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Optimal estimation of high-dimensional quantum states using locally gentle measurements

T0 review · 2 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Locally gentle measurements raise the minimax cost of estimating a rank-r quantum state to order rd²/(nα²), with a gentleness penalty that scales only with ambient dimension d.

desk verdict Solid minimax theory for gentle tomography with a real dimensional surprise, but the matching lower bound only holds for α bounded away from 1/2. read the letter →

arxiv 2607.24491 v1 pith:AKQYMJPU submitted 2026-07-27 math.ST quant-phstat.TH

Cristina Butucea , Jan Johannes , Henning Stein This is my paper
classification math.STquant-phstat.TH MSC 62C2081P5062F12
keywords quantumstatetomographygentlemeasurementslocaldifferentialprivacyminimaxratesmutuallyunbiasedbasesFrobeniusnormdata-processinginequalitylow-rankdensitymatrices
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 how much harder it becomes to learn a d-dimensional quantum state when every measurement is forced to disturb the state by at most a fixed amount α. Without that constraint the best Frobenius risk scales like rd/n for rank-r states; with local gentleness the optimal rate becomes rd²/(nα²). The extra factor d/α² is surprising because classical local differential privacy typically multiplies risk by the full number of parameters, here order rd. The authors construct explicit gentle measurements—built from mutually unbiased bases and a quantum label-switch kernel—that attain the new rate, show they can be realized by entangling the system with an ancillary register via controlled-NOT gates, and prove matching lower bounds via a new quantum data-processing inequality on carefully chosen families of states. The result quantifies the unavoidable information–disturbance trade-off for high-dimensional quantum tomography and shows that the geometry of quantum state space can be kinder to privacy than its classical counterpart.

What carries the argument

A quantum Assouad reduction that converts estimation into simultaneous testing of 2^D local alternatives, controlled by a new data-processing inequality: the sum of symmetrized KL divergences of outcome laws after any locally α-gentle product measurement is bounded by a multiple of nα² times a sum of squared trace distances between neighboring states. Matching upper bounds are obtained from gentle-ized mutually unbiased bases (or MUMs) followed by projected least-squares and spectral thresholding.

What would settle it

Exhibit either a locally α-gentle product measurement and estimator whose Frobenius risk is o(rd²/(nα²)) for a sequence of rank-r states, or a family of rank-r states on which every such measurement incurs risk ω(rd²/(nα²)), for α bounded away from 1/2.

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Extended reading notes

Core claim

Under the constraint that each of n independent measurements may change the unknown state by at most α in trace distance, the minimax risk of estimating a rank-r density matrix on ℂ^d in Frobenius norm is of order rd²/(nα²) (up to logs). The same rate holds for the full-rank case r = d, giving d³/(nα²). Thus the statistical price of gentleness is a multiplicative factor d/α² that depends on the ambient Hilbert-space dimension rather than on the intrinsic number of free parameters rd.

Load-bearing premise

The matching lower bound requires the gentleness parameter α to stay a fixed distance below one-half, so that the constants that blow up as α approaches one-half remain bounded.

Editorial extensions

If this is right

  • Any quantum algorithm that re-uses post-measurement states under a strict disturbance budget of α must pay at least a d/α² sample overhead for full tomography.
  • The same gentle MUB construction yields an optimal estimator of the probability vector of a pure state with risk of order d/(nα²).
  • Classical local DP mechanisms can be lifted to quantum measurements via Naimark dilation with an ancillary register and controlled-NOT gates while preserving both privacy and gentleness.
  • When a complete set of MUMs with efficiency κ ~ 1 exists, the same rate holds in every finite dimension, not only prime-power dimensions.

Reading between the lines

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

  • The ambient-dimension penalty suggests that other quantum estimation tasks with local gentleness (e.g., shadow tomography or entanglement witnessing) may likewise escape the classical parameter-count barrier.
  • Because the lower-bound construction works for any derived parameter that separates under Hamming distance on the same hypercube, the framework immediately yields gentleness lower bounds for many linear functionals of ρ.
  • If the α-away-from-1/2 restriction can be removed, the same rates would describe the entire gentle regime down to nearly non-disturbing measurements.
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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

2 major / 6 minor

Summary. The paper studies estimation of a d-dimensional, rank-r quantum state from n separately measured copies when each measurement is locally α-gentle, meaning that every post-measurement state is within trace distance α of the input state. The authors construct a quantum label-switch measurement by gently randomizing each basis in a complete set of mutually unbiased bases, extend it using mutually unbiased measurements, and analyze an unbiased linear estimator followed by projection onto the state space and spectral thresholding. They obtain a Frobenius mean-squared-error upper bound of order rd² log(d/ε)/(nα²), together with a rank-consistency guarantee. Lower bounds are developed through an Assouad reduction and a new data-processing inequality for locally gentle measurements, with separate packings for full-rank, low-rank, and pure states, giving a matching rate up to a logarithmic factor when α is bounded away from 1/2. The paper also treats gentle estimation of the probability vector of a pure state and gives an ancilla-and-controlled-gate implementation of the basic label-switch mechanism.

Significance. If the stated optimality holds over the claimed parameter range, the paper provides a sharp quantification of the information–disturbance trade-off in high-dimensional quantum tomography. The ambient-dimensional gentleness penalty d/α², rather than a penalty proportional to the number of parameters rd, is a notable contrast with classical local differential privacy. The manuscript has substantial strengths: an explicit unbiased estimator; implementable gentle label-switch measurements; MUB/MUM constructions; matrix-concentration analysis with projection and rank-consistent spectral thresholding; and a new gentle data-processing inequality combined with full-rank, low-rank, and pure-state packings. These ideas should be useful beyond tomography. The significance is currently tempered by the two scope gaps identified below.

major comments (2)
  1. [Theorem 10; Proposition 5; Theorem 11; Abstract] The advertised optimality is not established over the full range α∈(0,1). Proposition 5 requires α<1/2, and Theorem 11 inherits this restriction; Theorem 10 therefore assumes α bounded away from 1/2 and its constants contain (1−2α)^4. For α∈[1/2,1) no matching lower bound is given, and monotonicity only implies that the risk is no larger than at a smaller α. At α=1, where all product measurements are allowed, the claimed rate is rd²/n, but the manuscript does not prove or identify a matching Frobenius-loss product-measurement lower bound; the cited trace-norm lower bound [26] does not yield this by norm equivalence alone. Thus the Abstract's unqualified optimal rate and sharp 1/α² dependence are currently proved only for α≤1/2−c. Please either restrict the main claims accordingly or provide a lower-bound argument covering α∈[1/2,1], and clarify the behavior as α↑1/2.
  2. [§5.3, Proposition 9] Theorem 8 uses a complete set of MUBs, which is guaranteed here only in prime-power dimensions. For arbitrary d, Proposition 9 uses MUMs but the upper bound becomes rd²/(nα²κ²). The text says the optimal rate extends when κ∼1, yet no construction or cited theorem is given ensuring complete MUMs with efficiency κ bounded below by a dimension-independent constant. If κ(d) tends to zero, the displayed all-d rate is not attained by the proposed estimator. Please supply a dimension-uniform lower bound on κ for a concrete MUM family, restrict the main theorem to dimensions where this is known, or state the additional κ-dependent penalty in the main claims.
minor comments (6)
  1. [Proposition 9] The displayed threshold t²(ϵ)=92 log(d/ϵ)d²/(κnα) appears to be missing squares. Appendix D says to replace α by ακ in Theorem 8, which would give t²(ϵ) proportional to d²/(nα²κ²), not d²/(κnα).
  2. [Appendix C, Eqs. (37)–(38)] With σ²≤10d²/(nα²) and R≤3d/(nα), the stated Matrix-Bernstein theorem appears to give an exponent −t²nα²/(24d²) (for t≤2), rather than the displayed −t²nα²/(4d²). This does not affect the rate, but the constants should be reconciled. The threshold constant also changes between Theorem 8 (92) and Proposition 14 (16).
  3. [Theorem 10] The phrase “α bounded away from 1/2” is ambiguous because it can include α>1/2. Since Theorem 11 applies only for α<1/2, write α∈(0,1/2−c] explicitly.
  4. [§4.4; Abstract] Section 4.4 gives an ancilla-plus-CNOT implementation for the probability-vector measurement. If the same physical implementability is claimed for the full MUB/MUM tomography measurement, add a sentence or proposition explaining the reduction via a basis change before the label-switch circuit.
  5. [General] There are several typographical/grammatical errors, including “proof is allows,” “a again,” “standing state tomography,” “thought” for “through,” “finish of,” “chosed,” and “meausrements.” In Proposition 9, Pρ(rank(ρ̂n=r) is also missing a closing parenthesis.
  6. [Definition 7 and Theorem 11] The notation ρ_{±j,i}, ρ_{ν,i}, and ρ_{ν^{j±}} is quite close and denotes different averaged and padded states. A short displayed reminder before Theorem 11 would make the lower-bound chain easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: minimax rates come from explicit gentle estimators and independent Assouad/DPI lower bounds, not from fitted inputs or load-bearing self-definition.

full rationale

The paper’s central claim—that the Frobenius minimax rate under locally α-gentle product measurements is of order rd²/(nα²) (up to logs)—is established by two independent routes that do not reduce to each other by construction. Upper bounds (Thm. 8 / Prop. 9) are proved from an explicit MUB/MUM-based gentle measurement (label-switch operators (9)/(18)/(23)), unbiased projected least-squares, Matrix-Bernstein concentration, and spectral thresholding; the risk bound is derived from these operators’ second moments and is not assumed. Lower bounds (Thm. 10) use a new multi-hypothesis gentle data-processing inequality (Thm. 11) plus concrete packings (full-rank Gell-Mann perturbations, low-rank unitary Lie-algebra perturbations, pure-state constructions) inside Assouad’s method; the Hamming separations and trace-distance sums are computed from the packings, not imported as the target rate. Self-citations to the authors’ qubit gentleness [13] and certification [14] papers supply background tools (qDP–gentleness link, label-switch kernel idea) but are not uniqueness theorems that force the high-d rate; the high-d DPI and packings are developed in-place. There is no data fitting, no parameter tuned to the claimed rate, and no renaming of a known empirical pattern. The skeptic’s α<1/2 scope limitation on the lower bound is a correctness/regime gap, not circularity: the paper states the restriction explicitly and does not claim a matching lower bound for α≥1/2 by definitional sleight. Derivation chain is self-contained theory.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The work sits inside standard finite-dimensional quantum mechanics, classical minimax theory (Assouad, Matrix-Bernstein), and the existing gentle-measurement / quantum-DP dictionary. No free parameters are fitted; the only modeling choices are the local (product) gentleness model, the Frobenius loss, and the restriction α ≲ 1/2 − c. Invented objects are definitional (the concrete gentle POVM and the new DPI) rather than ontological.

assumptions (6)
  • domain assumption Finite-dimensional quantum mechanics: states are density matrices, measurements are Kraus/POVM operators with the standard Born rule and post-measurement update (Def. 1–2).
    Foundational; used throughout Sections 2–6.
  • domain assumption Local (product) gentleness: each of n copies is measured separately by an α-gentle instrument; coherent/entangled measurements across copies are disallowed.
    Stated in Def. 5 and the problem formulation; matches near-term hardware constraints but is a modeling choice.
  • standard math Equivalence (up to optimal constants) between α-gentleness and δ-quantum differential privacy for positive measurement operators, with δ = 4 arctanh(α) (Cor. 4, Lem. 2).
    Proved in the paper from Kantorovich and spectral arguments; builds on Aaronson–Rothblum.
  • standard math Existence of a complete set of d+1 mutually unbiased bases when d is a prime power, and of mutually unbiased measurements with efficiency κ in every dimension.
    Invoked in §5.1–5.3; standard in quantum information (Klappenecker–Rötteler, Kalev–Gour).
  • standard math Matrix-Bernstein concentration and Assouad’s lemma as black-box probabilistic tools.
    Used for upper bounds (Thm 13 / Tropp) and lower-bound reduction (§6.1).
  • ad hoc to paper α bounded away from 1/2 by a universal constant in the lower-bound theorem so that (1−2α)^−4 remains O(1).
    Explicit hypothesis of Theorem 10; without it the leading constant diverges and the rate statement needs re-phrasing.
invented entities (2)
  • MUB-wise (or MUM-wise) gentle label-switch measurement (eqs. 18, 23) independent evidence
    purpose: Achieve the optimal gentle tomography rate by gentle-izing each basis separately rather than a full 2-design.
    Explicit POVM built from the classical multinomial privacy kernel lifted to each MUB; not previously analyzed for high-d tomography.
  • Quantum data-processing inequality for sums of symmetrized KL divergences under local gentleness (Theorem 11) independent evidence
    purpose: Decouple Assouad coordinates and obtain tight lower bounds for gentle measurements.
    New information-theoretic inequality specialized to α-gentle product instruments; reduces to earlier single-hypothesis gentle DPI when D=1.

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Pith. "Pith review of Optimal estimation of high-dimensional quantum states using locally gentle measurements." pith.science (2026). https://pith.science/paper/AKQYMJPU

@misc{pith2026260724491,
  author       = {Pith},
  title        = {Pith review of: Optimal estimation of high-dimensional quantum states using locally gentle measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKQYMJPU}},
  note         = {Machine review of arXiv:2607.24491}
}
abstract

We study the task of estimating a $d-$dimensional quantum state $\rho$ under the constraint that the measurement is $\alpha-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = \omega$ containing statistical information and a post-measurement state $\rho_{M \to \omega}$ such that $\|\rho_{M\to \omega} - \rho\|_{Tr} \leq \alpha$. We describe gentle measurements and their connection to quantum differential privacy. Our results show that the optimal minimax estimation rate in Frobenius norm is of order $d^3/(n \alpha^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\leq d$ we prove that the optimal minimax rate is $rd^2/(n \alpha^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/\alpha^2$ scales with the ambient dimension of the Hilbert space, rather than the number of parameters $rd$, typically seen in classical differential privacy. We propose optimal gentle measurements and indicate how they can be physically implemented using an ancillary state and a CNOT gate to entangle it with the initial state. We notice that the resulting random variable has a likelihood that satisfies local differential privacy. Lower bounds are proven through a new quantum information-theoretic inequality applied to well chosen families of states in the manifold of (small-rank) quantum states.

Figures

Figures reproduced from arXiv: 2607.24491 by the authors.

Figure 1
Figure 1. Schematic representation of the d-dimensional Label Switch mechanism. Proposition 7 Let δ > 0, |ψ⟩ ∈ Spure(C d ) and ρ = |ψ⟩ ⟨ψ|. For |ϕ0⟩ = |0⟩ ⊗ ... ⊗ |0⟩ define σ = |ϕ0⟩ ⟨ϕ0| on (C 2 ) d . Then there exists a unitary operation U˜ δ, comprised of qubit rotations Rδ and qudit controlled NOT-gates U1, . . . , Ud, and a basis measurement M˜ B˜ = (1 ⊗ |ω⟩ ⟨ω|)ω∈Ω on the second system alone such that for ρδ,U˜ = U˜ δ(ρ… view at source ↗

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Works this paper leans on

65 extracted references · 7 linked inside Pith

  1. [26]

    Sample-Optimal Tomography of Quantum States.IEEE Transactions on Information Theory, 63(9):5628–5641, 2017

    JeongwanHaah, AramW.Harrow, ZhengfengJi, XiaodiWu, andNengkunYu. Sample-Optimal Tomography of Quantum States.IEEE Transactions on Information Theory, 63(9):5628–5641, 2017

    2017

  2. [1]

    Rothblum

    Scott Aaronson and Guy N. Rothblum. Gentle measurement of quantum states and differential privacy. InSTOC’19—Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 322–333. ACM, New York, 2019

    2019

  3. [2]

    Huggins, Ramis Movassagh, Dar Gilboa, and Jarrod R

    Amira Abbas, Robbie King, Hsin-Yuan Huang, William J. Huggins, Ramis Movassagh, Dar Gilboa, and Jarrod R. McClean. On quantum backpropagation, information reuse, and cheating measurement collapse. InProceedings of the 37th International Conference on Neural Information Processing Systems, NIPS ’23, pages 44792–44819, Red Hook, NY, USA, 2023. Curran Associates Inc

    2023

  4. [3]

    Quantum t-designs: t-wise Independence in the Quantum World

    Andris Ambainis and Joseph Emerson. Quantum t-designs: t-wise Independence in the Quantum World. InTwenty-Second Annual IEEE Conference on Computational Complexity (CCC’07), pages 129–140, 2007. ISSN: 1093-0159

    2007

  5. [4]

    L. M. Artiles, R. D. Gill, and M. I. Gut¸ă. An invitation to quantum tomography.Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(1):109–134, 2005. _eprint: https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2005.00491.x

    arXiv 2005

  6. [5]

    Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G

    Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, St...

    2019

  7. [6]

    K. M. R. Audenaert, M. Nussbaum, A. Szkoła, and F. Verstraete. Asymptotic Error Rates in Quantum Hypothesis Testing.Communications in Mathematical Physics, 279(1):251–283, 2008

    2008

  8. [7]

    Bagan, M

    E. Bagan, M. A. Ballester, R. D. Gill, A. Monras, and R. Muñoz-Tapia. Optimal full estimation of qubit mixed states.Physical Review A, 73(3), 2006

    2006

Show all 65 references
  1. [8]

    Braunstein and Carlton M

    Samuel L. Braunstein and Carlton M. Caves. Statistical distance and the geometry of quantum states.Physical Review Letters, 72(22):3439–3443, 1994. 30

    1994

  2. [9]

    Entanglement is necessary for optimal quantum property testing

    Sebastien Bubeck, Sitan Chen, and Jerry Li. Entanglement is necessary for optimal quantum property testing. In2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pages 692–703. IEEE Computer Soc., Los Alamitos, CA, 2020

    2020

  3. [10]

    Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data.The Annals of Statistics, 35(2), 2007

    Cristina Butucea, Madalin Guţa, and Luis Artiles. Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data.The Annals of Statistics, 35(2), 2007. arXiv:math/0504058

    2007 arXiv

  4. [11]

    Spectral thresholding quantum tomography for low rank states.New Journal of Physics, 17(11):113050, 2015

    Cristina Butucea, Mădălin Guţă, and Theodore Kypraios. Spectral thresholding quantum tomography for low rank states.New Journal of Physics, 17(11):113050, 2015

    2015

  5. [12]

    Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise.The Annals of Statistics, 46(6B), 2018

    Cristina Butucea, Mădălin Guţă, and Michael Nussbaum. Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise.The Annals of Statistics, 46(6B), 2018

    2018

  6. [13]

    Sample-optimal learning of quantum states using gentle measurements, 2025

    Cristina Butucea, Jan Johannes, and Henning Stein. Sample-optimal learning of quantum states using gentle measurements, 2025. arXiv:2505.24587 [quant-ph]

    2025 arXiv

  7. [14]

    Locally Gentle State Certification for High Dimensional Quantum Systems, 2026

    Cristina Butucea, Jan Johannes, and Henning Stein. Locally Gentle State Certification for High Dimensional Quantum Systems, 2026. arXiv:2602.04550 [quant-ph]

    2026 arXiv

  8. [15]

    Interactive versus noninteractive locally differentially private estimation: two elbows for the quadratic functional.The Annals of Statistics, 51(2):464–486, 2023

    Cristina Butucea, Angelika Rohde, and Lukas Steinberger. Interactive versus noninteractive locally differentially private estimation: two elbows for the quadratic functional.The Annals of Statistics, 51(2):464–486, 2023

    2023

  9. [16]

    Tony Cai, Donggyu Kim, Yazhen Wang, Ming Yuan, and Harrison H. Zhou. Optimal large-scale quantum state tomography with Pauli measurements.The Annals of Statistics, 44(2), 2016. arXiv:1603.07559 [math]

    2016 arXiv

  10. [17]

    Candes and Terence Tao

    Emmanuel J. Candes and Terence Tao. The Power of Convex Relaxation: Near-Optimal Matrix Completion.IEEE Transactions on Information Theory, 56(5):2053–2080, 2010

  11. [18]

    Quantum Overlapping Tomography.Physical Review Letters, 124(10):100401, 2020

    Jordan Cotler and Frank Wilczek. Quantum Overlapping Tomography.Physical Review Letters, 124(10):100401, 2020. arXiv:1908.02754 [quant-ph]

    2020 arXiv

  12. [19]

    Efficient projections onto thel 1 -ball for learning in high dimensions

    John Duchi, Shai Shalev-Shwartz, Yoram Singer, and Tushar Chandra. Efficient projections onto thel 1 -ball for learning in high dimensions. InProceedings of the 25th international conference on Machine learning - ICML ’08, pages 272–279, Helsinki, Finland, 2008. ACM Press

    2008

  13. [20]

    Duchi, Michael I

    John C. Duchi, Michael I. Jordan, and Martin J. Wainwright. Local privacy and statistical minimax rates. In2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1592–1592, 2013

    2013

  14. [21]

    Duchi, Michael I

    John C. Duchi, Michael I. Jordan, and Martin J. Wainwright. Minimax Optimal Procedures for Locally Private Estimation.Journal of the American Statistical Association, 113(521):182–201,

  15. [22]

    Calibrating Noise to Sensitivity in Private Data Analysis

    Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating Noise to Sensitivity in Private Data Analysis. In Shai Halevi and Tal Rabin, editors,Theory of Cryptography, pages 265–284, Berlin, Heidelberg, 2006. Springer

    2006

  16. [23]

    Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators.New Journal of Physics, 14(9):095022, 2012

    Steven T Flammia, David Gross, Yi-Kai Liu, and Jens Eisert. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators.New Journal of Physics, 14(9):095022, 2012

    2012

  17. [24]

    Quantum error correction below the surface code threshold.Nature, 638(8052):920–926, 2025

    Google Quantum AI and Collaborators. Quantum error correction below the surface code threshold.Nature, 638(8052):920–926, 2025. ISBN: 1476-4687

    2025

  18. [25]

    M. Guţă, J. Kahn, R. Kueng, and J. A. Tropp. Fast state tomography with optimal error bounds.J. Phys. A, 53(20):204001, 28, 2020

    2020

  19. [27]

    Hall.Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, volume 222 ofGraduate Texts in Mathematics

    Brian C. Hall.Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, volume 222 ofGraduate Texts in Mathematics. Springer International Publishing, Cham, 2015

    2015

  20. [28]

    Hannemann, D

    Th. Hannemann, D. Reiss, Ch. Balzer, W. Neuhauser, P. E. Toschek, and Ch. Wunderlich. Self-learning estimation of quantum states.Physical Review A, 65(5):050303, 2002

    2002

  21. [29]

    Harrow, Avinatan Hassidim, and Seth Lloyd

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum Algorithm for Linear Systems of Equations.Physical Review Letters, 103(15):150502, 2009

    2009

  22. [30]

    Helstrom

    Carl W. Helstrom. Quantum detection and estimation theory.Journal of Statistical Physics, 1:231–252, 1969

    1969

  23. [31]

    Holevo.Statistical Structure of Quantum Theory, volume 67 ofLecture Notes in Physics Monographs

    Alexander S. Holevo.Statistical Structure of Quantum Theory, volume 67 ofLecture Notes in Physics Monographs. Springer, Berlin, Heidelberg, 2001

    2001

  24. [32]

    Häffner, W

    H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt. Scalable multiparticle entanglement of trapped ions.Nature, 438(7068):643–646, 2005

    2005

  25. [33]

    Local Asymptotic Normality for Finite Dimensional Quantum Systems.Communications in Mathematical Physics, 289(2):597–652, 2009

    Jonas Kahn and Mădălin Guţă. Local Asymptotic Normality for Finite Dimensional Quantum Systems.Communications in Mathematical Physics, 289(2):597–652, 2009

    2009

  26. [34]

    Mutually unbiased measurements in finite dimensions.New Journal of Physics, 16(5):053038, 2014

    Amir Kalev and Gilad Gour. Mutually unbiased measurements in finite dimensions.New Journal of Physics, 16(5):053038, 2014

    2014

  27. [35]

    On the Chernoff bound for efficiency of quantum hypothesis testing.Ann

    Vladislav Kargin. On the Chernoff bound for efficiency of quantum hypothesis testing.Ann. Statist., 33(2):959–976, 2005. 32

    2005

  28. [36]

    Klappenecker and M

    A. Klappenecker and M. Rotteler. Mutually unbiased bases are complex projective 2-designs. Proceedings. International Symposium on Information Theory, 2005. ISIT 2005., pages 1740– 1744, 2005. Conference Name: Proceedings. International Symposium on Information Theory,

    2005

  29. [37]

    Von Neumann entropy penalization and low-rank matrix estimation

    Vladimir Koltchinskii. Von Neumann entropy penalization and low-rank matrix estimation. The Annals of Statistics, 39(6):2936–2973, 2011

    2011

  30. [38]

    Tsybakov

    Vladimir Koltchinskii, Karim Lounici, and Alexandre B. Tsybakov. Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion.The Annals of Statistics, 39(5):2302– 2329, 2011

    2011

  31. [39]

    Optimal estimation of low rank density matrices.J

    Vladimir Koltchinskii and Dong Xia. Optimal estimation of low rank density matrices.J. Mach. Learn. Res., 16(1):1757–1792, 2015

    2015

  32. [40]

    Low rank matrix recovery from rank one measurements.Applied and Computational Harmonic Analysis

    Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low rank matrix recovery from rank one measurements.Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, 42(1):88–116, 2017

    2017

  33. [41]

    Minimax estimation of low-rank quantum states and their linear functionals.Bernoulli, 30(1):610–635, 2024

    Samriddha Lahiry and Michael Nussbaum. Minimax estimation of low-rank quantum states and their linear functionals.Bernoulli, 30(1):610–635, 2024

    2024

  34. [42]

    The role of randomness in quantum state certification with unentangled measurements

    Yuhan Liu and Jayadev Acharya. The role of randomness in quantum state certification with unentangled measurements. InProceedings of Thirty Seventh Conference on Learning Theory, pages 3523–3555. PMLR, 2024

    2024

  35. [43]

    Universal Quantum Simulators.Science, 273(5278):1073–1078, 1996

    Seth Lloyd. Universal Quantum Simulators.Science, 273(5278):1073–1078, 1996

    1996

  36. [44]

    Computing partial traces and reduced density matrices.International Journal of Modern Physics C, 28(1):1750005, 2017

    Jonas Maziero. Computing partial traces and reduced density matrices.International Journal of Modern Physics C, 28(1):1750005, 2017

    2017

  37. [45]

    J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, and Seth Lloyd. Josephson Persistent-Current Qubit.Science, 285(5430):1036–1039, 1999

    1999

  38. [46]

    Recent developments of the operator Kantorovich inequality.Expo

    Mohammad Sal Moslehian. Recent developments of the operator Kantorovich inequality.Expo. Math., 30(4):376–388, 2012

    2012

  39. [47]

    Myrvold and Joy Christian, editors.Quantum reality, relativistic causality, and closing the epistemic circle, volume 73 ofWestern Ontario Series in Philosophy of Science

    Wayne C. Myrvold and Joy Christian, editors.Quantum reality, relativistic causality, and closing the epistemic circle, volume 73 ofWestern Ontario Series in Philosophy of Science. Springer, Dordrecht, 2009

    2009

  40. [48]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000

    2000

  41. [49]

    Efficient quantum tomography

    Ryan O’Donnell and John Wright. Efficient quantum tomography. InProceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’16, pages 899–912, New York, NY, USA, 2016. Association for Computing Machinery. 33

    2016

  42. [50]

    Quantum spectrum testing.Communications in Mathe- matical Physics, 387(1):1–75, 2021

    Ryan O’Donnell and John Wright. Quantum spectrum testing.Communications in Mathe- matical Physics, 387(1):1–75, 2021

    2021

  43. [51]

    Tsybakov

    Angelika Rohde and Alexandre B. Tsybakov. Estimation of high-dimensional low-rank matrices. The Annals of Statistics, 39(2), 2011. arXiv:0912.5338 [math]

    2011 arXiv

  44. [52]

    Romero-Isart, E

    O. Romero-Isart, E. Bagan, M.A. Ballester Sánchez, R.D. Gill, and R. Muñoz-Tapia.Optimal estimation of qubit mixed states with local measurements. Report Eurandom. Eurandom, Eindhoven, 2005

    2005

  45. [53]

    P.W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 124–134, 1994

    1994

  46. [54]

    D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani. Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum.Physical Review Letters, 70(9):1244–1247, 1993

    1993

  47. [55]

    Smolin, Jay M

    John A. Smolin, Jay M. Gambetta, and Graeme Smith. Efficient Method for Computing the Maximum-Likelihood Quantum State from Measurements with Additive Gaussian Noise. Physical Review Letters, 108(7):070502, 2012

    2012

  48. [56]

    Efficiency in local differential privacy.Annals of Statistics, 52(5):2139–2166, 2024

    Lukas Steinberger. Efficiency in local differential privacy.Annals of Statistics, 52(5):2139–2166, 2024

    2024

  49. [57]

    Joel A. Tropp. An Introduction to Matrix Concentration Inequalities.Foundations and Trends®in Machine Learning, 8(1-2):1–230, 2015

    2015

  50. [58]

    Joel A. Tropp. ACM 204: Matrix Analysis. 2024

    2024

  51. [59]

    Asymptotic equivalence of quantum state tomography and noisy matrix completion.The Annals of Statistics, 41(5), 2013

    Yazhen Wang. Asymptotic equivalence of quantum state tomography and noisy matrix completion.The Annals of Statistics, 41(5), 2013. arXiv:1311.4976 [math]

    2013 arXiv

  52. [60]

    A. Winter. Coding theorem and strong converse for quantum channels.IEEE Transactions on Information Theory, 45(7):2481–2485, 1999

    1999

  53. [61]

    Sample Efficient Identity Testing and Independence Testing of Quantum States

    Nengkun Yu. Sample Efficient Identity Testing and Independence Testing of Quantum States. In James R. Lee, editor,12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 11:1– 11:20,...

    2021

  54. [62]

    DX m=1 Hm |vm⟩ ⟨vm| # = α d+ 1 ρ+ α d+ 1 +β 1 47 Proof.We calculate Eρ

    Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, Peng Hu, Xiao-Yan Yang, Wei-Jun Zhang, Hao Li, Yuxuan Li, Xiao Jiang, Lin Gan, Guangwen Yang, Lixing You, Zhen Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-...

    2020

  55. [65]

    Rearranging this formula gives ∥A12∥2 F = Tr [A12A∗ 12]≤Tr[A 11]−Tr A2 11 = rX m=1 λm −λ 2 m ≤ r 4 where(λ m)r m=1 ⊆[0,1]are the eigenvalues ofA 11

    Since this sub-matrix is also positive, for its trace we have Tr[A11]−Tr A2 11 −Tr [A12A∗ 12]≥0. Rearranging this formula gives ∥A12∥2 F = Tr [A12A∗ 12]≤Tr[A 11]−Tr A2 11 = rX m=1 λm −λ 2 m ≤ r 4 where(λ m)r m=1 ⊆[0,1]are the eigenvalues ofA 11. This gives the bound DX j=1 1 2...

  56. [2005]

    ISBN: 9780780391512

    ISIT 2005. ISBN: 9780780391512

    2005

  57. [2018]

    _eprint: https://doi.org/10.1080/01621459.2017.1389735. 31

    2017

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.