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Compressed sensing quantum state tomography for qudits: A comparison of Gell-Mann and Heisenberg-Weyl observable bases

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Pith's one-line read Compressed sensing quantum state tomography for qudits is more measurement-efficient with the Heisenberg-Weyl observable basis than with the generalized Gell-Mann basis, and the advantage grows with the qudit dimension.

desk verdict Solid, clearly described numerical comparison of GGM vs HWO bases for two-qudit CS-QST on rank-1 states, but the abstract overgeneralizes beyond what was simulated. read the letter →

arxiv 2505.10462 v2 pith:6JKVEUHM submitted 2025-05-15 quant-ph

Yoshiyuki Kakihara , Daisuke Yamamoto , Giacomo Marmorini This is my paper · ORCID
classification quant-ph PACS 03.67.-a03.65.Wj
keywords compressedsensingquantumstatetomographyquditHeisenberg-WeylobservablebasisgeneralizedGell-Manncoherencesingularvaluethresholdingfidelitymeasurementselection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the choice of measurement-operator basis changes how many measurements compressed sensing quantum state tomography (CS-QST) needs to reconstruct an unknown qudit state. By numerically reconstructing random pure states of two qudits under 5% depolarizing and Gaussian measurement noise, it establishes that the Heisenberg-Weyl observable (HWO) basis reaches a mean fidelity minus one standard deviation of 0.95 with fewer randomly selected measurements than the generalized Gell-Mann (GGM) basis, and that the gap grows roughly as $k^2$ with qudit dimension $k$. The paper attributes this to coherence: the GGM basis has minimum coherence $\nu_{\min}$ scaling as $(k-1)^N$, while the HWO basis stays near $2^N$. If correct, the result gives a concrete rule for choosing bases in high-dimensional quantum tomography, while leaving GGM as a workable alternative at small $k$.

What carries the argument

The load-bearing object is the minimum coherence $\nu_{\min}\equiv d\max_a\|B_a\|_2$ defined in Eq. (17) from compressed sensing theory, which measures how concentrated tomographic weight is across the basis elements. For the GGM basis, the diagonal operators have spectral norms that grow with $k$, giving $\nu_{\min}=(k-1)^N$, while for the HWO basis all non-identity elements share the same spectral norm, giving $\nu_{\min}\approx 2^N$ (exactly $2^N$ when $k$ is a multiple of 8, and exactly $1$ for $k=4$). The paper uses this coherence difference to predict and explain why HWO requires fewer random measurements, alongside the singular value thresholding (SVT) algorithm used for reconstruction.

What would settle it

Repeat the same protocol for two-qudit states of ranks $r=2$ and $r=3$ at $k=9,12,15$ and for $N=3$ or $4$ qudits, keeping the same noise model, and compute $m_{95\%}^{\mathrm{GGM}}-m_{95\%}^{\mathrm{HWO}}$; if the gap stops growing as $k^2$, or reverses, the central claim holds only within the simulated regime.

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Extended reading notes

Core claim

The central claim is that in CS-QST of two-qudit systems, the HWO basis becomes more efficient than the GGM basis as the qudit dimension $k$ increases: the minimum number $m_{95\%}$ of randomly selected measurement operators needed to keep mean fidelity minus one standard deviation at or above 95% is smaller for HWO, and the gap between the two bases grows approximately as $k^2$ (Eqs. 28-29, Fig. 3). The paper explains this through the minimum coherence $\nu_{\min}=d\max_a\|B_a\|_2$: for GGM it grows as $(k-1)^N$, whereas for HWO it is nearly constant in $k$, taking values close to $2^N$ and equalling $2^N$ exactly when $k$ is a multiple of 8. The same coherence argument accounts for the paper's fixed-dimension comparison at $d=16$, where HWO-based qudit tomography matches Pauli-based qubit tomography while GGM degrades with increasing $k$.

Load-bearing premise

The central claim rests on simulations limited to rank-1 pure states of two qudits with a fixed 5% depolarizing channel and Gaussian measurement noise of standard deviation $0.1/d$; the general conclusion that HWO becomes more efficient as qudit dimension increases assumes this ordering survives for mixed low-rank states, more qudits, and other noise levels.

Editorial extensions

If this is right

  • For two-qudit systems, the number of measurements needed to reach 95% mean fidelity minus one standard deviation grows more slowly with $k$ for the HWO basis than for the GGM basis, and the gap grows roughly as $k^2$.
  • At a fixed Hilbert space dimension $d=16$, qudit tomography using the HWO basis matches the efficiency of four-qubit tomography using the Pauli basis, while GGM efficiency degrades as $k$ increases.
  • The coherence analysis predicts that GGM becomes less reliable at small $m/d^2$ because its diagonal operators have larger spectral norms, and the simulations confirm larger error bars for GGM in that regime.
  • Both bases eventually achieve high fidelity, so an experiment that can more easily implement GGM measurements can still use GGM at low or moderate $k$; high-dimensional implementations should prefer HWO.

Reading between the lines

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

  • A direct testable extension is to run the same comparison for mixed low-rank states ($r=2,3$); if the coherence argument is the true driver, the HWO advantage should persist because coherence is a property of the basis rather than of the state rank.
  • The coherence analysis suggests a mitigation for GGM: sampling measurement operators with probabilities weighted by their spectral norms, or always including the diagonal operators, could reduce the gap; this is worth testing.
  • The same basis-efficiency logic likely applies to other compressed-sensing recovery tasks in quantum information, such as channel or process tomography, where the measurement basis is also chosen rather than fixed.
  • Because the paper fixes the SVT algorithm and its hyperparameters, the size of the gap may shift with a different reconstruction algorithm; repeating the comparison with another matrix-completion method would separate basis effects from algorithm effects.
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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 / 5 minor

Summary. The paper numerically compares compressed sensing quantum state tomography (CS-QST) for qudit systems using two operator bases: the generalized Gell-Mann (GGM) basis and the Heisenberg-Weyl observable (HWO) basis. The authors generate rank-1 Ginibre states, apply 5% depolarizing noise and Gaussian measurement noise, reconstruct via singular value thresholding, and evaluate fidelity and trace distance. For two-qudit systems (N=2) with qudit dimension k=3,6,9,12,15, they report that the HWO basis requires fewer randomly selected measurements to achieve a mean fidelity minus one standard deviation of at least 0.95, and they fit the measured m_95% to functions of k (Eqs. 28-29), claiming the GGM-HWO gap grows approximately as k^2. They also compare fixed-dimension d=16 systems and attribute the observed differences to the coherence properties of the two bases, defined via a 'minimum coherence' ν_min (Eq. 17). The abstract states without qualification that 'the HWO basis becomes more efficient as the qudit dimension increases.'

Significance. If the reported trend is robust, the paper offers practically useful guidance for choosing measurement bases in qudit CS-QST, an area relevant to high-dimensional quantum information processing. The numerical protocol is clearly described (Ginibre states, SVT, AIC model selection), and the paper provides explicit fitting formulas that can be tested against other simulations or experiments. The main theoretical support, however, rests on a coherence calculation that, as written, is inconsistent with the paper's own definition of ν_min, and the empirical claims are based on a restricted set of simulations (rank-1, N=2) without error bars on the m_95% values. These issues do not, by themselves, invalidate the numerical comparison, but they make the central claim stronger than the evidence presented.

major comments (3)
  1. [Eq. (17), Eq. (31), Table I, Fig. 1] The coherence measure ν_min defined in Eq. (17) as d max_a ||B_a||_2 is inconsistent with the values reported in Eq. (31) and Table I for the HWO basis. With B_a = (ω_{i1}⊗...⊗ω_{iN})/√(k^N) as in Eq. (2), which the text calls an orthonormal basis, the spectral norm of B_a is (max_i ||ω_i||_2)^N / k^{N/2}, so ν_min = k^{N/2} (max_i ||ω_i||_2)^N, not (max_i ||ω_i||_2)^N as stated in Sec. II C. For the HWO basis, using the eigenvalue expression in Eq. (30), this gives ν_min = k^{N/2} (max_n ε_n^2)^{N/2}, not (max_n ε_n^2)^N as in Eq. (31). For example, for k=3, max_n ε_n^2 = 1.866 but ||W(0,1)||_2 = 1.366, so the value 1.866 reported in Table I is a squared eigenvalue, not a spectral-norm coherence. Under the corrected definition, HWO ν_min grows as k^{N/2}(√2)^N for k a multiple of 8, rather than remaining constant. This invalidates the statement in Sec. II C that ν_min for the HWO basis is 'nearly constant regardless of k' and undercuts the coherence-based explanation in Sec. III C. The authors should correct the normalization in Eq. (31) and revise Fig. 1, Table I, and the related discussion accordingly.
  2. [Abstract and Secs. III B, III C 1] The abstract and Sec. IV state without qualification that 'the HWO basis becomes more efficient as the qudit dimension increases,' but the numerical evidence is obtained only for N=2 two-qudit systems with rank-1 pure states (Sec. III B: 'we consider only pure states with rank r=1'), a fixed 5% depolarizing channel, and Gaussian measurement noise with standard deviation 0.1/d. No simulations for mixed states (r>1) or for N>2 are reported, so the unqualified claim is an extrapolation from a narrow regime. The abstract and summary should either explicitly restrict the claim to the simulated conditions (e.g., 'for two-qudit pure states under the noise model considered') or the authors should add simulations for other ranks and numbers of qudits to support the general statement.
  3. [Sec. III C 1, Eqs. (28)-(29), Fig. 3] The quantitative claim that the GGM-HWO gap in m_95% grows roughly as k^2 is based on least-squares fits to m_95% values, but the m_95% data points are reported without error bars. Each m_95% is determined from 50 repetitions (Sec. III B), so the sample mean and standard deviation used in the condition 'mean fidelity minus one standard deviation ≥ 0.95' carry statistical uncertainty that propagates to m_95%. Since no uncertainties are reported for the fitted parameters in Eqs. (28)-(29), the fitted k^2 scaling is not statistically grounded. The authors should provide error estimates for m_95% (e.g., via bootstrapping) and report confidence intervals for the fit parameters, or otherwise demonstrate that the observed trend is robust to the sampling uncertainty.
minor comments (5)
  1. [Sec. III B] The text describes the target state as a rank-1 pure state but then applies depolarizing noise to obtain \tilde{ρ}, which has full rank. Since the fidelity in Eq. (25) is computed against the original ρ, the effective reconstruction target is a pure state corrupted by noise, not the depolarized state itself; this is a reasonable protocol, but it would be clearer to state explicitly that ρ, not \tilde{ρ}, is the target for the fidelity and trace distance.
  2. [Sec. II C] The sentence 'Section II C presents our numerical simulations' is a typo; it should refer to Section III.
  3. [Acknowledgments] The sentence 'The work of was supported' is incomplete; the author names are missing.
  4. [Sec. III C 1] The connection between the fitted m_95% curves and the theoretical scaling m ∼ O(rd log^2 d) is only qualitative; the fitted forms have p_1 >> p_0, so the leading behavior is m_95% ∼ p_1 k^2, and the paper would benefit from a brief explanation of why the logarithmic factor is subleading in the simulated range.
  5. [Appendix B] Equation (31) uses the notation (max_n ε_n^2)^N without explicitly stating that ε_n are the eigenvalues of the single-qudit HWO operator; relating this to the corrected definition of ν_min discussed above would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HWO-vs-GGM comparison is produced by symmetric numerical simulation, and the m95% scaling curves are presented as descriptive fits, not predictions.

full rationale

The central derivation is self-contained. In Sec. III B-C, GGM and HWO reconstructions are generated from the same Ginibre rank-1 states with the same 5% depolarizing noise and Gaussian measurement noise, and processed by the same SVT algorithm, so the observed HWO advantage is an emergent simulation result, not an input. Eqs. (28)-(29) are explicitly least-squares/AIC fits to the simulated m95% data (Sec. III C 1 and Appendix V D), and the paper describes them as fitting curves rather than as predictions, so no fitted parameter is renamed as a prediction. The coherence mechanism in Sec. II C is imported from external references (Gross [27]; Asadian et al. [24]) and is used as an explanatory hypothesis tested by the independent numerics; there is no self-citation chain and no uniqueness theorem from the authors' prior work forcing the choice of basis. The paper itself states the scope limitation to rank-1, N=2 states (Sec. III B: 'we consider only pure states with rank r=1'), which narrows generality but is not a circular dependency. The apparent normalization mismatch between the nu_min definition in Eq. (17) and its evaluation in Eq. (31) is a consistency/correctness concern in the theoretical explanation, but it does not make the numerical comparison circular, since the simulations do not use nu_min as an input.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

No invented entities. The numerical study's central comparison depends on the modeling choices listed as axioms. The fitted m95% curves depend on eight least-squares coefficients, four per basis. The fixed SVT and noise parameters are disclosed but not varied, so their influence on the GGM-versus-HWO gap is not explored.

free parameters (8)
  • GGM p0 = -0.117
    Least-squares/AIC fit coefficient in Eq. (28) for m95%/d^2 for the GGM basis.
  • GGM p1 = 18.865
    Least-squares/AIC fit coefficient in Eq. (28) for m95%/d^2 for the GGM basis.
  • GGM p2 = -53.044
    Least-squares/AIC fit coefficient in Eq. (28) for m95%/d^2 for the GGM basis.
  • GGM p3 = 42.449
    Least-squares/AIC fit coefficient in Eq. (28) for m95%/d^2 for the GGM basis.
  • HWO p0 = -0.123
    Least-squares/AIC fit coefficient in Eq. (29) for m95%/d^2 for the HWO basis.
  • HWO p1 = 13.897
    Least-squares/AIC fit coefficient in Eq. (29) for m95%/d^2 for the HWO basis.
  • HWO p2 = -31.882
    Least-squares/AIC fit coefficient in Eq. (29) for m95%/d^2 for the HWO basis.
  • HWO p3 = 20.894
    Least-squares/AIC fit coefficient in Eq. (29) for m95%/d^2 for the HWO basis.
assumptions (5)
  • domain assumption The SVT algorithm (Ref [5]) reconstructs the low-rank state from the random subset P_Omega(M) under the simulations' sampling model.
    The entire reconstruction pipeline (Sec. III B and Appendix V C) assumes SVT with tau=5, delta=0.1d^2/m, t_max=2*t_0, and epsilon=1e-7 converges to a good approximation of the target for both bases.
  • domain assumption Rank-1 pure states generated from the Ginibre ensemble, with 5% depolarizing noise and Gaussian measurement noise N(0,0.1/d), are representative targets for low-rank CS-QST.
    Sec. III B states "we consider only pure states with rank r=1" and fixes the noise levels; the generality of the conclusion depends on this representativeness.
  • domain assumption The minimum coherence nu_min (Eq. 17) is a meaningful basis-dependent predictor of practical reconstruction efficiency.
    Sec. II C invokes Ref [27]'s coherence conditions to interpret the numerical ordering; the paper does not derive a direct theorem linking nu_min to m95% for these bases.
  • standard math The normalized basis B_a=(omega_{i1} tensor ... tensor omega_{iN})/sqrt(d) with Tr[omega_i omega_j]=k delta_{ij} is a valid orthonormal Hermitian basis for expanding density matrices (Eq. 2).
    This follows from the definitions in Sec. II B and is used for the expansion and for constructing the sample matrix in Eq. (24).
  • domain assumption The AIC-selected functional form in Eqs. (28)-(29) is an adequate model for the m95% data over the simulated range.
    Appendix V D selects the model on the same 14 data points used in Fig. 3; no held-out validation is provided, so the fitted curves are descriptive rather than predictive.

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Pith. "Pith review of Compressed sensing quantum state tomography for qudits: A comparison of Gell-Mann and Heisenberg-Weyl observable bases." pith.science (2026). https://pith.science/paper/6JKVEUHM

@misc{pith2026250510462,
  author       = {Pith},
  title        = {Pith review of: Compressed sensing quantum state tomography for qudits: A comparison of Gell-Mann and Heisenberg-Weyl observable bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JKVEUHM}},
  note         = {Machine review of arXiv:2505.10462}
}
read the original abstract

Quantum state tomography (QST) is an essential technique for reconstructing the density matrix of an unknown quantum state from measurement data, crucial for quantum information processing. However, conventional QST requires an exponentially growing number of measurements as the system dimension increases, posing a significant challenge for high-dimensional systems. To mitigate this issue, compressed sensing quantum state tomography (CS-QST) has been proposed, significantly reducing the required number of measurements. In this study, we investigate the impact of basis selection in CS-QST for qudit systems, which are fundamental to high-dimensional quantum information processing. Specifically, we compare the efficiency of the generalized Gell-Mann (GGM) and Heisenberg-Weyl observable (HWO) bases by numerically reconstructing density matrices and evaluating reconstruction accuracy using fidelity and trace distance metrics. Our results demonstrate that, while both bases allow for successful density matrix reconstruction, the HWO basis becomes more efficient as the qudit dimension increases. Furthermore, we find the best fitting curves that estimate the number of measurement operators required to achieve a fidelity of at least 95%. These findings highlight the significance of basis selection in CS-QST and provide valuable insights for optimizing measurement strategies in high-dimensional quantum state tomography.

Figures

Figures reproduced from arXiv: 2505.10462 by the authors.

Figure 1
Figure 1. The minimum coherence νmin, defined by Eq. (17), is plotted as a function of the dimension d = k N , comparing the GGM and HWO bases. Panels (a) and (b) correspond to N = 1 and N = 2, respectively, with qudit dimensions ranging from k = 2 to k = 16. The insets show magnified views of the HWO results in each panel. Note that for k = 2, both bases reduce to the Pauli basis. Table I. Numerical values of the minimum coh… view at source ↗
Figure 2
Figure 2. Fidelity F(ρ, X˜ ) and trace distance D(ρ, X˜ ) between the target two-qudit state ρ and the matrix X˜ obtained via CS-QST, for qudit dimensions (a) k = 3, (b) 6, (c) 9, (d) 12, and (e) 15. The horizontal axis represents the number of randomly selected measurement operators m, scaled by the total number of basis elements d 2 = k 4 . basis, the overall difference between the two becomes smaller. This convergence in r… view at source ↗
Figure 3
Figure 3. (a) The relative measurement fraction m95%/d2 , obtained by normalizing the number of measurements m95% required to achieve a fidelity of at least 95% in CS-QST of two-qudit states by the total number of measurement operators d 2 = k 4 , is plotted as a function of the qudit dimension k, along with the corresponding fitted curves. The fitted parameters for the GGM basis are p0 = −0.117, p1 = 18.865, p2 = −53.044, an… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of CS-QST performance for quantum [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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Pith tools

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