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  • Open Access

Triangle Criterion: A Mixed-State Magic Criterion with Applications in Distillation and Detection

Zhenhuan Liu1,*,†, Tobias Haug1,*,‡, Qi Ye2,3,§, Zi-Wen Liu4,∥, and Ingo Roth1,¶

  • *These authors contributed equally to this work.
  • †Contact author: qubithuan@gmail.com
  • ‡Contact author: tobias.haug@u.nus.edu
  • §Contact author: yeq22@mails.tsinghua.edu.cn
  • ∥Contact author: zwliu0@tsinghua.edu.cn
  • Contact author: ingo.roth@tii.ae
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PRX Quantum 7, 033016 – Published 24 July, 2026

DOI: https://doi.org/10.1103/rcpf-8nh9

Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the positive partial transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidence. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

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Popular Summary

Quantum computers promise capabilities far beyond those of classical machines, but realizing this power requires carefully managing fragile quantum resources. One such resource is magic, which enables universal quantum computation when combined with otherwise classically simulable operations. In realistic quantum devices, noise is unavoidable, so an important challenge is to understand how magic can be identified and used in noisy, mixed quantum states.

In this work, we draw direct inspiration from entanglement theory, where simple yet powerful criteria have played a central role in understanding complex quantum correlations. Motivated by these ideas, we introduce a new method for detecting quantum magic in mixed states, called the Triangle Criterion. Much like well-known entanglement tests, this criterion has a clear geometric interpretation and a direct operational meaning: it tells us exactly when a noisy multi-qubit state can be transformed, using standard fault-tolerant operations, into a basic magic state useful for computation.

Using this approach, we uncover new features of how magic behaves in multi-qubit systems. In particular, we show that processing several states together can be fundamentally more powerful than treating them individually. Our results also reveal intrinsic limits on how mixed a magic state can be and show that some magic states can be surprisingly hard to detect. Taken together, these findings suggest deep structural parallels, and important differences, between magic and entanglement, pointing toward a more unified understanding of quantum resources.

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References (54)

  1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  2. V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer computation, New J. Phys. 16, 013009 (2014).
  3. M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 1 (2007).
  4. O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009).
  5. D. Gottesman, The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006.
  6. S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  7. V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computation, New J. Phys. 14, 113011 (2012).
  8. M. Howard and E. Campbell, Application of a resource theory for magic states to fault-tolerant quantum computing, Phys. Rev. Lett. 118, 090501 (2017).
  9. J. R. Seddon and E. T. Campbell, Quantifying magic for multi-qubit operations, Proc. R. Soc. A 475, 20190251 (2019).
  10. X. Wang, M. M. Wilde, and Y. Su, Efficiently computable bounds for magic state distillation, Phys. Rev. Lett. 124, 090505 (2020).
  11. L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer Rényi entropy, Phys. Rev. Lett. 128, 050402 (2022).
  12. Z.-W. Liu and A. Winter, Many-body quantum magic, PRX Quantum 3, 020333 (2022).
  13. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  14. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  15. M. Horodecki, P. Horodecki, and R. Horodecki, Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature?, Phys. Rev. Lett. 80, 5239 (1998).
  16. G. Vidal and R. F. Werner, A computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  17. L. Gurvits and H. Barnum, Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A 66, 062311 (2002).
  18. K. Audenaert, M. B. Plenio, and J. Eisert, The entanglement cost under operations preserving the positivity of partial transpose, Phys. Rev. Lett. 90, 027901 (2003).
  19. X. Wang and M. M. Wilde, Exact entanglement cost of quantum states and channels under PPT-preserving operations, Phys. Rev. A 107, 012429 (2023).
  20. A. L. Shaw, Z. Chen, J. Choi, D. K. Mark, P. Scholl, R. Finkelstein, A. Elben, S. Choi, and M. Endres, Benchmarking highly entangled states on a 60-atom analogue quantum simulator, Nature (London) 628, 71 (2024).
  21. P. Calabrese, J. Cardy, and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109, 130502 (2012).
  22. B. W. Reichardt, Improved magic states distillation for quantum universality, Quantum Inf. Process. 4, 251 (2005).
  23. B. W. Reichardt, Quantum universality by state distillation, Quantum Inf. Comput. 9, 1030 (2009).
  24. C. Okay, M. Zurel, and R. Raussendorf, On the extremal points of the Λ-polytopes and classical simulation of quantum computation with magic states, Quantum Inf. Comput. 21, 1091 (2021).
  25. H. J. García, I. L. Markov, and A. W. Cross, On the geometry of stabilizer states, Quantum Inf. Comput. 14, 683 (2014).
  26. J. Gray, L. Banchi, A. Bayat, and S. Bose, Machine-learning-assisted many-body entanglement measurement, Phys. Rev. Lett. 121, 150503 (2018).
  27. A. Elben et al., Mixed-state entanglement from local randomized measurements, Phys. Rev. Lett. 125, 200501 (2020).
  28. K. Wang, Z. Song, X. Zhao, Z. Wang, and X. Wang, Detecting and quantifying entanglement on near-term quantum devices, npj Quantum Inf. 8, 52 (2022).
  29. R. Zhang, Z. Liu, C. Yang, Y.-Y. Fei, X.-F. Yin, Y. Mao, L. Li, N.-L. Liu, Y.-A. Chen, and J.-W. Pan, Entanglement detection with variational quantum interference: Theory and experiment, arXiv:2505.24764.
  30. P. S. Tarabunga and T. Haug, (2025). Quantifying mixed-state entanglement via partial transpose and realignment moments, arXiv:2507.13840.
  31. M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Rev. Math. Phys. 15, 629 (2003).
  32. E. T. Campbell and D. E. Browne, Bound states for magic state distillation in fault-tolerant quantum computation, Phys. Rev. Lett. 104, 030503 (2010).
  33. K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A:Math. Gen. 34, 7111 (2001).
  34. P. Liu, Z. Liu, S. Chen, and X. Ma, Fundamental limitation on the detectability of entanglement, Phys. Rev. Lett. 129, 230503 (2022).
  35. N. Bansal, W.-K. Mok, K. Bharti, D. E. Koh, and T. Haug, Pseudorandom density matrices, PRX Quantum 6, 020322 (2025).
  36. J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. Chwalla, C. F. Roos, M. Hennrich, and R. Blatt, Experimental multiparticle entanglement dynamics induced by decoherence, Nat. Phys. 6, 943 (2010). 10.1038/nphys1781
  37. G. Aubrun, S. J. Szarek, and D. Ye, Phase transitions for random states and a semicircle law for the partial transpose, Phys. Rev. A 85, 030302 (2012).
  38. A. Bakshi, A. Liu, A. Moitra, and E. Tang, High-temperature Gibbs states are unentangled and efficiently preparable, in 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) (2024), pp. 1027–1036.
  39. A. Palhares, S. Zamora, R. A. Macêdo, T. S. Sarubi, J. M. Varela, G. W. Rocha, D. A. Moreira, and R. Chaves, A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems, Phys. Lett. A 576, 131417 (2026).
  40. L. Leone, J. Eisert, and S. F. Oliviero, The unbearable hardness of deciding about magic, arXiv:2602.22330.
  41. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, Multiphoton entanglement and interferometry, Rev. Mod. Phys. 84, 777 (2012).
  42. M. Weilenmann, B. Dive, D. Trillo, E. A. Aguilar, and M. Navascués, Entanglement detection beyond measuring fidelities, Phys. Rev. Lett. 124, 200502 (2020).
  43. O. Gühne, Y. Mao, and X.-D. Yu, Geometry of faithful entanglement, Phys. Rev. Lett. 126, 140503 (2021).
  44. M. Zurel and J. Davis, Basis-independent stabilizerness and maximally noisy magic states, arXiv:2602.22336.
  45. I. Bengtsson and Å. Ericsson, Mutually unbiased bases and the complementarity polytope, Open Syst. Inf. Dyn. 12, 107 (2005).
  46. H. Zhu, Mutually unbiased bases as minimal Clifford covariant 2-designs, Phys. Rev. A 91, 060301 (2015).
  47. C. Palazuelos and J. I. de Vicente, Genuine multipartite entanglement of quantum states in the multiple-copy scenario, Quantum 6, 735 (2022).
  48. T. Haug and P. S. Tarabunga, Efficient witnessing and testing of magic in mixed quantum states, npj Quantum Inf. 12, 40 (2026).
  49. X.-D. Yu, S. Imai, and O. Gühne, Optimal entanglement certification from moments of the partial transpose, Phys. Rev. Lett. 127, 060504 (2021).
  50. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Complete family of separability criteria, Phys. Rev. A 69, 022308 (2004).
  51. Y. Shi, Both Toffoli and controlled-not need little help to do universal quantum computation, Quantum Inf. Comput. 3, 84 (2002).
  52. J. Haah and M. B. Hastings, Codes and protocols for distilling T, controlled-S, and Toffoli Gates, Quantum 2, 71 (2018).
  53. A. Kenfack and K. Życzkowski, Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B:Quantum Semiclassical Opt. 6, 396 (2004).
  54. D. Gross, Hudson’s theorem for finite-dimensional quantum systems, J. Math. Phys. (N.Y.) 47, 122107 (2006).

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PRX Quantum 7, 033016– Published 24 July, 2026
  • Received 23 December 2025
  • Accepted 5 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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