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REVIEW 3 major objections 4 minor 39 references

Highly-entangled, highly-doped states that are efficiently cross-device verifiable

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

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Pith's one-line read This paper introduces a class of highly entangled real quantum states and proves that two parties can efficiently estimate the inner product of any two such states via Bell sampling, without sharing classical descriptions.

desk verdict Nice family of states and clean hardness proofs, but the CDF vanishing claim is wrong for odd n, leaving the main efficiency theorem unproven as written (though likely repairable). read the letter →

arxiv 2501.11688 v2 pith:L2BJORMG submitted 2025-01-20 quant-ph

Janek Denzler , Santiago Varona , Tommaso Guaita , Jose Carrasco This is my paper · ORCID
classification quant-ph MSC 81P4581P68 PACS 03.67.-a
keywords cross-deviceverificationdistributedinnerproductestimationBellsamplingPauliClifford-rotatedW-statesrealCliffordgroupt-dopedstabilizerstatesRényientanglemententropy
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 introduces a family of real quantum states—real Clifford rotations of the W-state—and proves that two parties holding copies of any states in this family can efficiently estimate their overlap using Bell sampling and classical communication, a task called distributed inner product estimation. The estimates require only polynomially many two-copy Bell measurements and single-copy Pauli measurements, and the sample count can be determined experimentally in advance. This matters because previous efficient schemes relied on learning a classical description of each state and sharing it, an approach that fails for states with high entanglement or many non-Clifford gates. The paper shows the W-state class contains states with linearly growing half-system entanglement and states that cannot be approximated by circuits with few T-gates, yet their overlap is still efficiently verifiable. The protocol is also robust to preparation noise up to a small trace-distance error.

What carries the argument

The key object is the state class CW(n) = {C|W_n> : C ∈ rCl(n)}, where rCl(n) is the real Clifford group and |W_n> is the n-qubit W-state. The protocol is rDIPE, a Bell-sampling estimator of the normalized overlap c(rho,$\sigma$) that uses N1 two-copy Bell measurements to sample from the mixture of Bell distributions and N1 N2 single-copy Pauli measurements to estimate the Pauli expectations at the sampled strings. The proof that rDIPE is efficient on CW rests on two facts: for real pure states the Pauli distribution p_rho equals the Bell distribution q_rho, making the total-variation term $\Delta$ vanish; and the CDF of the Pauli expectation values of a W-state satisfies F_rho(eps)=0 for eps < 4/$n^{2}$, since its only nonzero expectations are 0, ±2/n, and 1-2z/n. These bounds turn the protocol's error estimate into a polynomially decaying exponential. The entanglement and non-approximability results use the real Clifford group being an orthogonal 2-design and a counting argument over Pauli strings with large expectation.

What would settle it

Randomly sample real Clifford unitaries C on n qubits, compute the half-system second Renyi entropy of C|W_n>, and average: if the average does not grow linearly in n, Proposition 2 is false. Likewise, an explicit search for an (n/4)-doped stabilizer state within trace distance 1/4 of a CW state would refute Proposition 3.

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Extended reading notes

Core claim

The paper's central claim is that for every sequence of states taken from CW(n) = {C|W_n> : C ∈ rCl(n)}, the rDIPE protocol—the real-state version of the Pauli-sampling inner-product estimator—is efficient: for any accuracy epsilon and failure probability $e^{{-delta}}$, polynomially many samples suffice to estimate c(rho,$\sigma$)=tr(rho $\sigma$)/$\sqrt$(tr $rho^{2}$ tr $sigma^{2}$) within epsilon. Efficiency relies on two features of CW. Because the states are real and pure, the Pauli distribution equals the Bell distribution, so the distribution-mismatch term $\Delta$ vanishes exactly. And because the underlying W-state has Pauli expectation values only 0, ±2/n, or 1-2z/n, its CDF vanishes below 4/$n^{2}$, making the error bound polynomial. The same class contains, by an orthogonal-2-design averaging argument, states whose second Renyi entropy across a half-system cut grows linearly in n, and, by a stabilizer-counting argument, states that any t-doped stabilizer approximation must miss by trace distance at least 1/4 unless t > n/4.

Load-bearing premise

The entire efficiency argument assumes the states have real amplitudes, so that p_rho equals q_rho and the distribution-mismatch term $\Delta$ vanishes.

Editorial extensions

If this is right

  • Cross-device verification no longer requires the states being compared to have efficient classical descriptions; highly entangled, highly doped states can be verified with polynomial local measurements.
  • The protocol is implementable in the near term, since each party only needs to prepare two copies of its state in a Bell measurement and perform single-copy Pauli measurements; the total sample complexity scales polynomially in n and 1/epsilon.
  • The required number of samples can be computed in advance from the parties' own measurement data using the empirical CDF and the Dvoretzky–Kiefer–Wolfowitz inequality, so the protocol is self-calibrating.
  • The protocol remains useful under realistic noise: states prepared within trace distance tau of a real CW state yield overlap estimates accurate up to O(tau), with polynomial resources.
  • The construction generalizes to Dicke states with constant excitation number and to superpositions of poly(n) computational basis states under real Cliffords, so the class of verifiable states is broader than the W-state itself.

Reading between the lines

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

  • If the realness condition can be relaxed by randomized compiling (which the paper explicitly does not rely on), the same Bell-sampling identity would apply to a wider class of near-real states; we would predict that the O(tau) robustness bound is tight in the imaginary-error direction, i.e., that small coherent Y-type errors are the dominant deviation.
  • The counting argument suggests a general trade-off: any family of states with a limited number of large Pauli expectation values will be hard for t-doped stabilizer approximation but easy for rDIPE, so searching for other such families (e.g., states from Haar-random circuits with real amplitudes) could extend the result.
  • Since the protocol reveals only sampled Pauli expectations, it provides a route to confidential verification—parties can certify they hold the same state without communicating enough information to reconstruct it; this is an implicit connection to quantum cryptography that the paper leaves as a future direction.
  • A testable extension: run the numerical simulation from Fig. 2 on the CW class itself for n up to ~30 and compare the empirical CDF with the analytical 4/n^2 threshold; a mismatch would indicate either an implementation error or a subtlety in the real-Clifford twirl assumption.
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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 paper introduces the class CW(n) of states obtained by applying real Clifford unitaries to the n-qubit W state, and claims that for any two sequences in CW the rDIPE protocol of Ref. [18] estimates the normalized inner product tr(ρσ)/sqrt(trρ²trσ²) with polynomially many copies. The authors argue that CW contains states with linear half-system second Rényi entropy and states that need more than n/4 T gates to approximate, so the protocol goes beyond the usual learn-and-share regime. They also claim robustness to trace-distance noise and propose an experimental procedure to pre-estimate the required number of samples.

Significance. If the main claims are correct, the paper is a valuable conceptual advance: it gives an explicit family of states that are highly entangled and highly non-Clifford, yet for which distributed inner product estimation is efficient via Pauli sampling, and it provides a concrete near-term measurement schedule. The class CW is simple, the Clifford-invariance observation is clean, and the paper correctly identifies that the key quantity is the Pauli CDF gap. The direct calculations of Pauli expectations for W-like states and the average entanglement computation are useful. However, the central efficiency theorem has a parity-dependent gap in its CDF argument, and the mixed-state concentration bound needed for the robustness statement is imported without proof; both issues must be fixed before the claims can be regarded as established.

major comments (3)
  1. [Appendix C, Eq. (C6); Proposition 1] The claimed CDF gap F(ε)=0 for ε<4/n² is false for odd n. For the n-qubit W state with n odd, Pauli strings consisting only of Z factors with weight (n±1)/2 have expectation ±1/n and hence squared expectation 1/n², so F(1/n²)>0. Consequently the proof of Proposition 1, which sets ε₂=min{(ε/8)², 3/n²} and uses '3/n² < 4/n²' to conclude F(ε₂)=0, is invalid for odd n: the inequality 4ε₁+4√ε₂+2F(ε₂)≤ε is not established. The same erroneous gap is used in Eq. (C7), where Lemma 3 is applied with 2ε₂<4/n², and in Appendix D, where ε₂=1/n² requires F(2/n²)=0. As written, the main efficiency claim and the robustness theorem are unproven for all odd n. The error appears repairable by choosing ε₂<1/n² for odd n (or by treating finitely many small n separately), but the proof must be corrected.
  2. [Section II, Eq. (4); Appendix D] The paper relies on a mixed-state version of the rDIPE concentration bound, Eq. (4), but does not prove it. The sentence 'The proof of this statement applies with minor modifications also to our case of mixed states' is not a derivation, and the robustness theorem is stated precisely for mixed states with trρ²,trσ²>1/2. Since Eq. (9) is obtained by substituting into Eq. (4), the mixed-state extension is load-bearing. In addition, Algorithm 1 and Eq. (4) assume that the purities A and B are known exactly; the text in Appendix A acknowledges this but does not incorporate the estimation error into the concentration bound. Please provide a precise statement and proof of the mixed-state bound, with the purity-estimation error included or explicitly accounted for in the theorem hypotheses.
  3. [Proposition 3 proof] The step 'Therefore, there exists at least one Pauli operator P with |⟨P⟩_ρ−⟨P⟩_ρ′|≥1/4' does not follow from the two displayed cardinality inequalities. The high-expectation set for ρ has size <2^{3n/4} and the set of ±1-expectation Paulis for ρ′ has size ≥2^{3n/4}, but this only shows that a Pauli with |⟨P⟩_ρ|>3/4 either lies in the ±1 set of ρ′ (which would give the desired gap) or lies outside it; in the latter case no bound on |⟨P⟩_ρ′| is provided. Additional structural information about t-doped states is needed to rule out the possibility that all large-expectation Paulis of ρ have intermediate expectation in ρ′. This proof gap affects the 'highly-doped' part of the paper's central claim and should be closed.
minor comments (4)
  1. [Algorithm 1, input line] The input specification reads 'computed by using usingN1 copies' and should read 'using N1 copies'.
  2. [Section V] The sample pre-estimation procedure explicitly assumes that the prepared state is real, which is consistent with the paper's stated realness limitation, but this assumption should be restated in the theorem or proposition that summarizes Section V so that it is not read as applying to arbitrary noisy inputs.
  3. [Eq. (C6)] Even after the parity fix, Eq. (C6) should state the gap separately for even and odd n: for even n the gap is 4/n², while for odd n it is 1/n². The current unified statement is misleading.
  4. [Appendix D, Eq. (D1)] The bound TV(q_{ρ′},q_ρ)≤∥ρ−ρ′∥_tr is quoted from the data-processing inequality; the derivation would be clearer if the factor 1/2 in the trace-distance convention were stated explicitly, since the final constant k=29 depends on the convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central efficiency claim is derived from a direct computation of Pauli expectation values for Clifford orbits of the W state, with the cited rDIPE bound used as a state-agnostic tool.

full rationale

The derivation of Proposition 1 is self-contained with respect to the target states. The key input, Eq. (C6), is obtained by a direct calculation of the Pauli expectation values of the W state, and the Clifford invariance of the CDF then extends it to all of CW(n). No parameter is fitted to data, no quantity is defined in terms of the prediction, and the sample-complexity bound follows by substituting the computed vanishing CDF into the general rDIPE inequality, Eq. (4). That inequality is imported from Ref. [18], which shares an author, but it is a state-agnostic theorem about the rDIPE estimator and does not assume the CW property or the desired conclusion; it is therefore independent support rather than a load-bearing self-citation. The self-citations to Refs. [18,21] concern respectively a general estimator bound and the real Clifford 2-design property, both external to this paper's construction. The robustness argument in Appendix D similarly uses only the computed CDF bound, Lemma 3, and triangle inequalities, with no fitted input. The acknowledged realness limitation is explicit and not circular. The skeptical objection that Eq. (C6) fails for odd n because W-state Pauli expectations can equal 1/n^2 is a correctness gap in the proof as written, not a circularity: even if the bound is wrong, it is not true by construction or by self-citation. For the circularity question, the derivation chain does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on imported concentration bounds from Ref [18] and on the realness condition p_rho = q_rho. No free parameters are fitted to data. The only ad hoc element is the mixed-state extension of Eq. (4), which is assumed without proof. The CW state family is a mathematical construction, not a newly postulated physical entity.

assumptions (6)
  • standard math Real Clifford group rCl(n) is an orthogonal 2-design
    Used in Proposition 2 to replace the Haar average over the Clifford group by a 2-fold twirl over the orthogonal group; imported from Ref [21].
  • standard math Commutant basis of O(2^n) x O(2^n) is spanned by Psym, Pasym, and B
    Used in the entropy calculation in Appendix B; taken from Refs [36,37].
  • domain assumption For real pure states, the Pauli distribution p_rho equals the Bell distribution q_rho
    Used throughout, especially to set Delta=0 in Proposition 1 and in the robustness proof; from Ref [19].
  • ad hoc to paper The concentration bound in Eq. (4) from Ref [18] holds for mixed states with tr(rho^2), tr(sigma^2) > 1/2 after minor modifications
    Load-bearing for the robustness theorem in Appendix D, but the paper does not show the modifications or provide a proof.
  • domain assumption The estimator in Algorithm 1 uses exact values of the purities A=tr(rho^2) and B=tr(sigma^2)
    Acknowledged in Appendix A; in practice A and B need to be estimated, adding an error term that is only sketched.
  • standard math Trace-norm closeness implies TV(q_rho', q_rho) <= ||rho - rho'||_tr via data processing
    Used in the robustness proof; standard inequality for quantum channels.

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Cite this review

Pith. "Pith review of Highly-entangled, highly-doped states that are efficiently cross-device verifiable." pith.science (2026). https://pith.science/paper/L2BJORMG

@misc{pith2026250111688,
  author       = {Pith},
  title        = {Pith review of: Highly-entangled, highly-doped states that are efficiently cross-device verifiable},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L2BJORMG}},
  note         = {Machine review of arXiv:2501.11688}
}
abstract

In this paper, we introduce a class of highly entangled real quantum states that cannot be approximated by circuits with $\log$-many non-Clifford gates and prove that Bell sampling enables efficient cross-device verification (or distributed inner product estimation) for these states. That is, two remote parties can estimate the inner product ${\rm tr}(\rho\sigma)$, each having black-box access to copies of a state $\rho$ (or respectively $\sigma$) in this class. This is significant because it is clear that this task can be achieved in those cases (such as low entanglement or low non-Clifford gate count) where one can independently learn efficient classical descriptions of each state using established techniques and share the description to compute the overlap. Instead, our results demonstrate that this is possible even in more complex scenarios where these "learn and share" methods are insufficient. Our proposal is scalable, as it just requires a number of two-copy Bell measurements and single-copy Pauli measurements that grows polynomially with both the number of qubits and the desired inverse-error, and can be implemented in the near term. Moreover, the required number of samples can be efficiently experimentally determined by the parties in advance, and our findings are robust against preparation errors. We anticipate that these results could have applications in quantum cryptography and verification.

Figures

Figures reproduced from arXiv: 2501.11688 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Works this paper leans on

39 extracted references · 17 canonical work pages

  1. [18]

    Carrasco, Efficient distributed inner product estima- tion via Pauli sampling (2024), arXiv:2405.06544 [quant- ph]

    M.Hinsche, M.Ioannou, S.Jerbi, L.Leone, J.Eisert,and J. Carrasco, Efficient distributed inner product estima- tion via Pauli sampling (2024), arXiv:2405.06544 [quant- ph]

    arXiv 2024

  2. [1]

    Eisert, D

    J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum cer- tification and benchmarking, Nature Reviews Physics2, 382 (2020)

    2020

  3. [2]

    O’Donnell and J

    R. O’Donnell and J. Wright, Efficient quantum tomog- raphy, in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing , STOC ’16 (Asso- ciation for Computing Machinery, New York, NY, USA,

  4. [3]

    M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Prac- tical characterization of quantum devices without tomog- raphy, Physical Review Letters107, 210404 (2011)

    2011

  5. [4]

    S. T. Flammia and Y.-K. Liu, Direct Fidelity Estimation from Few Pauli Measurements, Physical Review Letters 106, 230501 (2011)

    work page 2011

  6. [5]

    Schuch, M

    N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, En- tropy scaling and simulability by matrix product states, Phys. Rev. Lett.100, 030504 (2008)

    2008

  7. [6]

    Chia, C.-Y

    N.-H. Chia, C.-Y. Lai, and H.-H. Lin, Efficient learning of t-doped stabilizer states with single-copy measurements, Quantum 8, 1250 (2024)

    work page 2024

  8. [7]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Learning t-doped stabilizer states, Quantum8, 1361 (2024)

    2024

Show all 39 references
  1. [8]

    Mahadev, Classical verification of quantum computa- tions, in2018 IEEE 59th Annual Symposium on Founda- tions of Computer Science (FOCS) (2018) pp

    U. Mahadev, Classical verification of quantum computa- tions, in2018 IEEE 59th Annual Symposium on Founda- tions of Computer Science (FOCS) (2018) pp. 259–267

    2018

  2. [9]

    Brakerski, V

    Z. Brakerski, V. Koppula, U. Vazirani, and T. Vidick, Simplerproofsofquantumness,in 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020) , Leibniz International Pro- ceedings in Informatics (LIPIcs), Vol. 158 (Dagstuhl, Germany, 2020)...

    2020

  3. [10]

    Brakerski, P

    Z. Brakerski, P. Christiano, U. Mahadev, U. Vazirani, and T. Vidick, A cryptographic test of quantumness and certifiable randomness from a single quantum device, Journal of the ACM68, 1 (2021)

    2021

  4. [11]

    Elben, B

    A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. Joshi, R. Blatt, C. F. Roos, andP.Zoller,Cross-PlatformVerificationofIntermediate Scale Quantum Devices, Physical Review Letters124, 010504 (2020)

    2020

  5. [12]

    Carrasco, A

    J. Carrasco, A. Elben, C. Kokail, B. Kraus, and P. Zoller, Theoretical and Experimental Perspectives of Quantum Verification, PRX Quantum2, 010102 (2021)

    2021

  6. [13]

    Anshu, Z

    A. Anshu, Z. Landau, and Y. Liu, Distributed quantum inner product estimation, inProceedings of the 54th An- nual ACM SIGACT Symposium on Theory of Computing (2022) pp. 44–51

    2022

  7. [14]

    Knörzer, D

    J. Knörzer, D. Malz, and J. I. Cirac, Cross-platform veri- fication in quantum networks, Phys. Rev. A107, 062424 (2023)

    2023

  8. [15]

    W. Gong, J. Haferkamp, Q. Ye, and Z. Zhang, On the samplecomplexityofpurityandinnerproductestimation (2024), arXiv:2410.12712 [quant-ph]

    2024 arXiv

  9. [16]

    Arunachalam and L

    S. Arunachalam and L. Schatzki, Distributed inner prod- uct estimation with limited quantum communication (2024), arXiv:2410.12684 [quant-ph]

    2024 arXiv

  10. [17]

    Y. Qian, Y. Du, Z. He, M.-H. Hsieh, and D. Tao, Multimodal deep representation learning for quantum cross-platform verification, Phys. Rev. Lett.133, 130601 (2024)

    2024

  11. [19]

    Montanaro, Learning stabilizer states by bell sampling (2017), arXiv:1707.04012 [quant-ph]

    A. Montanaro, Learning stabilizer states by bell sampling (2017), arXiv:1707.04012 [quant-ph]

    2017 arXiv

  12. [20]

    Here and in what follows,P will denote the probability with respect to the internal randomness of rDIPE for fixed inputs

  13. [21]

    A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman, Real Randomized Benchmarking, Quantum2, 85 (2018)

    2018

  14. [22]

    R. Ruiz, A. Sopena, B. Pozsgay, and E. López, Effi- cient eigenstate preparation in an integrable model with Hilbert space fragmentation (2024), arXiv:2411.15132 [quant-ph]

    2024

  15. [23]

    Hauschild, J

    J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel, K. Hémery, W. Kadow, J. Kemp, N. Kirchner, V. S. Liu, G. Möller, D. Parker, M. Rader, A. Romen, S. Scalet, L. Schoonderwoerd, M. Schulz, T. Soejima, P. Thoma, Y. Wu,...

    2024

  16. [24]

    The proof is analogous for arbitrary non-Clifford gates acting on a constant number of qubits

  17. [25]

    Aharonov, A simple proof that Toffoli and Hadamard are quantum universal (2003), arXiv:quant-ph/0301040 [quant-ph]

    D. Aharonov, A simple proof that Toffoli and Hadamard are quantum universal (2003), arXiv:quant-ph/0301040 [quant-ph]

    2003 arXiv

  18. [26]

    Shi, Both Toffoli and Controlled-NOT need lit- tle help to do universal quantum computation (2002), arXiv:quant-ph/0205115 [quant-ph]

    Y. Shi, Both Toffoli and Controlled-NOT need lit- tle help to do universal quantum computation (2002), arXiv:quant-ph/0205115 [quant-ph]

    2002 arXiv

  19. [27]

    J.J.WallmanandJ.Emerson,Noisetailoringforscalable quantum computation via randomized compiling, Physi- 9 cal Review A94, 10.1103/physreva.94.052325 (2016)

    2016 doi

  20. [28]

    Dvoretzky, J

    A. Dvoretzky, J. Kiefer, and J. Wolfowitz, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, The Annals of Mathematical Statistics27, 642 (1956)

    1956

  21. [29]

    Massart, The Tight Constant in the Dvoretzky-Kiefer- Wolfowitz Inequality, The Annals of Probability18, 1269 (1990)

    P. Massart, The Tight Constant in the Dvoretzky-Kiefer- Wolfowitz Inequality, The Annals of Probability18, 1269 (1990)

    1990

  22. [30]

    Huang, M

    H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Quantum advantage in learning from experiments, Science376, 1182 (2022)

    2022

  23. [31]

    Schmid, N

    C. Schmid, N. Kiesel, W. Wieczorek, H. Weinfurter, F. Mintert, and A. Buchleitner, Experimental direct ob- servation of mixed state entanglement, Phys. Rev. Lett. 101, 260505 (2008)

    2008

  24. [32]

    Miller, K

    D. Miller, K. Levi, L. Postler, A. Steiner, L. Bittel, G. A. L. White, Y. Tang, E. J. Kuehnke, A. A. Mele, S. Khatri, L. Leone, J. Carrasco, C. D. Marciniak, I. Pogorelov, M. Guevara-Bertsch, R. Freund, R. Blatt, P. Schindler, T. Monz, M. Ringbauer, and J. Eisert, Ex- periment...

    2024 arXiv

  25. [33]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature 528, 77–83 (2015)

    2015

  26. [34]

    Bluvstein, H

    D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin, A quantum processor based on coherent transport of en- tangled atom arrays, Nature604, 451–456 (2022)

    2022

  27. [35]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kali- nowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Luki...

    2023

  28. [36]

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

    2024

  29. [37]

    Collins and P

    B. Collins and P. Śniady, Integration with respect to the haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics 264, 773–795 (2006)

    2006

  30. [38]

    Galvin, Three tutorial lectures on entropy and count- ing (2014), arXiv:1406.7872 [math.CO]

    D. Galvin, Three tutorial lectures on entropy and count- ing (2014), arXiv:1406.7872 [math.CO]

    2014 arXiv

  31. [39]

    Indeed, TV(qρ′,qρ) ⩽∥ρ⊗ρ−ρ′⊗ρ′∥tr/2 since the Bell distribution is the (diagonal) output of a quantum chan- nel acting on two copies of a state, and∥ρ⊗ρ−ρ′⊗ρ′∥tr ⩽ 2∥ρ−ρ′∥tr

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