

  44. [52]

    Paviglianiti, M

    A. Paviglianiti, M. Seclì, E. Tirrito, and V . Savona, The true cost of factoring: Linking magic and number- theoretic complexity in shor’s algorithm, arXiv preprint arXiv:2605.05347 (2026)

    2026 arXiv

  45. [53]

    Choi, Completely positive linear maps on com- plex matrices, Linear algebra and its applications10, 285 (1975)

    M.-D. Choi, Completely positive linear maps on com- plex matrices, Linear algebra and its applications10, 285 (1975)

    1975

  46. [54]

    Jamiołkowski, Linear transformations which pre- serve trace and positive semidefiniteness of operators, Reports on mathematical physics3, 275 (1972)

    A. Jamiołkowski, Linear transformations which pre- serve trace and positive semidefiniteness of operators, Reports on mathematical physics3, 275 (1972)

    1972

  47. [55]

    Mari and J

    A. Mari and J. Eisert, Positive wigner functions render classical simulation of quantum computation efficient, Physical Review Letters109, 230503 (2012)

    2012

  48. [56]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Recon- struction of gaussian quantum mechanics from liouville mechanics with an epistemic restriction, Physical Re- view A86, 10.1103/physreva.86.012103 (2012)

    2012 doi

  49. [57]

    Eggeling and R

    T. Eggeling and R. F. Werner, Separability properties of tripartite states with U⊗U⊗U symmetry, Physical Review A63, 042111 (2001)

    2001

  50. [58]

    A. W. Harrow, The church of the symmetric subspace, arXiv preprint arXiv:1308.6595 (2013). 7

    2013 arXiv

  51. [59]

    Khatri and M

    S. Khatri and M. M. Wilde, Principles of quantum com- munication theory: A modern approach, arXiv preprint arXiv:2011.04672 (2020)

    2011 arXiv

  52. [60]

    A. A. Mele, Introduction to haar measure tools in quan- tum information: A beginner’s tutorial, Quantum8, 1340 (2024)

    2024

  53. [61]

    A Nonstabilizerness Resource Law for Universal Quantum State Purification

    K. Markov, I. Patel, and J. Hayes, Optimal synthesis of linear reversible circuits, Quantum Information and Computation8, 0282 (2008). Appendix — A Nonstabilizerness Resource Law for Universal Quantum State Purification In this Supplemental Material, we provide the detailed de...

    2008

  54. [62]

    For prime dimensiond, the unitary boost and shift operatorsX, Z∈ L(H)are defined by X|j⟩=|j⊕1⟩, Z|j⟩=ω j|j⟩,(S1) whereω=e 2πi/d and⊕denotes addition modulod

    The discrete Wigner function We recall the discrete Wigner representation used in this work [25, 28, 32]. For prime dimensiond, the unitary boost and shift operatorsX, Z∈ L(H)are defined by X|j⟩=|j⊕1⟩, Z|j⟩=ω j|j⟩,(S1) whereω=e 2πi/d and⊕denotes addition modulod. The Heisenber...

  55. [63]

    P u Au/d=I; 3.Tr[A uAu′ ] =dδ(u,u ′), whereδ(a, b)is the discrete Dirac delta function; 4.Tr[A u] = 1; 5.H= P u WH(u)Au; 6.{A u}u ={(A u)T }u

  56. [64]

    A Hermitian operatorHhas non-negative discrete Wigner functions if∀u, W H(u)≥0

    For the composite systemH A ⊗ HB, the phase space point operators are the tensor product of the subsystem phase space point operatorsA uA⊕uB =A uA ⊗A uB. A Hermitian operatorHhas non-negative discrete Wigner functions if∀u, W H(u)≥0. For odd dimensions, according to the discre...

  57. [65]

    In particular, one has to either exclude some Clifford operations from the set of free operations [33, 34], or lose compatibility with tensor products [35]

    The robustness of magic For qubit systems, however, the discrete phase space approach cannot be applied in the same way as in odd dimensions. In particular, one has to either exclude some Clifford operations from the set of free operations [33, 34], or lose compatibility with ...

  58. [66]

    Here we use the partially transposed permutation algebra, which gives a finite-dimensional representation of the walled Brauer algebra

    Tripartite states with partially transposed permutation symmetry Positivity in permutation-type algebras is useful in symmetry-reduced quantum information problems. Here we use the partially transposed permutation algebra, which gives a finite-dimensional representation of the...

  59. [67]

    +a l 1(ak 2 −a j 2) . Thus WE(v|u) =µ 1(δ(u1,v) +δ(u 2,v)) + µ2 d (Tr [Au1 Au2 Av] + Tr [Au2 Au1 Av]) =µ 1(δ(u1,v) +δ(u 2,v)) + 2 µ2 d Re(Tr[Au1 Au2 Av]) =µ 1(δ(u1,v) +δ(u 2,v)) + 2 µ2 d cos 4π d C(u1,u 2,v) . (S21) 18 zis calculated as z= max u X v |WE(v|u)| = max u1,u2 X v µ...

  60. [68]

    This completes the verification of dual feasibility

    Thus we show that−αQ T3 A2 I AO − βRT3 A2 I AO +C A2 I AO ≥0. This completes the verification of dual feasibility. Combining the primal upper bound and the dual lower bound, we conclude thatM Dδ(2, f, p, d) = log (KM(f−λ 0) + 1), and the proof is complete. ■ Appendix D: Proof ...

  61. [69]

    Hence these are stabilizer states, Clifford equivalent to a one-qubit Bell state tensored with computational basis states

    Ifx= (k, i, k)∈ S + i , y= (ℓ, i, ℓ)∈ S + i , k, ℓ̸=i, k̸=ℓ,thenϵ i(x)ϵi(y) = +1, and the two states are |χ± x,y⟩= |k, i, k⟩ ± |ℓ, i, ℓ⟩√ 2 = |k, k⟩AI,1 AO ± |ℓ, ℓ⟩AI,1 AO √ 2 ⊗ |i⟩AI,2 (S23) The second registerA I,2 is fixed to|i⟩, while the first and third registers form a t...

  62. [70]

    The same argument as in the first case shows that these are stabilizer states

    Ifx= (i, k, k)∈ S − i , y= (i, ℓ, ℓ)∈ S − i , k, ℓ̸=i, k̸=ℓ,then againϵ i(x)ϵi(y) = +1. The same argument as in the first case shows that these are stabilizer states

  63. [71]

    Hence |χ± x,y⟩= |x⟩ ∓ |y⟩√ 2 .(S24) It is less clear to observe that they are stabilizer states

    Ifx= (k, i, k)∈ S + i , y= (i, ℓ, ℓ)∈ S − i ,thenϵ i(x)ϵi(y) =−1. Hence |χ± x,y⟩= |x⟩ ∓ |y⟩√ 2 .(S24) It is less clear to observe that they are stabilizer states. Regardxandyas binary strings inF 3m 2 . Apply the PauliX-operator corresponding to the bit stringx,X x :=N3m r=1 X...

  64. [72]

    The estimates above give x12 ≤1, x 23 ≤1, x 13 ≤1,Re(x 123)≤1.(S78) Therefore −2 + (d−1)x 12 +x 23 +x 13 + 2 Re(x123)≤ −2 + (d−1) + 1 + 1 + 2 =d+ 1

    First supposem 12 ≥m 13, m 12 ≥m 23. The estimates above give x12 ≤1, x 23 ≤1, x 13 ≤1,Re(x 123)≤1.(S78) Therefore −2 + (d−1)x 12 +x 23 +x 13 + 2 Re(x123)≤ −2 + (d−1) + 1 + 1 + 2 =d+ 1. (S79)

  65. [73]

    Next supposem 23 ≥m 12 ≥m 13. Then x12 ≤2 m12−m23 , x 23 ≤2 m23−m12 , x 13 ≤2 m13−m23 ,Re(x 123)≤2 (m13−m12)/2 ≤1.(S80) For simplicity, we denote r= 2 m23−m12 ≥1, s= 2 m13−m12 ≤1, h= 2 mGHZ+2m12+m13+mloc ≥1,(S81) then we have rh= 2 mGHZ+m12+m13+m23+mloc =d, x 12 ≤ 1 r , x 23 ≤...

  66. [74]

    This case is symmetric to Case 2 ofm 23 ≥m 12 ≥m 13, withm 13 andm 23 interchanged, and thusx 13 andx 23 interchanged

    Supposem 13 ≥m 12 ≥m 23. This case is symmetric to Case 2 ofm 23 ≥m 12 ≥m 13, withm 13 andm 23 interchanged, and thusx 13 andx 23 interchanged. Similarly, we set r= 2 m13−m12 , s= 2 m23−m12 , h= 2 mGHZ+2m12+m23+mloc .(S85) Repeating the same argument as in Case 2 gives −2 + (d...

  67. [75]

    Then x12 ≤2 m12−m23 ≤2 m13−m23 , x 23 ≤2 m23−m13 , x 13 ≤2 m13−m23 ,Re(x 123)≤1.(S86) We denoter= 2 m23−m13, then1≤r≤d

    Supposem 23 ≥m 13 ≥m 12. Then x12 ≤2 m12−m23 ≤2 m13−m23 , x 23 ≤2 m23−m13 , x 13 ≤2 m13−m23 ,Re(x 123)≤1.(S86) We denoter= 2 m23−m13, then1≤r≤d. Thus −2 + (d−1)x 12 +x 23 +x 13 + 2 Re(x123)≤ −2 + (d−1)2 m13−m23 + 2m23−m13 + 2m13−m23 + 2 = d r +r ≤d+ 1. (S87)

  68. [76]

    1 + d2 +d(−2(δ−2)δ−3) + 2(δ−2)δ+ 4 (d(δ+f−1)−δ) 2(d−1)(δ−1)δ(d(δ−1)−δ) # +c +s+ +c −s− =p

    Supposem 13 ≥m 23 ≥m 12. This case is symmetric to Case 4, again by interchangingm 13 andm 23, and thusx 13 andx 23. Setr= 2 m13−m23 .Then1≤r≤d, and the same argument as in Case 4 gives −2 + (d−1)x 12 +x 23 +x 13 + 2 Re(x123)≤d+ 1.(S88) The above cases exhaust all possible ord...

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.