Pith. sign in

REVIEW 4 major objections 6 minor 3 cited by

Optimal entanglement witness of multipartite systems using support vector machine approach

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read SVM-trained hyperplanes become optimal entanglement witnesses

desk verdict The central claim of optimal SVM entanglement witnesses is unproven: the decision boundary is only validated on finite samples, so the operators need not be witnesses at all. read the letter →

arxiv 2504.18163 v1 pith:JCOUAVR2 submitted 2025-04-25 quant-ph

Mahmoud Mahdian , Zahra Mousavi This is my paper · ORCID
classification quant-ph MSC 81P4068T0581P68
keywords entanglementwitnesssupportvectormachineoptimalnon-decomposableBell-diagonalstatesGHZPPTentangledlearningdetection
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 claims that a support-vector-machine classifier, trained on the local Pauli expectation values of known separable and entangled states, produces a Hermitian operator that is an entanglement witness: nonnegative on every separable state and negative on the entangled states it targets. The authors construct such witnesses for two-, three-, and four-qubit Bell-diagonal and Werner/GHZ mixed states, and argue that whenever the SVM training succeeds the resulting hyperplane is optimal, being completely tangent to the convex set of separable states. They further report a three-qubit non-decomposable witness that detects a positive-partial-transpose entangled state, a type that the PPT criterion alone cannot identify. If the claim holds, the method is a numerical recipe for optimal entanglement witnesses that extends to N-qubit systems without requiring an analytic construction.

What carries the argument

The load-bearing object is the maximum-margin hyperplane of a soft-margin SVM in the space of local Pauli expectation values. The optimization minimizes $\frac{1}{2}\|w\|^2 + C\sum_i \xi_i$ subject to $y_i(w^T x_i + b)\ge 1-\xi_i$; the resulting coefficients $c_{\vec i}$ define $W$ by Eq. (5), and the bias $b$ places the boundary $\langle W\rangle=0$. Max-margin geometry is what makes the boundary tangent to the separable region, and the paper's optimality certificate is the vanishing of $\mathrm{Tr}(W|\nu\rangle\langle\nu|)$ on product states plus the claim that no positive operator lies in the kernel of $W$.

What would settle it

For each constructed witness, compute the minimum of $\mathrm{Tr}(W\rho)$ over all fully separable states: for two-qubit Bell-diagonal states use the exact octahedron separability condition, and for three- and four-qubit states use dense sampling of product states. A single separable state with negative expectation would show that the SVM hyperplane is not an entanglement witness.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the SVM decision boundary can be read directly as an entanglement witness. With features $x_{\vec i} = \mathrm{Tr}(\sigma_{i_1}\otimes\cdots\otimes\sigma_{i_N}\rho)$ and decision function $y = \sum_{\vec i} c_{\vec i} x_{\vec i} - b$, the weight vector becomes the witness $W = \sum_{\vec i} c_{\vec i}\sigma_{i_1}\otimes\cdots\otimes\sigma_{i_N}$; the separable class has $y\ge 0$ and the entangled class $y<0$. For the Werner families studied, the trained witnesses are reported to be completely tangent to the separable region: explicit product states $|\nu\rangle$ satisfy $\mathrm{Tr}(W|\nu\rangle\langle\nu|)=0$, and the absence of any positive operator that could be subtracted from $W$ is taken as proof of optimality. In the three-qubit case, a witness built for the PPT-entangled density matrix of the paper's reference [34] has negative expectation on that state, which the authors use to conclude that the witness is non-decomposable.

Load-bearing premise

The whole construction assumes that a hyperplane which fits the training data also keeps nonnegative expectation on every separable state the algorithm never saw, and the paper does not verify this on the full separable set.

Editorial extensions

If this is right

  • For the two-, three-, and four-qubit Werner/GHZ families studied, the trained SVM yields a single operator that detects every entangled state in the family above its threshold $p$.
  • The same local-measurement construction is claimed to generalize to $N$-qubit systems with equal local dimensions, giving numerical optimal witnesses without analytic guesses.
  • Non-decomposable witnesses produced this way can detect PPT-entangled (bound) states that escape the PPT criterion.
  • Whenever the SVM succeeds, the witnesses are automatically optimal in the standard sense, because the margin-maximizing boundary is completely tangent to the separable region.

Reading between the lines

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

  • A strict check of the claim would verify nonnegativity of $\mathrm{Tr}(W\rho)$ over the entire separable set, not just the training sample; the paper's tangency points are evidence, not a full certificate, for states far from those points.
  • For other state families, the same recipe should be expected to work only if the training sample densely covers the separable-entangled boundary; sparse or biased samples could produce a hyperplane that cuts into the separable region.
  • Because the witness is a sum of local Pauli terms, the trained coefficients translate directly into a measurement protocol, so the method is testable in the laboratory.
  • The approach could be benchmarked against exact separability criteria where they exist, for instance the octahedron condition for two-qubit Bell-diagonal states, to measure how close numerical witnesses come to optimality.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper proposes a support vector machine (SVM) method to construct entanglement witnesses (EWs) for two-, three-, and four-qubit systems. The authors train an SVM on separable and entangled Werner states (and on a three-qubit PPT entangled family), map the decision hyperplane to an operator W via a Pauli-basis expansion, and claim that the resulting EWs are optimal and completely tangent to the separable region. They also report a non-decomposable witness that detects a PPT entangled state.

Significance. If the central claims were correct, the paper would offer a practical, scalable numerical route to optimal EWs for multipartite systems, including PPT-entangled detection, which is of genuine interest to the quantum information community. The paper usefully provides an explicit algorithmic pipeline (Algorithm 1) and explicit numerical witnesses. However, the key assertions—that the SVM hyperplane is a valid EW and that it is optimal—are neither rigorously proven nor, in the two-qubit case, consistent with the reported expectation values. The concrete failure of the two-qubit witness to detect all entangled Werner states undermines the claimed central result.

major comments (4)
  1. [§3.1, Eqs. (12)–(13)] The two-qubit witness W1 fails to detect all entangled Werner states and therefore cannot be optimal. From the entries of W1 in Eq. (13), using ρ_W(p) = p|φ00><φ00| + (1-p)/4 I, one obtains Tr(W1ρ_W(p)) = 0.25 − 0.703p; this gives a detection threshold p ≈ 0.355, whereas ρ_W(p) is entangled for p > 1/3. The paper's statement that W1 'recognizes all the entangled states ... for all p's' is thus contradicted by its own numerical witness.
  2. [§3.1, Table 1 and Eq. (17)] The optimality proof is invalid. The condition Tr(W1|ν><ν|)=0 does not place |ν> in the kernel of W1, since W1 is an indefinite operator; the Lewenstein criterion for optimality requires product vectors |ν> with W1|ν>=0. Moreover, the argument that no |ω> is orthogonal to the four listed |ν_i> does not rule out all positive operators P, and if the four |ν_i> were indeed in the kernel and spanned the full space, they would force W1=0, which is false. The support of the claim 'we have proved that all the EW's are completely tangent ... and are therefore optimal' is therefore missing.
  3. [§2, Definition 2 and §3] No evidence is provided that any of the SVM-derived operators is a valid entanglement witness. The SVM optimization in Eq. (4) only enforces the margin constraints on the finite training set; a valid EW must satisfy Tr(Wρ_sep) ≥ 0 for every separable state ρ_sep. The manuscript does not verify this condition for W1, WGHZ, the PPTES witness, or the four-qubit witness, so the identification of the SVM hyperplane with an EW (Eqs. (5)–(9)) is unsubstantiated.
  4. [§3.2.1, Eqs. (22)–(23)] The non-decomposable witness claim rests on a single numerical example (a=0.3525, b=0.3196, c=0.81642, Tr(Wρ_ppt) = −0.0129). To establish that W is a non-decomposable EW, the authors must show both that W is a valid witness (nonnegative expectation on all separable states) and that it detects a PPT entangled state; the latter is demonstrated, but the former is not, and the single example does not constitute a proof.
minor comments (6)
  1. [Eq. (5)] The tensor product factor should read σ_{i1}⊗σ_{i2}...⊗σ_{iN}, not σ_{i1}⊗σ_{i1}...⊗σ_{iN}.
  2. [§3.1] The text cites 'matrix 18' for the two-qubit Werner state; the correct reference is Eq. (12).
  3. [Table 1] Rows 2 and 4 are identical, so only three distinct tangent states are listed despite the text referring to four product pure states.
  4. [§2, Eqs. (4), (8), (9)] The bias term b is defined in Eqs. (4) and (8) but is absent from the classifier in Eq. (9); the role of the constant term in the EW expectation value should be clarified.
  5. [Eq. (18)] The Werner parameter is restricted to 0.6 ≤ p ≤ 1 without explanation, while the two-qubit case used 0 ≤ p ≤ 1; the choice of training ranges should be justified.
  6. [General] There are numerous typographical errors, including 'seperable', 'classifid', 'matrice', and 'Hillbert', and the abstract's 'We drive and implement' should be 'We derive and implement'.

Circularity Check

3 steps flagged · score 6.0 of 10

SVM decision boundary is relabeled as an optimal EW; the 'tangent' states used in the optimality proof are the fit's own support vectors, so the central claim reduces to the training fit.

  1. fitted input called prediction [Section 2, Eqs. (4)-(9)]
    "According to the correspondence between SVM and the EW, we obtain a hyperplane that can separate the separable states, i.e., Tr(Wρsep)≥ 0 from the entangled states, i.e., Tr(Wρent)< 0."

    The SVM program in Eq. (4) constrains only the labeled training points via y_i(w·x_i+b)≥1−ξ_i; it never imposes Tr(Wρ_sep)≥0 on the full convex separable set required by Definition 2. W in Eq. (5) is the fitted weight vector, so the statement that this hyperplane separates separable from entangled states is the classifier's own decision rule restated as a physical claim. No independent validation or theorem connects the finite-sample separator to a valid EW.

  2. fitted input called prediction [Section 3.1, Table 1 and following optimality argument]
    "By selecting the following angles in Tab.1, we find that Tr(W1|ν⟩⟨ν|) = 0 ... which means that there is no positive operator in this case, so our EW is an optimal witness."

    The angles tabulated are chosen by imposing Tr(W1|ν⟩⟨ν|)=0 after W1 is already the SVM solution. These are the support vectors of the fitted hyperplane, so equality holds by construction of the margin; they are not independently established boundary points of the true separable set. The optimality argument then relies on these same fitted states to conclude that no positive operator can be subtracted, without ever proving W1≥0 on all separable states or that the four product states span the kernel. Thus the claimed optimality proof is circular in the sense that the fit supplies both the witness and the only points used to certify it.

1 more flagged steps
  1. self definitional [Appendix, Algorithm 1, Step 4]
    "Translate the SVM decision boundary into an entanglement witness. ... Output: The SVM decision boundary(w,b ) is used to define the entanglement witness W, ensuring it satisfies the necessary criteria for detecting quantum entanglement."

    This step defines W by construction as the SVM decision boundary and asserts that it satisfies the EW criteria. The 'ensuring' is not backed by any check on the full separable set; it is simply the output of the optimization relabeled as a witness. The central object of the paper is therefore identical to the fitted classifier, and the claimed optimality and tangency properties are asserted rather than derived from Definition 2.

full rationale

The paper is not burdened by self-citation loops; the numerical EWs are original outputs of an SVM fit. However, the headline claim that 'when the algorithm succeeds, the EWs are optimal and are completely tangent to the separable region' is not independently established. The optimization is performed on a finite training set; the product states used in the optimality proof are the equality points (support vectors) of that same fit, and thresholds such as p≥0.11 for four qubits follow directly from the fitted coefficients. There is no verification of Definition 2 over the full separable set and no externally validated benchmark showing that the hyperplane is a true EW. The PPTES example is an independent spot check, but it does not repair the global optimality claim. Hence the central claim partially reduces to the fitting procedure: the predictor is the fit, and the certification is done with the fit's own support vectors. Score 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation is a supervised fit; the key unproven assumption is that the fitted SVM boundary is a valid EW on the entire separable set. The optimality proof additionally assumes the finite set of tangent states spans the kernel, which is only checked numerically for a few states.

free parameters (3)
  • SVM regularization parameter C = not specified
    Controls the soft margin trade-off and thus the hyperplane coefficients that define the witness (Eq. 4).
  • Training set size and composition = not specified
    The numbers of separable and entangled samples, and their parameter ranges, are not given; these determine the fitted witness.
  • Training p ranges for each family = not specified
    For two-qubit Werner states, the p values used to generate entangled training data are not stated; the resulting detection boundary (p≈0.355) depends on this choice.
assumptions (4)
  • standard math The set of separable states is closed and convex.
    Used implicitly to justify the Hahn-Banach separation underlying witnesses (Section 1).
  • domain assumption Known separability criteria (p>1/3 for Werner, PPT for bound entangled states) correctly label the training data.
    The SVM training labels rely on these criteria, which are correct for the chosen families.
  • ad hoc to paper The finite training sample is representative of the full separable and entangled sets, so the maximum-margin hyperplane is a supporting hyperplane of the true separable set.
    The paper never verifies that the fitted witness is nonnegative on all separable states; this is the load-bearing assumption for the EW validity.
  • standard math A witness is optimal if the product states in its kernel span the Hilbert space (Lewenstein criterion).
    Used in Section 3.1 to prove optimality from a finite set of tangent product states.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal entanglement witness of multipartite systems using support vector machine approach." pith.science (2026). https://pith.science/paper/JCOUAVR2

@misc{pith2026250418163,
  author       = {Pith},
  title        = {Pith review of: Optimal entanglement witness of multipartite systems using support vector machine approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCOUAVR2}},
  note         = {Machine review of arXiv:2504.18163}
}
read the original abstract

An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We drive and implement a numerical method based on machine learning to create a multipartite EW. Using support vector machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three and four qubits in Bell-diagonal mixed states, which can be generalized to multipartite mixed states as GHZ states in systems where all modes have equal size. One of the important features of this method is that, when the algorithm succeeds, the EWs are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EWs that can detect positive partial transpose entangled states (PPTES).

Figures

Figures reproduced from arXiv: 2504.18163 by the authors.

Figure 1
Figure 1. The figure illustrates the convex separable states, PPT entangled [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Principal component analysis (PCA) plots of three principal com [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Separable states for different values of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Principal component analysis (PCA) plots of three principal com [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Principal component analysis (PCA) plots of three principal com [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bounding the entanglement of a state from its spectrum

    quant-ph 2026-04 unverdicted novelty 6.0 of 10

    Linear maps and their inverses yield compact spectral criteria that upper-bound negativity and Schmidt number of full-rank bipartite states under any global unitary.

  2. Hybrid Quantum-Classical Learning of Nonlinear Entanglement Witnesses via Continuous-Variable Quantum Neural Networks

    quant-ph 2025-09 reject novelty 4.0 of 10

    A hybrid CV-quantum neural network with a classical readout is shown in simulation to classify two- and three-mode states as separable or entangled with over 99% accuracy, and a universal-approximation theorem is stat...

  3. Scalable Entanglement Detection in Quantum Systems via Fisher Linear Discriminant Analysis

    quant-ph 2025-09 reject novelty 3.0 of 10

    Fisher Linear Discriminant Analysis, trained on Pauli expectation values from simulated two-, three-, and four-qubit states, classifies entangled versus separable states with reported accuracies between 33% and 100% d...

Reference graph

Works this paper leans on

35 extracted references · 34 canonical work pages · cited by 3 Pith papers

  1. [1]

    Can quantum-mechanical description of physical reality be considered complete?,

    B. P. A. Einstein and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,”Phys. Rev., vol. 47, no. 777, 1935

    work page 1935

  2. [2]

    Discussion of probability relations between separated systems,

    E. Schrodinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc., vol. 31, no. 555, 1935

    work page 1935

  3. [3]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information. Cambridge University Press, 2000

    work page 2000

  4. [4]

    Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,

    C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,”Phys. Rev. Lett, vol. 70, p. 1895, 1993

    work page 1993

  5. [5]

    Communication via oneand two- particle operators on einstein-podolsky-rosen states,

    C. H. Bennett and S. J. Wiesner, “Communication via oneand two- particle operators on einstein-podolsky-rosen states,” Phys. Rev. Lett, vol. 69, p. 2881, 1992

    work page 1992

  6. [6]

    Quantum cryptography based on bell’s theorem,

    A. K. Ekert, “Quantum cryptography based on bell’s theorem,”Phys. Rev. Lett, vol. 67, pp. 661–663, Aug 1991. 20

    work page 1991

  7. [7]

    Entanglement detection,

    O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports, vol. 474, no. 1, pp. 1–75, 2009

    work page 2009

  8. [8]

    Classical deterministic complexity of edmonds’ problem and quantum entanglement,

    L. Gurvits, “Classical deterministic complexity of edmonds’ problem and quantum entanglement,”In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC, vol. 03, no. 10-19, 2003

    work page 2003

Show all 35 references
  1. [9]

    Quantum entan- glement,

    M. H. R. Horodecki, P. Horodecki and K. Horodecki, “Quantum entan- glement,” Rev. Mod. Phys., vol. 81, p. 865, 2009

    2009

  2. [10]

    Criterion for k separability in mixed multipartite states,

    B. C. H. A. Gabriel and M. Huber, “Criterion for k separability in mixed multipartite states,”Quantum Inf. Comput., vol. 10, p. 0829, 2010

    2010

  3. [11]

    Separability criterion for density matrices,

    A. Peres, “Separability criterion for density matrices,”Phys. Rev. Lett., vol. 77, p. 1413, 1996

    1996

  4. [12]

    Separability of mixed states: necessaryandsufficientconditions,

    M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessaryandsufficientconditions,” Phys. Lett. A,vol.223, no.2, p. 1, 1996

    1996

  5. [13]

    Separability criterion and inseparable mixed states with positive partial transposition,

    P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A, vol. 232, no. 5, p. 333, 1997

    1997

  6. [14]

    Optimization of entanglement witnesses,

    M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,”Phys. Rev. A, vol. 62, p. 052310, 2000

    2000

  7. [15]

    Detecting some three-qubit mub diagonal entangled states via nonlinear optimal entan- glement witnesses,

    A. H. M.A. Jafarizadeh, M. Mahdian and K. Aghayar, “Detecting some three-qubit mub diagonal entangled states via nonlinear optimal entan- glement witnesses,”Eur. Phys. J. D, vol. 50, p. 107, 2008

    2008

  8. [16]

    Investigating a class of2N 2Nd bound entangled density ma- trices via linear and nonlinear entanglement witnesses constructed by exact convex optimization,

    M. A. Jafarizadeh, Y. Akbari, K. Aghayar, A. Heshmati, and M. Mah- dian, “Investigating a class of2N 2Nd bound entangled density ma- trices via linear and nonlinear entanglement witnesses constructed by exact convex optimization,”Phys. Rev. A, vol. 78, p. 032313, Sep 2008

    2008

  9. [17]

    Optimization of entanglement witnesses,

    M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,”Physical Review A, vol. 62, no. 5, p. 052310, 2000. 21

    2000

  10. [18]

    a survey of quantum learning the- ory.,

    R. G. c. Arunachalam, S. de Wolf, “a survey of quantum learning the- ory.,” SIGACT News, vol. 48, no. 41-67, 2017

    2017

  11. [19]

    Quantum machine learning,

    C. e. a. Ciliberto, “Quantum machine learning,”a classical perspective. Proc. R. Soc. Lond. A, vol. 474, no. 20170551, 2018

    2018

  12. [20]

    Quantum learning, in new directions in cognitive sci- ence: Proceedings of the international symposium, saariselka, finland, p. pylkknen and p. pylkk (editors),

    R. Chrisley, “Quantum learning, in new directions in cognitive sci- ence: Proceedings of the international symposium, saariselka, finland, p. pylkknen and p. pylkk (editors),” no. 77-89, 1995

    1995

  13. [21]

    Variational quantum approximate support vector machine with inference transfer,

    e. a. Siheon Park, “Variational quantum approximate support vector machine with inference transfer,”Scientific Reports., vol. 13, no. 3288, 2023

    2023

  14. [22]

    Neural-network quantum state tomography,

    G. e. a. Torlai, “Neural-network quantum state tomography,”Nat. Phys., vol. 14, no. 447–450, 2018

    2018

  15. [23]

    Reinforcement learning in different phases of quantum con- trol,

    M. Bukov, A. G. R. Day, D. Sels, P. Weinberg, A. Polkovnikov, and P. Mehta, “Reinforcement learning in different phases of quantum con- trol,” Phys. Rev. X, vol. 8, p. 031086, Sep 2018

    2018

  16. [24]

    Quantum topol- ogy identification with deep neural networks and quantum walks,

    Y. Ming, C.-T. Lin, S. D. Bartlett, and W.-W. Zhang, “Quantum topol- ogy identification with deep neural networks and quantum walks,”npj Comput. Mater, vol. 5, 2019

    2019

  17. [25]

    Solving the quantum many-body problem with artificial neural networks,

    G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,”Science,, vol. 355, no. 602–606, 2017

    2017

  18. [26]

    Entanglement detection with artificial neural net- works,

    e. a. Naema Asif, “Entanglement detection with artificial neural net- works,” Scientific Reports., vol. 13, no. 1562, 2023

    2023

  19. [27]

    Quantum entanglement recognition,

    J. Y. Khoo and M. Heyl, “Quantum entanglement recognition,”Phys. Rev. Res., vol. 3, p. 033135, Aug 2021

    2021

  20. [28]

    Mixed state entanglement classification using artificial neural networks,

    C. Harney, M. Paternostro, and S. Pirandola, “Mixed state entanglement classification using artificial neural networks,”New Journal of Physics, vol. 23, p. 063033, jun 2021

    2021

  21. [29]

    Separability-entanglementclassifierviamachinelearning,

    S. Lu, S. Huang, K. Li, J. Li, J. Chen, D. Lu, Z. Ji, Y. Shen, D. Zhou, andB.Zeng, “Separability-entanglementclassifierviamachinelearning,” Phys. Rev. A, vol. 98, p. 012315, Jul 2018. 22

    2018

  22. [30]

    Detecting entanglement with deep quantum neural networks,

    P.-H. Qiu, X.-G. Chen, and Y.-W. Shi, “Detecting entanglement with deep quantum neural networks,”IEEE Access, vol. 7, pp. 94310–94320, 2019

    2019

  23. [31]

    Support vector machine,

    D. A. Pisner and D. M. Schnyer, “Support vector machine,” inMachine learning, pp. 101–121, Elsevier, 2020

    2020

  24. [32]

    Cristianini and J

    N. Cristianini and J. Shawe-Taylor,An introduction to support vector machines and other kernel-based learning methods. Cambridgeuniversity press, 2000

    2000

  25. [33]

    Quantum state discrimination for supervised classification,

    R. Giuntini, H. Freytes, D. K. Park, C. Blank, F. Holik, K. L. Chow, and G. Sergioli, “Quantum state discrimination for supervised classification,” https://arxiv.org/abs/2104.00971, 2021

    2021 arXiv

  26. [34]

    Classification of mixed three-qubit states,

    A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, “Classification of mixed three-qubit states,”Phys. Rev. Lett., vol. 87, no. 4, 2001

    2001

  27. [35]

    Three qubits can be entangled in two inequivalent ways,

    W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,”Phys. Rev. A., vol. 62, no. 062314, 2000. 23

    2000

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.