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REVIEW 4 major objections 6 minor 38 references

Geometric Optimization over Quantum State Spaces: Tight Uncertainty Relations and Resource Certification

T0 review · 4 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper claims that optimal entropic uncertainty relations for arbitrary finite-dimensional quantum measurements can be computed to any desired precision using a support-function outer-approximation of the quantum probability space.

desk verdict A clean new reduction plus a working numerical tool for entropic URs, undercut by an unproven convergence claim and an abstract that promises sections that aren't there. read the letter →

arxiv 2602.00595 v2 pith:6H223OPJ submitted 2026-01-31 quant-ph

Ma-Cheng Yang , Cong-Feng Qiao This is my paper · ORCID
classification quant-ph MSC 81P4581P1590C26 PACS 03.65.Ta03.67.-a
keywords entropicuncertaintyrelationsquantumprobabilityspaceouterapproximationsupportfunctionPOVMsteeringconcaveminimizationresourcecertification
topics The Measurement Problem
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

Entropic uncertainty relations bound the total uncertainty of several measurements by a state-independent number, but finding the optimal (tight) bound is a non-convex minimization generally believed hard. This paper claims that for any finite set of POVMs the optimal bound can be computed to any desired precision: it combines all measurements into a single effective POVM, maps the accessible probability distributions to a low-dimensional convex set, and approximates that set from outside by polytopes. The crucial move is that the boundary of the quantum probability set is available through largest-eigenvalue computations, which serve both to produce a cut that separates an invalid candidate from the true set and to produce a valid physical upper bound. The paper shows on examples that standard analytical and majorization-based bounds are loose for generic measurements, and that the computed tight bounds materially improve steering detection thresholds.

What carries the argument

The load-bearing identity is sigma_P(u) = lambda_max(sum_i u_i E_i), which expresses the support function of the accessible probability set as an eigenvalue problem. The algorithm reduces the optimization to a low-dimensional Z-space via the singular-value decomposition of the affine Bloch map, then alternates between minimizing the entropy at vertices of an outer polytope (lower bound) and evaluating the entropy at the ground state of the gradient Hamiltonian (upper bound), adding a cutting plane from the support-function oracle until the gap closes.

What would settle it

For a single qubit and a fixed POVM, compute the algorithm's lower bound with the alpha-generalized entropy (alpha = 2) and compare it to the exact minimum obtained by scanning all pure states over the Bloch sphere; if the algorithm's bound ever exceeds the exact minimum, the concavity premise fails. For the convergence claim, a single run on a small POVM where the gap stagnates above the chosen tolerance would falsify the universal 'preassigned precision' statement.

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Extended reading notes

Core claim

The central claim is that the map from states to outcome probabilities for any POVM spans a convex set whose support function is the largest eigenvalue of an effective observable, and that this oracle turns concave minimization over quantum states into an outer-approximation scheme. Iterating vertex checks and spectral cuts produces certified lower and upper bounds whose gap can be driven below any prescribed tolerance. The paper demonstrates the method on random POVMs in dimension 100 and on qutrit two- and three-measurement settings, where it outperforms known analytical bounds.

Load-bearing premise

The certified lower bound assumes the entropy is concave over the outer polytope; that holds for Shannon and Tsallis entropies but not for the alpha-generalized entropy with alpha > 1, and the paper's additional claim that the cutting planes converge to the exact boundary in Hausdorff metric is asserted without proof.

Editorial extensions

If this is right

  • For any finite-dimensional POVM and any concave entropy in the listed family, the algorithm returns certified lower and upper bounds on the optimal uncertainty; the gap can be driven below any preassigned tolerance.
  • The known analytical and majorization-based entropic bounds are substantially non-tight for generic measurement settings, so previously reported EURs understate the actual uncertainty limit.
  • Tighter EURs directly lower the visibility threshold for steering detection: the qutrit example shows steerability certified against more white noise than the majorization-based threshold allows.
  • The method handles multiple measurements (N > 2) and asymmetric, non-ideal measurement settings without requiring new analytical formulas.
  • The same optimization framework is claimed to apply to other convex resources, such as certifying maximal athermality from restricted measurements.

Reading between the lines

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

  • Because the computational cost is governed by the rank of the affine probability map, not the Hilbert-space dimension, the method may scale to high-dimensional systems with few outcomes; the paper only demonstrates this on one random example, so this remains a promising inference.
  • The support-function oracle could plausibly be adapted to certify boundaries of other quantum convex sets, such as separable states, by replacing the largest-eigenvalue step with a semidefinite relaxation; this extension is not in the paper.
  • The concavity restriction marks a real boundary of applicability: for the alpha-generalized entropy with alpha > 1 the vertex lower bound is not guaranteed, so a different lower-bounding mechanism would be needed to extend the method to those functionals.
  • If the computed bounds are as tight as the examples indicate, they could be used to benchmark or replace analytical EURs in quantum-cryptography security proofs, where every bit of tightness matters.
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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

4 major / 6 minor

Summary. The manuscript introduces a support-function-based outer-approximation method for computing h(E) = inf_ρ H(p_E(ρ)), the minimal entropy of an effective POVM obtained by concatenating several measurements, and applies it to entropic uncertainty relations (EURs) for Shannon, Tsallis, and Rényi entropies. The main steps are: an affine SVD reduction of the quantum probability space P(E) to a lower-dimensional convex set Z; the spectral identity σ_P(u) = λ_max(Σ_i u_i E_i) for the support function; and an iterative algorithm that maintains an outer polytope P_k ⊇ Z, obtains a lower bound h− by minimizing H over the vertices of P_k, an upper bound h+ from the ground state of the gradient observable Ω(∇H), and refines P_k with a support-function cutting plane. The authors claim that Algorithm 1 yields tight EUR bounds 'with a preassigned numerical precision', benchmark it against Maassen–Uffink, Coles–Piani, and majorization bounds for qutrit measurements, and use the tighter bounds to improve steering detection thresholds. The abstract additionally promises recovery of the variance-based uncertainty relations of [32] and certification of maximal athermality resources; neither of these appears in the body.

Significance. The reformulation h(E) = min_{z∈Z} H(s+Qz) and the oracle σ_P(u) = λ_max(Ω(u)) are clean and correct, and the sandwich structure—valid lower and upper bounds at every iteration—is a genuine strength: the method is parameter-free and the code is promised in the linked repository [38]. The comparisons in Figs. 2–4 provide useful numerical evidence that standard analytical and majorization bounds are not tight for the examined asymmetric qutrit settings. If the convergence and differentiability gaps identified below are resolved, this would be a broadly applicable computational tool for quantum resource theories. As written, however, the central certification claim is not established, the treatment of Rényi α>1 rests on a false concavity premise, and the abstract announces content missing from the body. The significance of the paper is therefore conditional on substantial revision.

major comments (4)
  1. [Outer approximation algorithm, 'Convergence and discussion', Algorithm 1] The central claim that Algorithm 1 yields tight EUR bounds 'with a preassigned numerical precision' is not supported. The convergence paragraph argues that as the set of included directions u grows, the outer polytope converges to P in the Hausdorff metric. That is true for exhaustive direction sets, but the algorithm adds only the adaptive normal g = −∇H(p_poly) (lines 8 and 16–17; Eq. (27)); no proof is given that these adaptively generated normals expose all relevant faces of Z, nor that the upper-bound point p_real from the ground state of Ω(g) (line 10) satisfies H(p_real) → h(E). Even if P_k → Z in Hausdorff distance, h+ need not approach h(E), so the stopping criterion Gap ≤ ε is not guaranteed to be reached. The cut can also fail to remove the current vertex when that vertex attains the supporting-plane value min_P⟨g,·⟩ while remaining infeasible; no rule for this case is given,
  2. [Eqs. (4), (9), (25); Introduction] Eq. (25) justifies the vertex lower bound by asserting 'the entropy function is concave', and the Introduction states that the uncertainty functional is concave in ρ. This is false for Rényi entropy with α>1, which is explicitly included in Eqs. (4) and (9). Rényi entropy is quasi-concave on the probability simplex for all α>0 (for α>1, Σ_i p_i^α is convex, so the superlevel sets are convex). The vertex lower bound remains valid under quasi-concavity, and the gradient-based cut inherits the needed first-order condition, so the results are repairable. However, the stated premise is incorrect as written, the derivation of Eq. (25) must be corrected, and the claim that Eq. (4) satisfies all of Deutsch's requirements should be qualified.
  3. [Abstract] The abstract announces two results that do not appear in the body: recovery of the exact variance-based uncertainty relations of [32] and determination of the maximal athermality resource from restricted measurements. The manuscript contains neither a variance-based UR computation nor any athermality analysis; the applications actually reported are Shannon/Tsallis EURs and steering thresholds (Figs. 2–4). The abstract and conclusion must be reconciled with the actual content, either by adding the missing sections or by removing these claims.
  4. [Figs. 2–4; 'Application and discussion'] The curves labeled 'qOptimal' are presented as the true quantum-optimal bounds, but the paper does not report the tolerance ε, the number of iterations, the vertex counts, or the converged gaps for the parameter scans behind these figures. Given that the certified-precision statement is unsupported (see first major comment), the numerical 'optimality' labels should be backed by the actual termination data. I also recommend an independent sanity check in a case where the exact bound is known (e.g., mutually unbiased bases, or the Schwonnek variance benchmark mentioned in the abstract).
minor comments (6)
  1. [Eqs. (8), (10); Appendix A] The main text says 'A proof of Eq. (8) is provided in Section A' and cross-references are informal; use a consistent appendix citation and place it at Eq. (10) as well as Eq. (8).
  2. [References [15], [21]] Reference [15] misspells 'Sánchez-Ruiz' as 'Sánches-Ruiz'; reference [21] lacks standard volume/page information and has an irregular DOI listing.
  3. [Introduction, Eq. (4)] The statement that Eq. (4) 'satisfies all the requirements for an uncertainty measure proposed by Deutsch' needs qualification: for Shannon and Tsallis entropies concavity holds, but for Rényi α>1 only quasi-concavity is available (see major comment 2).
  4. [Fig. 1] The caption should state the tolerance ε used and clarify whether the red square marks the vertex minimizing H over the current outer polytope or the final returned point.
  5. [Steering inequality (p. 5)] The inequality with p^{(k)}_{ij} introduces assemblage notation without definition; readers need either a brief restatement or a precise pointer to the conventions of [13].
  6. [Algorithm 1; implementation details] The paper should specify the numerical implementation: vertex-enumeration library and method, floating-point tolerance for vertex computation, and the finite-gradient or subgradient rule used when the minimizing vertex has zero probability components (relevant also to major comment 1).

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the algorithm is self-contained; Rényi-concavity and convergence-gap are non-circular correctness concerns.

full rationale

The paper's derivation chain is self-contained. The central quantity h(E)=inf_ρ H(p_E(ρ)) is reduced to h(E)=inf_{z∈Z} H(s+Qz), and the feasible set Z is described exactly through support-function half-spaces (Eqs. (18)-(23)); no fitted parameter or previously assumed EUR bound is used to define the target. Algorithm 1 alternates a vertex lower bound with a spectral upper bound, and each cutting plane is exactly the support-function inequality in the direction of the negative gradient, so each outer polytope is a genuine outer approximation of Z. The final bounds are then compared against independent external benchmarks (Maassen-Uffink, Coles-Piani, Rudnicki-Puchała-Życzkowski, majorization bounds, and the Schwonnek variance result), so the claim that analytical/majorization bounds are loose is supported by an independently computed boundary, not by an input-to-output circularity. The only self-citation, Ref. [8] (Li-Qiao), is background on majorization and is not load-bearing for the algorithm or any central claim. Two mathematical gaps exist but are not circularity: (i) Eq. (25) asserts 'the entropy function is concave' for all listed entropies, which is false for Rényi entropy with α>1, so the vertex lower-bound certificate is unproved in that case; and (ii) the 'Convergence and discussion' paragraph asserts that as included directions grow, the outer polytope converges to P in Hausdorff metric and 'thereby closing the gap,' but the adaptive gradient-cut directions are not proved to expose all relevant faces of Z, so the 'preassigned numerical precision' guarantee is an unproved assertion. These are correctness or proof gaps, not reductions of the result to its own inputs, and therefore do not raise the circularity score.

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

The core argument depends on support-function duality, vertex optimality for concave minimization, and the concavity of the chosen entropy. The first and last are standard for Shannon/Tsallis, but the Rényi concavity premise is false for α>1, and the Hausdorff convergence premise is unproved. There are no fitted free parameters and no invented entities.

assumptions (5)
  • standard math Closed convex set equals intersection of supporting half-spaces (support-function dual representation).
    Used in Eqs. (22)-(23) to derive the dual description of Z.
  • standard math For a concave objective, the minimum over a polytope is attained at a vertex.
    Used in Eq. (25) to justify h−; the paper also applies this to Rényi α>1, where the premise fails.
  • ad hoc to paper Rényi entropy is concave for all α>1.
    Assumed when Eqs. (9)-(11) and the vertex lower bound are applied to Rényi entropy; false on the simplex for α>1 (e.g., α=2, p=(0.9,0.1,0)).
  • ad hoc to paper The cutting-plane sequence P_k converges to Z in the Hausdorff metric.
    Stated in 'Convergence and discussion' without proof; required for guaranteed termination with a certified gap.
  • standard math Shannon and Tsallis entropies are concave on the probability simplex.
    Background fact needed for the main numerical examples and the steering application.

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Cite this review

Pith. "Pith review of Geometric Optimization over Quantum State Spaces: Tight Uncertainty Relations and Resource Certification." pith.science (2026). https://pith.science/paper/6H223OPJ

@misc{pith2026260200595,
  author       = {Pith},
  title        = {Pith review of: Geometric Optimization over Quantum State Spaces: Tight Uncertainty Relations and Resource Certification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6H223OPJ}},
  note         = {Machine review of arXiv:2602.00595}
}
abstract

Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable non-convex optimization problem. We introduce a generic support-function-based outer-approximation framework for solving concave-minimization (or convex-maximization) problems over the quantum state space. By mapping the problem onto a reduced $\mathcal{Z}$-space, we characterize the exact quantum boundary through supporting half-spaces derived from the largest eigenvalues of effective observables. This yields an effective method that produces tight bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. As an initial application, we recover the exact variance-based uncertainty relations of [PRL \textbf{119}, 170404 (2017)] and efficiently compute optimal entropic uncertainty relations (EURs). Our results reveal that standard analytical and majorization-based EUR bounds are fundamentally loose for generic measurements, and we show that the resulting exact bounds directly enhance quantum steering detection under asymmetric settings. We further apply the framework to determine the maximal athermality resource certifiable from a restricted measurement scenario. Our method thus provides a universal computational tool for exploring the boundaries of quantum state space and the limits of quantum resources.

Figures

Figures reproduced from arXiv: 2602.00595 by the authors.

Figure 1
Figure 1. FIG. 1. Visualization of the convergence of the outer [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of entropic uncertainty bounds for the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of entropic uncertainty bounds for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Steering detection thresholds for isotropic states us [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Works this paper leans on

38 extracted references · 1 canonical work pages

  1. [32]

    Schwonnek, L

    R. Schwonnek, L. Dammeier, and R. F. Werner, State- independent uncertainty relations and entanglement de- tection in noisy systems, Phys. Rev. Lett.119, 170404 (2017)

    2017

  2. [38]

    Geo- metric optimization for tight entropic uncertainty re- lations

    M.-C. Yang and C.-F. Qiao, Source code for “Geo- metric optimization for tight entropic uncertainty re- lations”,https://github.com/yangmacheng/Optimal_ bound_entropy_UR(2026)

    2026

  3. [1]

    Heisenberg, ¨Uber den anschaulichen inhalt der quan- tentheoretischen kinematik und mechanik, Zeitschrift f¨ ur Physik33, 879 (1927)

    W. Heisenberg, ¨Uber den anschaulichen inhalt der quan- tentheoretischen kinematik und mechanik, Zeitschrift f¨ ur Physik33, 879 (1927)

    1927

  4. [2]

    E. H. Kennard, Zur quantenmechanik einfacher bewe- gungstypen, Zeitschrift f¨ ur Physik44, 326 (1927)

    1927

  5. [3]

    Weyl,Gruppentheorie und Quantenmechanik (Leipzig: Hirzel

    H. Weyl,Gruppentheorie und Quantenmechanik (Leipzig: Hirzel. (Transl. by Robertson, H.P. (1931) The Theory of Groups and Quantum Mechanics), 1928)

    1931

  6. [4]

    H. P. Robertson, The uncertainty principle, Phys. Rev. 34, 163 (1929)

    1929

  7. [5]

    Deutsch, Uncertainty in quantum measurements, Phys

    D. Deutsch, Uncertainty in quantum measurements, Phys. Rev. Lett.50, 631 (1983)

    1983

  8. [6]

    M. H. Partovi, Majorization formulation of uncertainty in quantum mechanics, Phys. Rev. A84, 052117 (2011)

    2011

Show all 38 references
  1. [7]

    Friedland, V

    S. Friedland, V. Gheorghiu, and G. Gour, Universal un- certainty relations, Phys. Rev. Lett.111, 230401 (2013)

    2013

  2. [8]

    Li and C.-F

    J.-L. Li and C.-F. Qiao, The optimal uncertainty relation, Ann Phys531, 1900143 (2019)

    2019

  3. [9]

    P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Entropic uncertainty relations and their applications, Rev. Mod. Phys.89, 10.1103/RevModPhys.89.015002 (2017)

    2017 doi

  4. [10]

    Giovannetti, Separability conditions from entropic un- certainty relations, Phys

    V. Giovannetti, Separability conditions from entropic un- certainty relations, Phys. Rev. A70, 012102 (2004)

    2004

  5. [11]

    G¨ uhne and M

    O. G¨ uhne and M. Lewenstein, Entropic uncertainty rela- tions and entanglement, Phys. Rev. A70, 022316 (2004)

    2004

  6. [12]

    Schneeloch, C

    J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti, and J. C. Howell, Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations, Phys. Rev. A87, 062103 (2013)

    2013

  7. [13]

    A. C. S. Costa, R. Uola, and O. G¨ uhne, Steering criteria from general entropic uncertainty relations, Phys. Rev. A98, 050104 (2018)

    2018

  8. [14]

    Maassen and J

    H. Maassen and J. B. Uffink, Generalized entropic uncer- tainty relations, Phys. Rev. Lett.60, 1103 (1988)

    1988

  9. [15]

    A. J. M. Garrett and S. F. Gull, Numerical study of the information uncertainty principle, Phys. Lett. A151, 453–458 (1990)

    1990

  10. [16]

    S´ anches-Ruiz, Optimal entropic uncertainty relation in two-dimensional hilbert space, Phys

    J. S´ anches-Ruiz, Optimal entropic uncertainty relation in two-dimensional hilbert space, Phys. Lett. A244, 189 (1998)

    1998

  11. [17]

    Ghirardi, L

    G. Ghirardi, L. Marinatto, and R. Romano, An optimal entropic uncertainty relation in a two-dimensional hilbert space, Phys. Lett. A317, 32–36 (2003)

    2003

  12. [18]

    J. I. de Vicente and J. S´ anchez-Ruiz, Improved bounds on entropic uncertainty relations, Phys. Rev. A77, 042110 (2008)

    2008

  13. [19]

    Pucha la, L

    Z. Pucha la, L. Rudnicki, and K. ˙Zyczkowski, Majoriza- tion entropic uncertainty relations, J. Phys. A: Math. Theor.46, 272002 (2013)

    2013

  14. [20]

    P. J. Coles and M. Piani, Improved entropic uncertainty relations and information exclusion relations, Phys. Rev. A89, 022112 (2014)

    2014

  15. [21]

    Y. Xiao, N. Jing, S.-M. Fei, and X. Li-Jost, Im- proved uncertainty relation in the presence of quantum memory, J. Phys. A: Math. Theor.49, 10.1088/1751- 8113/49/49/49lt01 (2016)

    2016 doi

  16. [22]

    S´ anchez, Entropic uncertainty and certainty relations for complementary observables, Phys

    J. S´ anchez, Entropic uncertainty and certainty relations for complementary observables, Phys. Lett. A173, 233 (1993)

    1993

  17. [23]

    S´ anchez-Ruiz, Improved bounds in the entropic uncer- tainty and certainty relations for complementary observ- ables, Phys

    J. S´ anchez-Ruiz, Improved bounds in the entropic uncer- tainty and certainty relations for complementary observ- ables, Phys. Lett. A201, 125 (1995)

    1995

  18. [24]

    S. Wu, S. Yu, and K. Mølmer, Entropic uncertainty re- lation for mutually unbiased bases, Phys. Rev. A79, 022104 (2009)

    2009

  19. [25]

    Rudnicki, Z

    L. Rudnicki, Z. Pucha la, and K. Zyczkowski, Strong ma- jorization entropic uncertainty relations, Phys. Rev. A 89, 052115 (2014)

    2014

  20. [26]

    Liu, L.-Z

    S. Liu, L.-Z. Mu, and H. Fan, Entropic uncertainty re- lations for multiple measurements, Phys. Rev. A91, 042133 (2015)

    2015

  21. [27]

    Zhang, Y

    J. Zhang, Y. Zhang, and C.-S. Yu, Entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory, Sci. Rep.5, 10.1038/srep11701 (2015)

    2015 doi

  22. [28]

    Y. Xiao, N. Jing, S.-M. Fei, T. Li, X. Li-Jost, T. Ma, and Z.-X. Wang, Strong entropic uncertainty relations for multiple measurements, Phys. Rev. A93, 042125 (2016)

    2016

  23. [29]

    Riccardi, C

    A. Riccardi, C. Macchiavello, and L. Maccone, Tight en- tropic uncertainty relations for systems with dimension three to five, Phys. Rev. A95, 032109 (2017)

    2017

  24. [30]

    Wehner and A

    S. Wehner and A. Winter, Entropic uncertainty relations- a survey, New J. Phys.12, 025009 (2010)

    2010

  25. [31]

    Horst and H

    R. Horst and H. Tuy,Global Optimization: Deterministic Approaches(Springer-Verlag, Berlin, 1996)

    1996

  26. [33]

    M. A. Nielsen and I. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cam- bridge University Press, Cambridge, 2010)

    2010

  27. [34]

    Tsallis, Possible generalization of boltzmann-gibbs statistics, J

    C. Tsallis, Possible generalization of boltzmann-gibbs statistics, J. Stat. Phys.52, 479 (1988)

    1988

  28. [35]

    R. T. Rockafellar,Convex Analysis(Princeton University Press, 2015)

    2015

  29. [36]

    R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G¨ uhne, Quantum steering, Rev. Mod. Phys.92, 015001 (2020). 8

    2020

  30. [37]

    Horodecki and P

    M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A59, 4206 (1999)

    1999

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