Pith. sign in

REVIEW 2 major objections 4 minor 79 references

A single two-condition criterion decides when quantum tomography reaches its optimal 1/N infidelity scaling.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-05 04:40 UTC pith:WZKYJ553

load-bearing objection A credible unified necessary-and-sufficient condition for optimal infidelity scaling in QST/QDT/QPT, with new adaptive algorithms and a real AAPT demonstration; main caveats are the modified fidelity metric, missing slope uncertainties, and no code/data. the 2 major comments →

arxiv 2509.05988 v1 pith:WZKYJ553 submitted 2025-09-07 quant-ph

Unified formalism and adaptive algorithms for optimal quantum state, detector and process tomography

Shuixin Xiao , Xiangyu Wang , Yuanlong Wang , Zhibo Hou , Jun Zhang , Ian R. Petersen , Wen-Zhe Yan , Hidehiro Yonezawa
show 3 more authors
Franco Nori Guo-Yong Xiang Daoyi Dong
This is my paper
classification quant-ph
keywords quantum state tomographyquantum detector tomographyquantum process tomographyinfidelity scalingadaptive tomographypositive semidefinite operatorsancilla-assisted process tomography
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that one pair of conditions determines whether any quantum tomography protocol—for states, detectors, or processes—attains the best possible accuracy scaling. With N copies of the system, the estimator must converge to the true positive semidefinite matrix in mean squared error at rate 1/N, and the eigenvalues it assigns to the target's null space must also shrink as 1/N. The authors prove this equivalence in arbitrary finite dimension, including degenerate spectra, and use it to design adaptive algorithms for state, detector, and process tomography that provably meet both conditions. They validate the process-tomography algorithm in photonic experiments, reporting the first experimental demonstration of optimal 1/N infidelity scaling in ancilla-assisted process tomography.

Core claim

The paper's central claim is Theorem 1: for any unknown positive semidefinite operator S with rank r, the infidelity E(1-F(Shat,S)) scales as O(1/N) if and only if C1 holds—the mean squared error E||Shat-S||^2 is O(1/N)—and C2 holds—the expected sum of the estimated eigenvalues in the null space, E sum_{j=r+1}^d \hat{lambda}_j, is O(1/N). The infidelity index is defined for arbitrary positive semidefinite operators by subtracting a trace-mismatch term from the normalized fidelity, so it applies uniformly to density matrices, POVM elements, and process matrices and reduces to ordinary state infidelity for quantum states. The paper argues this is the first equivalent characterization of optima

What carries the argument

The central object is the modified fidelity 1-F(Shat,S), defined for arbitrary positive semidefinite operators S by subtracting a trace-mismatch term from the normalized fidelity so that F=1 only when Shat=S; it reduces to ordinary state infidelity for density matrices and extends to POVM elements and process matrices. The load-bearing identity is Theorem 1: with S of rank r and estimated eigenvalues in non-increasing order, E(1-F)=O(1/N) iff E||Shat-S||^2=O(1/N) and E sum_{j=r+1}^d \hat{lambda}_j=O(1/N). The proof combines a second-order expansion of the infidelity with a Davis-Kahan sin-theta subspace perturbation bound, which controls how estimated eigenvectors leak into the null space. T

Load-bearing premise

The proofs assume the target's smallest nonzero eigenvalue is a fixed positive constant; the error bounds have that eigenvalue in the denominator, so the guaranteed 1/N scaling is not uniform as the spectral gap shrinks.

What would settle it

Take a fixed rank-deficient state and compare two protocols at increasing N: static maximum-likelihood or linear-regression estimation, which satisfies C1 but not C2, and the paper's two-step adaptive QST, which is designed to satisfy both. If the theorem is right, log-log slopes of the infidelity versus N should be about -1/2 for the static protocol and about -1 for the adaptive protocol. An adaptive slope that fails to approach -1 would refute the claimed equivalence.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Any tomography protocol, current or future, can be checked for optimal infidelity scaling by testing two error quantities rather than re-deriving task-specific bounds.
  • The unified criterion closes the gap between sufficiency and necessity for detector and process tomography, including non-trace-preserving processes, where the paper shows the older fidelity definition preserves sufficiency but not necessity.
  • The proposed adaptive QST and QDT algorithms use fewer measurement settings or probe states than earlier adaptive methods while still achieving O(1/N) infidelity.
  • For full-rank targets, condition C1 alone gives the optimal scaling; the extra null-eigenvalue condition C2 is exactly what rank-deficient targets require.
  • The adaptive ancilla-assisted process tomography algorithm is proven and experimentally demonstrated to reach O(1/N) infidelity for both unitary and non-unitary processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A practical diagnostic the paper does not spell out: experiments seeing 1/sqrt(N) infidelity can attribute the shortfall to whichever condition fails—global mean squared error or null-eigenvalue leakage—and target their fix accordingly.
  • The proof's Davis-Kahan bounds divide by the smallest nonzero eigenvalue of the target, so for nearly rank-deficient systems the constant in O(1/N) can be large; the optimal slope may only appear at very large N, a regime the paper does not quantify.
  • The same two-condition logic could transfer to other positive semidefinite estimation problems, such as quantum assemblage tomography or covariance estimation, wherever the loss is fidelity-like and the null space is the hard part.
  • The open resource-allocation fraction alpha invites a testable extension: the optimal split between the static and adaptive steps likely depends on the rank and spectral gap of the target, and numerical sweeps could reveal a closed-form tradeoff.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a unified treatment of infidelity scaling for quantum state, detector, and process tomography. It defines a modified fidelity F for positive semidefinite operators that reduces to the usual state fidelity and removes the distortion problem of earlier detector/process fidelities. The main result, Theorem 1, states that for a fixed unknown positive semidefinite operator S, the expected infidelity E(1-F(hat S,S)) is O(1/N) if and only if (C1) the mean squared error E||hat S-S||^2 is O(1/N) and (C2) the expected sum of estimated eigenvalues corresponding to the null space of S is O(1/N). Based on this characterization, the authors propose adaptive two- or three-step algorithms for QST, QDT, and AAPT, prove their optimal O(1/N) scaling, and support the claims with numerical simulations and photonic experiments, including what is reported as the first experimental demonstration of optimal infidelity scaling in ancilla-assisted process tomography.

Significance. If the results hold, this is a substantial advance. The equivalent characterization in Theorem 1 is genuinely new and covers degenerate cases, non-trace-preserving processes, and arbitrary finite dimensions. The adaptive algorithms are concrete and come with detailed analytic proofs (Theorems 3--5) rather than numerical heuristics, and they use fewer measurement settings or probe states than some earlier adaptive schemes. The AAPT experiment is also the first of its kind reported. The proof is not circular: the theorem is derived from standard perturbation bounds (Davis--Kahan), Fuchs--van de Graaf, and a Taylor expansion of the infidelity. The main weakness is that the experimental O(1/N) claim rests on visually read log-log slopes without quantitative reporting, and the proofs hide a strong dependence on the smallest positive eigenvalue lambda_r, which is not flagged as a practical limitation.

major comments (2)
  1. [Experimental results, Fig. 3 and Fig. S6] The central experimental claim that adaptive AAPT 'achieves O(1/N)' is inferred from log-log plots without reporting the fitted slopes, their standard errors, or goodness of fit. The non-adaptive comparison is similarly asserted to be O(1/sqrt N). This is load-bearing for the advertised first experimental demonstration. Please report, for each dataset (unitary and phase-damping, both allocations, adaptive and non-adaptive), the fitted exponent with uncertainty, the fitting range, and an R^2 or residual diagnostic; otherwise the scaling conclusion is not quantitatively supported.
  2. [Supplementary Section III-A, Theorem 3 and Corollary 1; Section III-C, Theorem 5] The proofs rely on Proposition 2 (Davis--Kahan) with denominators proportional to lambda_r, the smallest positive eigenvalue of S. The asymptotic O(1/N) statements are valid for each fixed S, but the hidden constants grow as lambda_r -> 0, and for a tiny spectral gap the asymptotic regime only begins at very large N (roughly N0 >> 1/lambda_r^2 for the eigenbasis to be aligned). The manuscript advertises algorithms for 'arbitrary' S without quantifying this. This is not a logical gap in Theorem 1, but it is a practical limitation that should be stated and quantified in the main text, not only implied in the supplement.
minor comments (4)
  1. [Section III-C] Typo: 'Schmit decomposition' should be 'Schmidt decomposition'.
  2. [Fig. 2 caption] Typo: 'cofiguration' should be 'configuration'.
  3. [Eq. (1) and surrounding text] The normalized fidelity F in Eq. (1) depends on the task-specific infimum f (1/d - 1 for QDT, -1 for QPT). For the claim that Theorem 1 applies to 'any positive semidefinite operator S', the value of f for a generic S is not defined. Please clarify that the theorem applies to the three named classes (or specify how f is assigned).
  4. [Eq. (S.51)] The Taylor expansion is cited to [35] but is central to both directions of Theorem 1. A short derivation or a pointer to the exact equation in [35] would improve verifiability.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is a substantive equivalence proved from standard inequalities; the self-citations found are not load-bearing reductions.

full rationale

Theorem 1 is not circular: the new fidelity F is defined independently of conditions C1 and C2, and the proof of necessity/sufficiency uses Fuchs–van de Graaf, the Davis–Kahan variant (Prop. 2), Weyl-type Lemma 1, and the infidelity expansion Eq. (S.51). C1 and C2 are not built into the metric by definition, and the equivalence is derived rather than assumed. The adaptive algorithms are supported by direct MSE/eigenvalue bounds (Theorems 3–5) that call on Proposition 2 with λ_r as a fixed spectral gap; for any fixed S with λ_r>0 the O(1/N) scaling follows, so the λ_r-dependence affects constants only and is a practical caveat, not a circular step. Self-citations exist—[60] for the two-stage correction and [37] for the distortion problem—but these are published results with independent proofs and do not assume the present infidelity-scaling claim. One flagged support-quality issue is Supplement I-D, where Ref. [63] is an in-preparation self-citation for AAPT MSE background, and another is Supplement III-C around Eq. (S.114), where the QPST infidelity scaling is asserted without a full proof. These are missing-support/presentation concerns, not circular reductions: no step was found where a prediction reduces by construction to a fitted parameter, a redefinition, or a self-citation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 7 axioms · 0 invented entities

The central claim rests on standard perturbation bounds, quantum estimation bounds, and domain assumptions about probes and measurement completeness. The only hand-chosen number is the resource split alpha. No new physical entities are introduced.

free parameters (1)
  • resource allocation alpha = alpha in (0,1); examples 0.5, 0.9
    The two-step and three-step adaptive algorithms split N into N0=alpha N and N-N0; the proof works for any constant alpha, but the achievable constant in the O(1/N) bound depends on alpha and its optimal value is left open.
axioms (7)
  • standard math Davis-Kahan sin-theta theorem (Proposition 2, Ref. [67])
    Used in the proof of Theorem 1 (Eqs. S.55-S.57) and Theorem 3 to bound eigenvector subspace errors.
  • standard math Fuch-van de Graaf inequalities (Ref. [68])
    Used to derive necessity of C1 and C2 from infidelity scaling in Theorem 1.
  • domain assumption Quantum Cramer-Rao bound: optimal MSE is O(1/N) for all parameters
    Used to identify C1 as optimal MSE scaling.
  • domain assumption Existence of full-rank MSE O(1/N) estimators (MLE/LRE)
    Step-1 of adaptive QST/QDT assumes a preliminary estimator with E||rhohat-rho||^2=O(1/N0); proved for LRE in supplement, cited for MLE.
  • domain assumption Pure input state with full Schmidt number in AAPT
    Theorem 5 assumes Sch(sigma_in)=d^2, which is required for the reconstruction formula Eq. (S.118).
  • domain assumption Informationally complete measurements/probe states
    Needed for Step-1 estimators to achieve O(1/N0) MSE.
  • domain assumption New fidelity range normalization f for QDT (1/d - 1) and QPT (-1)
    Used to define F in Eq. (S.48); values taken from Ref. [37] for QDT.

pith-pipeline@v1.3.0-alltime-deepseek · 38672 in / 16741 out tokens · 164329 ms · 2026-08-05T04:40:07.238271+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Unified formalism and adaptive algorithms for optimal quantum state, detector and process tomography." pith.science (2026). https://pith.science/paper/WZKYJ553

@misc{pith2026250905988,
  author       = {Pith},
  title        = {Pith review of: Unified formalism and adaptive algorithms for optimal quantum state, detector and process tomography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZKYJ553}},
  note         = {Machine review of arXiv:2509.05988}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Quantum tomography is a standard technique for characterizing, benchmarking and verifying quantum systems/devices and plays a vital role in advancing quantum technology and understanding the foundations of quantum mechanics. Achieving the highest possible tomography accuracy remains a central challenge. Here we unify the infidelity metrics for quantum state, detector and process tomography in a single index $1-F(\hat S,S)$, where $S$ represents the true density matrix, POVM element, or process matrix, and $\hat S$ is its estimator. We establish a sufficient and necessary condition for any tomography protocol to attain the optimal scaling $1-F= O(1/N) $ where $N$ is the number of state copies consumed, in contrast to the $O(1/\sqrt{N})$ worst-case scaling of static methods. Guided by this result, we propose adaptive algorithms with provably optimal infidelity scalings for state, detector, and process tomography. Numerical simulations and quantum optical experiments validate the proposed methods, with our experiments reaching, for the first time, the optimal infidelity scaling in ancilla-assisted process tomography.

Figures

Figures reproduced from arXiv: 2509.05988 by Daoyi Dong, Franco Nori, Guo-Yong Xiang, Hidehiro Yonezawa, Ian R. Petersen, Jun Zhang, Shuixin Xiao, Wen-Zhe Yan, Xiangyu Wang, Yuanlong Wang, Zhibo Hou.

Figure 1
Figure 1. Figure 1: FIG. 1. Three quantum tomography tasks: QST, QDT, and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Key components include a polarizing beam splitter (PBS), half-wave plates (HWPs), quarter-wave plates (QWPs), [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Experimental results for adaptive and non-adaptive [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

79 extracted references · 60 canonical work pages

  1. [1]

    Jullien, P

    T. Jullien, P. Roulleau, B. Roche, A. Cavanna, Y. Jin, and D. C. Glattli, Quantum tomography of an electron, Nature514, 603 (2014)

    2014

  2. [2]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2010)

    2010

  3. [3]

    J. Q. You and F. Nori, Atomic physics and quantum optics using superconducting circuits, Nature474, 589 (2011)

    2011

  4. [4]

    Gebhart, R

    V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezz` e, and C. Bonato, Learning quantum systems, Nature Re- views Physics5, 141 (2023)

    2023

  5. [5]

    Wang, Z.-W

    X.-B. Wang, Z.-W. Yu, J.-Z. Hu, A. Miranowicz, and F. Nori, Efficient tomography of quantum-optical Gaus- sian processes probed with a few coherent states, Physical Review A88, 022101 (2013)

    work page 2013

  6. [6]

    J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, Tomography of quantum de- tectors, Nature Physics5, 27 (2009)

    2009

  7. [7]

    R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wen- ner, M. Steffen, A. N. Cleland, and J. M. Martinis, Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits, Nature Physics6, 409 (2010)

    work page 2010

  8. [8]

    Burgarth, K

    D. Burgarth, K. Maruyama, and F. Nori, Indirect quan- tum tomography of quadratic Hamiltonians, New Journal of Physics13, 013019 (2011)

    work page 2011

  9. [9]

    Cramer, M

    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, 6 and Y.-K. Liu, Efficient quantum state tomography, Na- ture Communications1, 149 (2010)

    work page 2010

  10. [10]

    Miranowicz, K

    A. Miranowicz, K. Bartkiewicz, J. Peˇ rina, M. Koashi, N. Imoto, and F. Nori, Optimal two-qubit tomography based on local and global measurements: Maximal ro- bustness against errors as described by condition num- bers, Physical Review A90, 062123 (2014)

    work page 2014

  11. [11]

    Rambach, M

    M. Rambach, M. Qaryan, M. Kewming, C. Ferrie, A. G. White, and J. Romero, Robust and efficient high- dimensional quantum state tomography, Physical Review Letters126, 100402 (2021)

    work page 2021

  12. [12]

    Z. Hou, H. S. Zhong, Y. Tian, D. Dong, B. Qi, L. Li, Y. Wang, F. Nori, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Full reconstruction of a 14-qubit state within four hours, New Journal of Physics18, 083036 (2016)

    work page 2016

  13. [13]

    Zhu, Y.-D

    Y. Zhu, Y.-D. Wu, G. Bai, D.-S. Wang, Y. Wang, and G. Chiribella, Flexible learning of quantum states with generative query neural networks, Nature Communica- tions13, 6222 (2022)

    work page 2022

  14. [14]

    Miranowicz, S ¸

    A. Miranowicz, S ¸. K. ¨Ozdemir, J. Bajer, G. Yusa, N. Imoto, Y. Hirayama, and F. Nori, Quantum state to- mography of large nuclear spins in a semiconductor quan- tum well: Optimal robustness against errors as quanti- fied by condition numbers, Physical Review B92, 075312 (2015)

    work page 2015

  15. [15]

    Burgarth and K

    D. Burgarth and K. Yuasa, Quantum system identifica- tion, Physical Review Letters108, 080502 (2012)

    work page 2012

  16. [16]

    Gross, Y.-K

    D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis- ert, Quantum state tomography via compressed sensing, Physical Review Letters105, 150401 (2010)

    2010

  17. [17]

    Ferrie, Self-guided quantum tomography, Physical Re- view Letters113, 190404 (2014)

    C. Ferrie, Self-guided quantum tomography, Physical Re- view Letters113, 190404 (2014)

    work page 2014

  18. [18]

    Chen, P.-Y

    H.-B. Chen, P.-Y. Lo, C. Gneiting, J. Bae, Y.-N. Chen, and F. Nori, Quantifying the nonclassicality of pure de- phasing, Nature Communications10, 3794 (2019)

    work page 2019

  19. [19]

    Ahmed, C

    S. Ahmed, C. S´ anchez Mu˜ noz, F. Nori, and A. F. Kockum, Quantum state tomography with conditional generative adversarial networks, Physical Review Letters 127, 140502 (2021)

    work page 2021

  20. [20]

    Zheng, W

    R.-H. Zheng, W. Ning, Y.-H. Chen, J.-H. L¨ u, L.-T. Shen, K. Xu, Y.-R. Zhang, D. Xu, H. Li, Y. Xia, F. Wu, Z.- B. Yang, A. Miranowicz, N. Lambert, D. Zheng, H. Fan, F. Nori, and S.-B. Zheng, Observation of a superradi- ant phase transition with emergent cat states, Physical Review Letters131, 113601 (2023)

    work page 2023

  21. [21]

    K. M. R. Audenaert, J. Calsamiglia, R. Mu˜ noz Tapia, E. Bagan, L. Masanes, A. Acin, and F. Verstraete, Dis- criminating states: The quantum Chernoff bound, Phys- ical Review Letters98, 160501 (2007)

    work page 2007

  22. [22]

    J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample- optimal tomography of quantum states, IEEE Transac- tions on Information Theory63, 5628 (2017)

    work page 2017

  23. [23]

    D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohout, and A. M. Steinberg, Adaptive quan- tum state tomography improves accuracy quadratically, Physical Review Letters111, 183601 (2013)

    work page 2013

  24. [24]

    Zhu,Quantum State Estimation and Symmetric Infor- mationally Complete POMs, Ph.D

    H. Zhu,Quantum State Estimation and Symmetric Infor- mationally Complete POMs, Ph.D. thesis, National Uni- versity of Singapore (2012)

    work page 2012

  25. [25]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- enhanced measurements: Beating the standard quantum limit, Science306, 1330 (2004)

    2004

  26. [26]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics5, 222 (2011)

    2011

  27. [27]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Physical Review Letters96, 010401 (2006)

    2006

  28. [28]

    D. W. Berry and H. M. Wiseman, Optimal states and almost optimal adaptive measurements for quantum in- terferometry, Physical Review Letters85, 5098 (2000)

    work page 2000

  29. [29]

    T. A. Wheatley, D. W. Berry, H. Yonezawa, D. Nakane, H. Arao, D. T. Pope, T. C. Ralph, H. M. Wise- man, A. Furusawa, and E. H. Huntington, Adaptive op- tical phase estimation using time-symmetric quantum smoothing, Physical Review Letters104, 093601 (2010)

    work page 2010

  30. [30]

    Yonezawa, D

    H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, Quantum-enhanced optical-phase tracking, Science337, 1514 (2012)

    work page 2012

  31. [31]

    Pereira, D

    L. Pereira, D. Mart ´ ınez, G. Canas, E. G´ omez, S. Wal- born, G. Lima, and A. Delgado, High-accuracy adaptive quantum tomography for high-dimensional quantum sys- tems, arXiv preprint arXiv:2009.04791 (2020)

    Pith/arXiv arXiv 2009

  32. [32]

    Husz´ ar and N

    F. Husz´ ar and N. M. T. Houlsby, Adaptive Bayesian quantum tomography, Physical Review A85, 052120 (2012)

    work page 2012

  33. [33]

    G. I. Struchalin, I. A. Pogorelov, S. S. Straupe, K. S. Kravtsov, I. V. Radchenko, and S. P. Kulik, Experimen- tal adaptive quantum tomography of two-qubit states, Physical Review A93, 012103 (2016)

    work page 2016

  34. [34]

    B. Qi, Z. Hou, Y. Wang, D. Dong, H.-S. Zhong, L. Li, G.-Y. Xiang, H. M. Wiseman, C.-F. Li, and G.-C. Guo, Adaptive quantum state tomography via linear regres- sion estimation: Theory and two-qubit experiment, npj Quantum Information3, 19 (2017)

    work page 2017

  35. [35]

    Pereira, L

    L. Pereira, L. Zambrano, J. Cort´ es-Vega, S. Niklitschek, and A. Delgado, Adaptive quantum tomography in high dimensions, Physical Review A98, 012339 (2018)

    work page 2018

  36. [36]

    D. Ahn, Y. S. Teo, H. Jeong, F. Bouchard, F. Hufnagel, E. Karimi, D. Koutn` y, J.ˇReh´ aˇ cek, Z. Hradil, G. Leuchs, and L. L. S´ anchez-Soto, Adaptive compressive tomogra- phy with no a priori information, Physical Review Letters 122, 100404 (2019)

    work page 2019

  37. [37]

    S. Xiao, Y. Wang, D. Dong, and J. Zhang, Optimal and two-step adaptive quantum detector tomography, Auto- matica141, 110296 (2022)

    work page 2022

  38. [38]

    J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomogra- phy, Physical Review Letters90, 193601 (2003)

    work page 2003

  39. [39]

    S. H. Lie and H. Jeong, Faithfulness and sensitivity for ancilla-assisted process tomography, Physical Review Letters130, 020802 (2023)

    work page 2023

  40. [40]

    Mohseni and D

    M. Mohseni and D. A. Lidar, Direct characterization of quantum dynamics, Physical Review Letters97, 170501 (2006)

    work page 2006

  41. [41]

    H. Wang, W. Zheng, N. Yu, K. Li, D. Lu, T. Xin, C. Li, Z. Ji, D. Kribs, B. Zeng, X. Peng, and J. Du, Quan- tum state and process tomography via adaptive measure- ments, Science China Physics, Mechanics & Astronomy 59, 100313 (2016)

    work page 2016

  42. [42]

    Y. S. Teo, B.-G. Englert, J. ˇReh´ aˇ cek, and Z. Hradil, Adaptive schemes for incomplete quantum process to- mography, Physical Review A84, 062125 (2011)

    work page 2011

  43. [43]

    Bongioanni, L

    I. Bongioanni, L. Sansoni, F. Sciarrino, G. Vallone, and P. Mataloni, Experimental quantum process tomogra- phy of non-trace-preserving maps, Physical Review A82, 7 042307 (2010)

    work page 2010

  44. [44]

    [37] identifies the distortion problem as that Fd,p( ˆS, S) = 1 only guarantees ˆS/Tr( ˆS) =S/Tr(S) rather than ˆS=Swhen Tr(S) is unknown

    Ref. [37] identifies the distortion problem as that Fd,p( ˆS, S) = 1 only guarantees ˆS/Tr( ˆS) =S/Tr(S) rather than ˆS=Swhen Tr(S) is unknown. Consequently, the existing infidelity definition 1−F d,p is incomplete: if applied to QDT or non-trace-preserving QPT, it pre- serves the sufficiency of C1 and C2 in Theorem 1 but fails to maintain their necessity

    work page

  45. [45]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estima- tion, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019)

    2019

  46. [46]

    Hradil, Quantum-state estimation, Physical Review A 55, R1561 (1997)

    Z. Hradil, Quantum-state estimation, Physical Review A 55, R1561 (1997)

    work page 1997

  47. [47]

    G. A. M. Paris and J. Rehacek,Quantum State Estima- tion, Vol. 649 (Springer Science & Business Media, Berlin Heidelberg, 2004)

    work page 2004

  48. [48]

    Fiur´ aˇ sek, Maximum-likelihood estimation of quantum measurement, Physical Review A64, 024102 (2001)

    J. Fiur´ aˇ sek, Maximum-likelihood estimation of quantum measurement, Physical Review A64, 024102 (2001)

    work page 2001

  49. [49]

    B. Qi, Z. Hou, L. Li, D. Dong, G.-Y. Xiang, and G.- C. Guo, Quantum state tomography via linear regression estimation, Scientific Reports3, 3496 (2013)

    work page 2013

  50. [50]

    Hou, J.-F

    Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.- D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Determin- istic realization of collective measurements via photonic quantum walks, Nature Communications9, 1414 (2018)

    2018

  51. [51]

    J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise, Physical Review Letters108, 070502 (2012)

    work page 2012

  52. [52]

    But the converse is not true; i.e., Ref

    In fact, all the POVM elements in Step-2 of our two- step adaptive QST algorithm are also measured in the adaptive step of the algorithm in [31]. But the converse is not true; i.e., Ref. [31] requires strictly more POVM elements

    work page

  53. [53]

    B. Mu, H. Qi, I. R. Petersen, and G. Shi, Quantum to- mography by regularized linear regressions, Automatica 114, 108837 (2020)

    work page 2020

  54. [54]

    Blume-Kohout, Optimal, reliable estimation of quan- tum states, New Journal of Physics12, 043034 (2010)

    R. Blume-Kohout, Optimal, reliable estimation of quan- tum states, New Journal of Physics12, 043034 (2010)

    2010

  55. [55]

    Bhatia,Perturbation Bounds for Matrix Eigenvalues (Society for Industrial and Applied Mathematics, 2007)

    R. Bhatia,Perturbation Bounds for Matrix Eigenvalues (Society for Industrial and Applied Mathematics, 2007)

    work page 2007

  56. [56]

    Y. Wang, S. Yokoyama, D. Dong, I. R. Petersen, E. H. Huntington, and H. Yonezawa, Two-stage estimation for quantum detector tomography: Error analysis, numerical and experimental results, IEEE Transactions on Informa- tion Theory67, 2293 (2021)

    work page 2021

  57. [57]

    R. A. Horn and C. R. Johnson,Matrix Analysis(Cam- bridge University Press, 2012)

    2012

  58. [58]

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

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

    2018

  59. [59]

    Y. Wang, D. Dong, B. Qi, J. Zhang, I. R. Petersen, and H. Yonezawa, A quantum Hamiltonian identification algorithm: Computational complexity and error analy- sis, IEEE Transactions on Automatic Control63, 1388 (2018)

    work page 2018

  60. [60]

    S. Xiao, Y. Wang, J. Zhang, D. Dong, G. J. Mooney, I. R. Petersen, and H. Yonezawa, A two-stage solution to quantum process tomography: Error analysis and op- timal design, IEEE Transactions on Information Theory 71, 1803 (2025)

    work page 2025

  61. [61]

    M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D. Mortimer, T. J. Osborne, M. J. Bremner, A. W. Har- row, and A. Hines, Quantum dynamics as a physical re- source, Physical Review A67, 052301 (2003)

    2003

  62. [62]

    Mohseni, A

    M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Quantum-process tomography: Resource analysis of dif- ferent strategies, Physical Review A77, 032322 (2008)

    work page 2008

  63. [63]

    Xiao, Ancilla-assisted quantum process tomography, in preparation (2023)

    S. Xiao, Ancilla-assisted quantum process tomography, in preparation (2023)

    work page 2023

  64. [64]

    I. A. Pogorelov, G. I. Struchalin, S. S. Straupe, I. V. Radchenko, K. S. Kravtsov, and S. P. Kulik, Experimen- tal adaptive process tomography, Physical Review A95, 012302 (2017)

    work page 2017

  65. [65]

    Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications10, 285 (1975)

    M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications10, 285 (1975)

    1975

  66. [66]

    Zhang, A

    L. Zhang, A. Datta, H. B. Coldenstrodt-Ronge, X.-M. Jin, J. Eisert, M. B. Plenio, and I. A. Walmsley, Recursive quantum detector tomography, New Journal of Physics 14, 115005 (2012)

    2012

  67. [67]

    Y. Yu, T. Wang, and R. J. Samworth, A useful variant of the Davis–Kahan theorem for statisticians, Biometrika 102, 315 (2015)

    work page 2015

  68. [68]

    Fuchs and J

    C. Fuchs and J. van de Graaf, Cryptographic distin- guishability measures for quantum-mechanical states, IEEE Transactions on Information Theory45, 1216 (1999)

    work page 1999

  69. [69]

    Pepper, T

    A. Pepper, T. J. Baker, Y. Wang, Q.-C. Song, L. K. Shalm, V. B. Verma, S. W. Nam, N. Tischler, S. Slus- sarenko, H. M. Wiseman,et al., Scalable multiparty steering using a single entangled photon-pair, A VS Quan- tum Science6(2024)

    work page 2024

  70. [70]

    Villegas-Aguilar, Y

    L. Villegas-Aguilar, Y. Wang, A. Pepper, T. J. Baker, G. J. Pryde, S. Slussarenko, N. Tischler, and H. M. Wise- man, Quantum assemblage tomography, arXiv preprint arXiv:2408.15576 (2024)

    work page arXiv 2024

  71. [71]

    C. F. Loan, The ubiquitous Kronecker product, Jour- nal of Computational and Applied Mathematics123, 85 (2000)

    work page 2000

  72. [72]

    Chan and K.-H

    N. Chan and K.-H. Li, Diagonal elements and eigenval- ues of a real symmetric matrix, Journal of Mathematical Analysis and Applications91, 562 (1983)

    work page 1983

  73. [73]

    R. D. Gill and S. Massar, State estimation for large en- sembles, Physical Review A61, 042312 (2000)

    2000

  74. [74]

    Johnston, QETLAB: A MATLAB toolbox for quan- tum entanglement, version 0.9 (2016)

    N. Johnston, QETLAB: A MATLAB toolbox for quan- tum entanglement, version 0.9 (2016)

    work page 2016

  75. [75]

    Zyczkowski and M

    K. Zyczkowski and M. Kus, Random unitary matrices, Journal of Physics A: Mathematical and General27, 4235 (1994)

    work page 1994

  76. [76]

    Johansson, P

    J. Johansson, P. Nation, and F. Nori, QuTiP: An open- source Python framework for the dynamics of open quan- tum systems, Computer Physics Communications183, 1760 (2012)

    work page 2012

  77. [77]

    J. A. Miszczak, Generating and using truly random quan- tum states in Mathematica, Computer Physics Commu- nications183, 118 (2012). 8 Supplemental Material NOT A TION AND KEY SYMBOLS We present all relevant notations below, with the acronyms summarized in Table I, and the key symbols used in our proposed adaptive algorithms listed in Table II. Notation:T...

    work page 2012

  78. [78]

    For a non-trace-preserving process, after the two-step adaptive QPST, we obtain ˆσ out, where Tr (ˆσout)<1 and also reconstruct ˜X0 as Eq

    Therefore, using Theorem 1, we haveE 1−F ˆX, X =O(1/N) for trace-preserving processes. For a non-trace-preserving process, after the two-step adaptive QPST, we obtain ˆσ out, where Tr (ˆσout)<1 and also reconstruct ˜X0 as Eq. (S.118). Then using Eq. (S.23) in Step-3, we obtain the final estimate ˆX. Similar to trace-preserving processes, using Eqs. (S.118...

    work page

  79. [79]

    The results are shown in Fig. S2(b) where all the infidelities surpass the GM bound and are slightly smaller than that forN 0 = 0.5N, indicating that the resource distribution proportion affects the tomography error. Furthermore, as the rank increases, the mean infidelities in Fig. S2 also increase and a similar phenomenon was also observed in [35]. The r...

    work page