Pith. sign in

REVIEW 3 major objections 4 minor 109 references

Limits on broadcasting non-stabilizerness through unrestricted operations

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Quantum 'magic' cannot be broadcast universally: even unrestricted operations fail to clone non-stabilizerness beyond a fixed reference level.

desk verdict A correct first theorem and a suggestive numerics section, but the two headline no-broadcasting results for unrestricted operations rest on a false machine-orthogonality inference. read the letter →

arxiv 2501.15794 v1 pith:E5VD7EM2 submitted 2025-01-27 quant-ph

Rivu Gupta , Tanoy Kanti Konar , Ayan Patra , Aditi Sen De This is my paper · ORCID
classification quant-ph
keywords non-stabilizernessmagicbroadcastingquantumno-cloningstabilizeroperationsrobustnessofstate-dependentcloningresourcetheoryquditsystems
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

This paper asks whether 'magic'—the non-stabilizerness that makes quantum circuits hard to simulate classically—can be cloned or broadcast across many subsystems. It first proves that stabilizer operations, the free operations that cannot create the resource, cannot clone magic in any finite dimension, even when the auxiliary copy is only required to become non-stabilizer while the original keeps its magic. It then proves that the impossibility persists for unrestricted operations in a qualified sense: no fixed unitary can broadcast the magic of all qudit states, and a qubit unitary built to broadcast the magic of given reference states cannot broadcast any state with higher magic than those references. The paper closes with explicit conditions under which specific cloning machines broadcast magic perfectly, and numerical evidence that magic broadcasting demands more magic-generating power than ordinary state broadcasting.

What carries the argument

The load-bearing objects are the magic monotones used for each part of the argument: the order-2 stabilizer entropy and its extended version, which is additive under tensor products and monotone under partial trace, and the single-qubit robustness of magic $R(\rho)=\max\{1,\sum_j |m_j|\}$. The unifying mechanism is the broadcasting unitary $U(|\psi_i\rangle_S |0\rangle_A |\mu_0\rangle_M)=|\tilde{\psi}_i\rangle_{SA} |\mu_i\rangle_M$, together with the assumption that the machine states $|\mu_i\rangle_M$ are mutually orthogonal for orthogonal reference inputs. Orthogonality converts the output reduced states into convex mixtures of fixed reference outputs; convexity then caps the magic of any broadcast copy at the reference magic, which is exactly what blocks universal and above-reference broadcasting.

What would settle it

Search numerically over two-qubit unitary parameters for any single-qubit pure state whose input robustness of magic is larger than the reference-state robustness but whose output auxiliary copy, after the designed broadcasting unitary, still has the same magic as the input; finding one would refute Theorem 3, and measuring the inner product of machine states for two orthogonal reference inputs would check whether the assumed orthogonal-machine-state structure actually holds.

Watch

Extended reading notes

Core claim

The central claim is that non-stabilizerness is not freely replicable, even when no restriction is placed on the operations used. Theorem 1 rules out stabilizer cloning in all finite dimensions by showing that any successful clone would require the system's magic to fall below its initial value, contradicting the cloning condition. Theorem 2 rules out a universal qudit broadcasting unitary: because the output auxiliary state depends only on the squared amplitudes of the input superposition, it cannot reproduce the phase-dependent magic of every input. Theorem 3 sharpens this for qubits: convexity of the robustness of magic forces the broadcast output to have no more magic than the reference states the unitary was designed for, so an operation that broadcasts maximal-magic states still fails on generic lower-magic states. The same methods yield geometric conditions for perfect broadcasting and a characterization of how the original copying machine and the state-independent cloner broadcast magic.

Load-bearing premise

The weakest load-bearing step is the claim that orthogonal reference inputs force the machine states to be mutually orthogonal for every pair; without that, the output of the broadcasting unitary is not a convex mixture of the reference outputs and the proof does not go through.

Editorial extensions

If this is right

  • Stabilizer circuits cannot be used to bypass magic distillation in any finite dimension: a blank register plus free operations can never end with nonvanishing magic while preserving the input magic.
  • Every unrestricted magic broadcaster is state-specific and bounded by its reference magic, so practical protocols must match the target magic content to the machine design rather than aiming for a universal cloner.
  • The original copying machine, when set to a non-stabilizer reference state, can perfectly broadcast the magic of whole families of lower-magic states, not just the reference states themselves.
  • The state-independent cloner broadcasts maximal-magic states at a fixed 2/3 magic ratio, while low-magic states can be broadcast at ratios approaching 0.9 by tuning the machine parameters.
  • Perfect magic broadcasting and perfect state broadcasting are different tasks: magic can be copied with fidelity as low as 3 times 10^{-3}, and magic-broadcasting unitaries have higher average magic-generating power than state-broadcasting unitaries.

Reading between the lines

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

  • The phase-versus-amplitude mismatch behind Theorem 2 suggests a broader pattern: any resource measure that depends on complex phases of superpositions cannot be universally broadcast when outputs see only squared amplitudes, so the no-go may extend beyond magic to other coefficient-sensitive resources.
  • The geometric polytope condition for perfect qubit broadcasting may generalize to higher-dimensional stabilizer polytopes, giving explicit construction recipes for qutrit and qudit magic broadcasters; the paper does not work out this generalisation.
  • The numerical gap in magic-generating power between magic broadcasting and state broadcasting could be turned into a proof that magic-broadcasting circuits necessarily require more non-stabilizer gates than state-broadcasting ones, sharpening the resource-cost comparison with distillation.
  • If the no-go theorems are correct, the useful figure of merit for a magic-broadcasting protocol is not output fidelity but the pair of reference magic and input magic; protocols should be designed by first fixing the target magic level and then searching for unitaries within that level.
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

3 major / 4 minor

Summary. The manuscript studies broadcasting and cloning of non-stabilizerness ("magic") in quantum states. Theorem 1 claims that stabilizer operations cannot clone nonvanishing magic for all states in any finite dimension, using additivity and monotonicity of the extended stabilizer Rényi entropy after a Stinespring dilation. Theorems 2 and 3 claim that even with unrestricted operations there is no universal magic-broadcasting transformation for qudits, and that a transformation designed to broadcast the magic of specific orthogonal qubit states cannot broadcast states with higher magic. The paper then analyzes the Wootters-Zurek and Bužek-Hillery cloning machines, derives conditions for partial or perfect magic broadcasting, and presents numerical results comparing the magic-generating power of magic-broadcasting and state-broadcasting unitaries.

Significance. If the unrestricted-operation no-go results were valid, they would constitute a substantial extension of no-broadcasting ideas to the resource theory of non-stabilizerness, with implications for magic distillation and fault-tolerant quantum computation. The paper also contains a useful self-contained derivation of the single-qubit robustness-of-magic witness in Appendix A, and Theorem 1 appears sound: its proof uses Clifford invariance, additivity, and monotonicity of the Rényi measure without relying on the questionable orthogonality inference. However, the proofs of Theorems 2 and 3 contain a load-bearing logical gap concerning orthogonality of machine states. Until this gap is repaired, the central unrestricted-operation claims are not established as written.

major comments (3)
  1. [Section III, after Eq. (5)] The inference that orthogonal reference input states imply orthogonal machine states is invalid. Unitarity of U gives ⟨ψ̃_i|ψ̃_j⟩_{SA} ⟨µ_i|µ_j⟩_M = 0 for i ≠ j, and each zero product only forces at least one factor to vanish. The parenthetical "may not necessarily be orthogonal" does not rule out the case where the SA factors are orthogonal and the machine states are non-orthogonal. This matters because the proof of Theorem 2 uses machine orthogonality to assert ρ̃_Φ^{S(A)} = Σ_i |α_i|² ρ̃_i^{S(A)}; without machine orthogonality, cross terms proportional to ⟨µ_j|µ_i⟩ survive after tracing, so the convex-decomposition step is not justified and the subsequent coefficient-dependence argument collapses.
  2. [Section III, Eq. (6) and Theorem 3 proof] The expression ρ̃_ϕ^A = |α|² ρ̃_ψ^A + |β|² ρ̃_ψ'^A after tracing out S and M presupposes that the two product outputs |ψ̃⟩_SA |µ_1⟩_M and |ψ̃'⟩_SA |µ_2⟩_M are orthogonal in at least one factor, so that no cross term survives. The proof invokes machine-state orthogonality, which is not established. Consequently, the convexity bound R(ρ̃_ϕ^A) ≤ |α|² R(ρ̃_ψ^A) + |β|² R(ρ̃_ψ'^A) and the resulting conclusion that states with higher magic cannot be broadcast are not proven. A valid proof would need either a demonstration that machine states can be taken orthogonal without loss of generality or a different no-go argument.
  3. [Section III.A, Observation] The Observation that there exists no universal unitary broadcasting the magic of arbitrary non-stabilizer qubit states is justified only by the same convex-mixture assumption used in Theorem 3. Since that assumption is unsupported as described above, the Observation is likewise not established by the arguments given in the manuscript.
minor comments (4)
  1. [Section IV.B.2, Eq. (11)] The block of expressions after "2 α = ..." appears garbled: the quantities α and β are not clearly defined, and the displayed formula mixes undefined notation with what seems to be an attempted expression for a state or a parameter. Please rewrite this passage.
  2. [Section IV.A, conditions (1) and (2)] Condition (2) reads |⟨ψ|ρ̃_S(A)|ψ⟩|² ≤ 1 - ε, but the text describes it as corresponding to "perfect state broadcasting." For state broadcasting one would expect the fidelity to be close to 1, not bounded away from 1; please clarify whether the inequality direction is a typographical error and explain the intended numerical protocol.
  3. [Throughout] There is a minor typo in Section III where "the natural question to as is" should be "the natural question to ask is," and the phrase "Schwartz inequality" should be "Cauchy-Schwarz inequality."
  4. [References] References [54] and [95] are the same paper (Zhang, Feng, and Luo, Phys. Rev. A 110, 012462 (2024)); one duplicate citation should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the no-go results are derived from independent magic-measure properties and convexity, not from assumptions equivalent to the conclusions.

full rationale

The derivation chain is self-contained. Theorem 1 uses Clifford invariance, additivity and monotonicity of the stabilizer Rényi entropy (properties from Leone–Hamma and Leone–Bittel, not authors of this paper) and an additivity/monotonicity contradiction; no fitted parameter is renamed as a prediction. Theorems 2 and 3 are no-go statements derived from the assumed unitary broadcasting form (Eq. 5), the orthogonality of reference states, and convexity of the robustness of magic; their outputs are bounds or contradictions, not restatements of the inputs. The geometric conditions in Sec. III.B and the Wootters–Zurek/Buzek–Hillery analyses are explicit algebra from the cloning transformations, not fits masquerading as discoveries. The numerical section optimizes unitaries to satisfy stated broadcasting conditions and then compares magic-generating power; this is an optimization, not a fit of the theorem's conclusions. No load-bearing self-citation occurs: the cited properties of M2, cM2, and R come from independent groups. One non-circular concern should be recorded: the inference in Sec. III that ⟨ψ_i|ψ_j⟩=0 forces machine states ⟨µ_i|µ_j⟩=0 is not established by unitarity (unitarity only makes the full products orthogonal), and both Theorem 2's convex decomposition and Eq. (6) rely on it. That is a correctness/rigor gap, not a circularity, so it does not raise the circularity score.

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

The central analytical claims rest on two types of inputs: standard properties of the stabilizer Renyi entropy and robustness of magic, which are cited from the literature, and an ad hoc assumption about the orthogonality of machine states, which is not justified and is in fact false as stated. The numerical section introduces optimization parameters but these are not fitted constants on which a predictive claim depends.

assumptions (4)
  • ad hoc to paper The reference input states being orthogonal implies the machine states are mutually orthogonal.
    This is asserted before Theorem 2 with the justification that the output states may not be orthogonal, but that justification is logically false; orthogonality of product outputs does not require orthogonality of the machine factors.
  • standard math The stabilizer Renyi entropy M2 is additive over tensor products and the extended measure cM2 is monotonic under partial trace.
    Used in the proof of Theorem 1, Eqs. (4)-(5); these properties are established in Refs [64,69].
  • domain assumption Any stabilizer operation can be dilated to a Clifford unitary acting on the input plus a pure stabilizer environment state.
    Used in the proof of Theorem 1, around Eq. (3); standard for completely stabilizer-preserving operations but not proven here.
  • standard math Robustness of magic is convex for qubit states.
    Used in the proof of Theorem 3; convexity is established in Ref [79].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Limits on broadcasting non-stabilizerness through unrestricted operations." pith.science (2026). https://pith.science/paper/E5VD7EM2

@misc{pith2026250115794,
  author       = {Pith},
  title        = {Pith review of: Limits on broadcasting non-stabilizerness through unrestricted operations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5VD7EM2}},
  note         = {Machine review of arXiv:2501.15794}
}
read the original abstract

In the resource theory of non-stabilizerness, we prove that stabilizer operations cannot replicate or broadcast the "magic" resource of all quantum states in an arbitrary finite dimension. Moreover, we show that even in unrestricted scenarios, there are fundamental limits on cloning the non-stabilizerness content of quantum states. When using an auxiliary system as part of the cloning process, we show that it is impossible to broadcast the non-stabilizerness of qubits that possess a greater degree of non-stabilizerness than the known states on which the transformations are based. We also derive the conditions to broadcast magic perfectly using unrestricted operations. Furthermore, we compare non-stabilizerness broadcasting with traditional state cloning methods, such as state-dependent and -independent cloners, which can achieve perfect broadcasting of states with known magic content. Our findings reveal that state-dependent cloning unitaries designed for specific states have lower average non-stabilizerness-generating power than magic-generating unitaries.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

109 extracted references · 64 canonical work pages

  1. [1]

    |R( ˜ρS(A)) − R(|ψ⟩S⟨ψ|)| ≤ϵ

  2. [2]

    |⟨ψ| ˜ρS(A)|ψ⟩|2 ≤ 1 − ϵ, 6 where the first and the second conditions rep- resent perfect magic and state broadcasting re- spectively, up to numerical accuracy ϵ. The two-qubit unitary is parameterized as USA = U(1) S (κ1, λ1, ν1) ⊗ U(2) A (κ2, λ2, ν2) ⊗ ˆUSA (α, β, δ) ⊗ U(3) S (κ3, λ3, ν3) ⊗ U(4) A (κ4, λ4, ν4), where U(i) S(A) ∈ SU(2) and ˆUSA (α, β, δ)...

  3. [3]

    reference states

    The Wootters-Zurek copying machine - The seminal work on quantum cloning by Wootters and Zurek [ 78] proposes a transformation wherein the final output states have the form ρ′ S = σA = cos2 θ 2 ρψS + sin2 θ 2 ρψ⊥ S , ( 8) where we have considered that the input state is |ϕ⟩ = cos θ 2 |ψ⟩S + eiζ sin θ 2 ψ⊥ S and the machine can perfectly copy the state |ψ⟩...

  4. [4]

    The output copy has magic R(σA) = 1√ 2 (3 + √ 6)1/2 > √

  5. [5]

    Note that this does not violate Theorem 2, which does not guarantee the broadcasting of magic lower than that of the reference state but only says that it is possible

    This is because the unitary transformation is not necessarily a stabilizer operation and can itself generate magic. Note that this does not violate Theorem 2, which does not guarantee the broadcasting of magic lower than that of the reference state but only says that it is possible. How- ever, in this case, despite the state |H⟩ having less non- stabilize...

  6. [6]

    The Buzek-Hillery optimal cloning transformation - Let us now consider the state-independent cloner de- signed by Buzek and Hillery [ 62], whose optimality was established independently in Ref. [ 63]. The nov- elty of the state-independent transformation lies in the fact that it can prepare approximate clones of the in- put state with fidelity 2/3. In fac...

  7. [7]

    Such states are characterized by θ → 0 such that R → 1+

    In the limit ξ → 0, states with low amounts of magic can be broadcast with a high ratio. Such states are characterized by θ → 0 such that R → 1+. For example, at θ = 0.1 and ξ ≤ 0.02, we can obtain Mratio ∈ [0.88, 0.91], which is a marked improvement over the ratio furnished by the state- independent model

  8. [8]

    In fact, this is the only family of states whose magic can be broad- cast at a fixed ratio using the transformation rules just mentioned

    When θ = π/2, the magic of the family of states with different values of ζ can be broadcast with ratio Mratio = 1/ √ 2 which again overtakes the state-independent threshold. In fact, this is the only family of states whose magic can be broad- cast at a fixed ratio using the transformation rules just mentioned

Show all 109 references
  1. [9]

    INFOSYS scholarship for senior students

    For maximal magic states, the ratio of the final to the initial magic is optimum, Mratio = 2/3 when ξ = 1/6. The cloner can broadcast magic with a monotonically increasing ratio as the non- stabilizerness of the input state decreases. Although the broadcasting of states and ma...

    2022

  2. [10]

    Nielsen and I

    M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000)

    2000

  3. [11]

    Preskill, in Lecture Notes for Physics 229:Quantum Infor- mation and Computation (CreateSpace Independent Pub- lishing Platform, 2015)

    J. Preskill, in Lecture Notes for Physics 229:Quantum Infor- mation and Computation (CreateSpace Independent Pub- lishing Platform, 2015)

    2015

  4. [12]

    M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013)

    2013

  5. [13]

    Watrous, The Theory of Quantum Information (Cam- bridge University Press, 2018)

    J. Watrous, The Theory of Quantum Information (Cam- bridge University Press, 2018)

    2018

  6. [14]

    L. K. Grover, in Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC ’ 96 (ACM Press, 1996) pp. 212–219

    1996

  7. [15]

    P . W. Shor, SIAM Journal on Computing26, 1484 (1997)

    1997

  8. [16]

    Jozsa, Proceedings of the Royal Society of London

    R. Jozsa, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sci- ences 454, 323 (1998)

    1998

  9. [17]

    Horodecki, P

    R. Horodecki, P . Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009)

    2009

  10. [18]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017)

    2017

  11. [19]

    Shor (IEEE Comput

    P . Shor (IEEE Comput. Soc. Press, 1996) pp. 56–65

    1996

  12. [21]

    Gottesman, Phys

    D. Gottesman, Phys. Rev. A 54, 1862 (1996)

    1996

  13. [22]

    D. P . DiVincenzo and P . W. Shor, Phys. Rev. Lett.77, 3260 (1996)

    1996

  14. [23]

    Gottesman, Phys

    D. Gottesman, Phys. Rev. A 57, 127 (1998)

    1998

  15. [24]

    D. E. Koh, Quantum Information and Computation 17, 262 (2017)

    2017

  16. [25]

    Bouland, J

    A. Bouland, J. F. Fitzsimons, and D. E. Koh, in 33rd Computational Complexity Conference (CCC 2018), Leib- niz International Proceedings in Informatics (LIPIcs), Vol. 102, edited by R. A. Servedio (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany,

    2018

  17. [26]

    Yoganathan, R

    M. Yoganathan, R. Jozsa, and S. Strelchuk, Proceedings of the Royal Society A: Mathematical, Physical and En- gineering Sciences 475, 20180427 (2019)

    2019

  18. [27]

    Eastin and E

    B. Eastin and E. Knill, Phys. Rev. Lett. 102, 110502 (2009)

    2009

  19. [28]

    K. Bu, R. J. Garcia, A. Jaffe, D. E. Koh, and L. Li, Com- munications in Mathematical Physics 405, 161 (2024)

    2024

  20. [29]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P . B. Ataides, N. Maskara, I. Cong, X. Gao, P . S. Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V . Vuleti´ c, and M. D. Lukin, Na...

    2024

  21. [30]

    S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, npj Quantum Information 8, 148 (2022)

    2022

  22. [31]

    Haug and M

    T. Haug and M. Kim, PRX Quantum 4, 010301 (2023)

    2023

  23. [32]

    T. Haug, S. Lee, and M. S. Kim, Phys. Rev. Lett. 132, 240602 (2024)

    2024

  24. [33]

    Lami and M

    G. Lami and M. Collura, Phys. Rev. Lett. 131, 180401 (2023)

    2023

  25. [34]

    Odavi´ c, T

    J. Odavi´ c, T. Haug, G. Torre, A. Hamma, F. Franchini, and S. M. Giampaolo, SciPost Physics 15, 131 (2023)

    2023

  26. [35]

    P . S. Tarabunga, E. Tirrito, T. Chanda, and M. Dalmonte, PRX Quantum 4, 040317 (2023)

    2023

  27. [36]

    Haug and L

    T. Haug and L. Piroli, Quantum 7, 1092 (2023)

    2023

  28. [37]

    A. Gu, S. F. E. Oliviero, and L. Leone, Phys. Rev. A 110, 062427 (2024)

    2024

  29. [38]

    P . S. Tarabunga, E. Tirrito, M. C. Bañuls, and M. Dal- monte, Phys. Rev. Lett. 133, 010601 (2024)

    2024

  30. [39]

    Leone, S

    L. Leone, S. F. E. Oliviero, Y. Zhou, and A. Hamma, Quantum 5, 453 (2021)

    2021

  31. [40]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. A 107, 022429 (2023)

    2023

  32. [41]

    Turkeshi, M

    X. Turkeshi, M. Schirò, and P . Sierant, Phys. Rev. A 108, 10 042408 (2023)

    2023

  33. [42]

    R. J. Garcia, K. Bu, and A. Jaffe, Proceedings of the National Academy of Sciences 120 (2023), 10.1073/pnas.2217031120

    2023 doi

  34. [43]

    Leone, S

    L. Leone, S. F. E. Oliviero, G. Esposito, and A. Hamma, Phys. Rev. A 109, 032403 (2024)

    2024

  35. [44]

    A. Gu, L. Leone, S. Ghosh, J. Eisert, S. F. Yelin, and Y. Quek, Phys. Rev. Lett. 132, 210602 (2024)

    2024

  36. [45]

    Bansal, W

    N. Bansal, W. Mok, K. Bharti, D. E. Koh, and T. Haug, arXiv:2306.11607 (2024)

    2024 arXiv

  37. [46]

    T. Haug, K. Bharti, and D. E. Koh, arXiv: 2306.11677 (2023)

    2023 arXiv

  38. [47]

    Landauer, Physics Today 44, 23 (1991)

    R. Landauer, Physics Today 44, 23 (1991)

    1991

  39. [48]

    Landauer, Physica A: Statistical Mechanics and its Applications 263, 63 (1999)

    R. Landauer, Physica A: Statistical Mechanics and its Applications 263, 63 (1999)

    1999

  40. [49]

    J. L. Park, Foundations of Physics 1, 23 (1970)

    1970

  41. [51]

    Dieks, Physics Letters A 92, 271 (1982)

    D. Dieks, Physics Letters A 92, 271 (1982)

    1982

  42. [52]

    H. P . Yuen, Physics Letters A 113, 405 (1986)

    1986

  43. [53]

    Barnum, C

    H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev. Lett. 76, 2818 (1996)

    1996

  44. [54]

    Barnum, J

    H. Barnum, J. Barrett, M. Leifer, and A. Wilce, Phys. Rev. Lett. 99, 240501 (2007)

    2007

  45. [55]

    Piani, P

    M. Piani, P . Horodecki, and R. Horodecki, Phys. Rev. Lett. 100, 090502 (2008)

    2008

  46. [56]

    M. A. Nielsen and I. L. Chuang, Phys. Rev. Lett. 79, 321 (1997)

    1997

  47. [57]

    Mayers, Phys

    D. Mayers, Phys. Rev. Lett. 78, 3414 (1997)

    1997

  48. [58]

    A. K. Pati and S. L. Braunstein, Nature 404, 164 (2000)

    2000

  49. [59]

    S. L. Braunstein and A. K. Pati, Phys. Rev. Lett. 98, 080502 (2007)

    2007

  50. [60]

    K. Modi, A. K. Pati, A. Sen(De), and U. Sen, Phys. Rev. Lett. 120, 230501 (2018)

    2018

  51. [62]

    Patel, S

    D. Patel, S. Patro, C. Vanarasa, I. Chakrabarty, and A. K. Pati, Phys. Rev. A 103, 022422 (2021)

    2021

  52. [64]

    J. Son, R. Ganardi, S. Minagawa, F. Buscemi, S. H. Lie, and N. H. Y. Ng, arXiv:2412.06900 (2024)

    2024

  53. [65]

    Veitch, S

    V . Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New Journal of Physics 16, 013009 (2014)

    2014

  54. [66]

    Bravyi and A

    S. Bravyi and A. Kitaev, Physical Review A 71, 022316 (2005)

    2005

  55. [67]

    Jones, Phys

    C. Jones, Phys. Rev. A 87, 042305 (2013)

    2013

  56. [68]

    O’Gorman and E

    J. O’Gorman and E. T. Campbell, Phys. Rev. A 95, 032338 (2017)

    2017

  57. [69]

    M. B. Hastings and J. Haah, Phys. Rev. Lett. 120, 050504 (2018)

    2018

  58. [70]

    Krishna and J.-P

    A. Krishna and J.-P . Tillich, Phys. Rev. Lett. 123, 070507 (2019)

    2019

  59. [71]

    Bužek and M

    V . Bužek and M. Hillery, Physical Review A - Atomic, Molecular, and Optical Physics 54, 1844 (1996)

    1996

  60. [72]

    D. Bruß, D. P . DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Physical Review A 57, 2368 (1998)

    1998

  61. [73]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. Lett. 128, 050402 (2022)

    2022

  62. [74]

    Kliuchnikov, D

    V . Kliuchnikov, D. Maslov, and M. Mosca, Quantum Info. Comput. 13, 607–630 (2013)

    2013

  63. [75]

    Gheorghiu, M

    V . Gheorghiu, M. Mosca, and P . Mukhopadhyay, npj Quantum Information 8, 141 (2022)

    2022

  64. [76]

    Lostaglio and M

    M. Lostaglio and M. P . Müller, Phys. Rev. Lett. 123, 020403 (2019)

    2019

  65. [77]

    Scarani, S

    V . Scarani, S. Iblisdir, N. Gisin, and A. Acín, Rev. Mod. Phys. 77, 1225 (2005)

    2005

  66. [78]

    Leone and L

    L. Leone and L. Bittel, Phys. Rev. A 110, L040403 (2024)

    2024

  67. [79]

    Grassl, A

    M. Grassl, A. Klappenecker, and M. Rotteler, in Proceed- ings IEEE International Symposium on Information Theory, (IEEE, 2002) p. 45

    2002

  68. [80]

    Hostens, J

    E. Hostens, J. Dehaene, and B. De Moor, Phys. Rev. A 71, 042315 (2005)

    2005

  69. [81]

    Gheorghiu, Physics Letters A 378, 505 (2014)

    V . Gheorghiu, Physics Letters A 378, 505 (2014)

    2014

  70. [82]

    W. F. Stinespring, Proceedings of the American Mathe- matical Society 6, 211 (1955)

    1955

  71. [83]

    Gottesman, in Group22: Proceedings of the XXII In- ternational Colloquium on Group Theoretical Methods in Physics, edited by S

    D. Gottesman, in Group22: Proceedings of the XXII In- ternational Colloquium on Group Theoretical Methods in Physics, edited by S. P . Corney, R. Delbourgo, and P . D. Jarvis (Cambridge, MA, International Press, 1999) pp. 32–43

    1999

  72. [84]

    Ahmadi, H

    M. Ahmadi, H. B. Dang, G. Gour, and B. C. Sanders, Phys. Rev. A 97, 062332 (2018)

    2018

  73. [85]

    X. Wang, M. M. Wilde, and Y. Su, New Journal of Physics 21, 103002 (2019)

    2019

  74. [86]

    Gour, arXiv: 2402.05474 (2024)

    G. Gour, arXiv: 2402.05474 (2024)

    2024

  75. [87]

    W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982)

    1982

  76. [88]

    Howard and E

    M. Howard and E. Campbell, Phys. Rev. Lett. 118, 090501 (2017)

    2017

  77. [89]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Phys. Rev. X 10, 011058 (2020)

    2020

  78. [90]

    Gisin and S

    N. Gisin and S. Massar, Physical Review Letters 79, 2153 (1997)

    1997

  79. [91]

    Bruss, A

    D. Bruss, A. Ekert, and C. Macchiavello, Physical Re- view Letters 81, 2598 (1998)

    1998

  80. [92]

    R. F. Werner, Physical Review A 58, 1827 (1998)

    1998

  81. [93]

    Keyl and R

    M. Keyl and R. F. Werner, Journal of Mathematical Physics 40, 3283 (1999)

    1999

  82. [94]

    Niu and R

    C.-S. Niu and R. B. Griffiths, Physical Review A 58, 4377 (1998)

    1998

  83. [95]

    N. J. Cerf, Acta Phys. Slov 48, 115 (1998)

    1998

  84. [96]

    Buzek, M

    V . Buzek, M. Hillery, and R. Bednik, Acta Phys. Slov 48, 177 (1998)

    1998

  85. [97]

    N. J. Cerf, Journal of Modern Optics 47, 187 (2000)

    2000

  86. [98]

    S. L. Braunstein, V . Bužek, and M. Hillery, Physical Re- view A 63, 052313 (2001)

    2001

  87. [99]

    Iblisdir, A

    S. Iblisdir, A. Acin, and N. Gisin, arXiv:quant- ph/0505152 (2005)

    2005

  88. [100]

    Iblisdir, A

    S. Iblisdir, A. Acín, N. J. Cerf, R. Filip, J. Fiurášek, and N. Gisin, Physical Review A 72, 042328 (2005)

    2005

  89. [101]

    Vatan and C

    F. Vatan and C. Williams, Phys. Rev. A69, 032315 (2004)

    2004

  90. [102]

    T. P . Runarsson and X. Yao, IEEE Transactions on Sys- tems, Man, and Cybernetics, Part C (Applications and Reviews) 35, 233 (2005)

    2005

  91. [103]

    Gisin, G

    N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002)

    2002

  92. [104]

    Zhang, L

    Z. Zhang, L. Feng, and S. Luo, Phys. Rev. A 110, 012462 (2024)

    2024

  93. [105]

    Monroe, R

    C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P . Maunz, L.-M. Duan, and J. Kim, Physical Review A 89, 022317 (2014)

    2014

  94. [106]

    Van Meter and S

    R. Van Meter and S. J. Devitt, Computer 49, 31 (2016)

    2016

  95. [107]

    Takeuchi, K

    Y. Takeuchi, K. Fujii, R. Ikuta, T. Yamamoto, and N. Imoto, Physical Review A 93, 052307 (2016)

    2016

  96. [108]

    J. F. Fitzsimons, npj Quantum Information 3, 23 (2017)

    2017

  97. [109]

    Sheng and L

    Y.-B. Sheng and L. Zhou, Physical Review A 98, 052343 (2018)

    2018

  98. [110]

    Sanderson and R

    C. Sanderson and R. Curtin, The Journal of Open Source 11 Software 1, 26 (2016)

    2016

  99. [111]

    Sanderson and R

    C. Sanderson and R. Curtin, Mathematical and Compu- tational Applications 24, 70 (2019)

    2019

  100. [112]

    Quantum information and computation library (qiclib),

    T. Chanda, “Quantum information and computation library (qiclib),” https://github.com/titaschanda/ QIClib (2017)

    2017

  101. [113]

    The NLopt nonlinear-optimization package,

    S. G. Johnson, “The NLopt nonlinear-optimization package,” https://github.com/stevengj/nlopt (2007)

    2007

Pith tools

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