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REVIEW 3 major objections 4 minor 71 references

Witness based nonlinear detection of quantum entanglement

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Reordering the subsystems of multiple copies of a state lets existing entanglement witnesses detect entanglement that each witness alone—and the standard aligned multi-copy construction—misses.

desk verdict Crossed multi-copy witness pairing is a genuine new detection idea with explicit examples; proof gaps are real but likely repairable. read the letter →

arxiv 2502.02868 v1 pith:OKZB5LNF submitted 2025-02-05 quant-ph

Yiding Wang , Tinggui Zhang , Xiaofen Huang , Shao-Ming Fei This is my paper · ORCID
classification quant-ph MSC 81P42 PACS 04.70.Dy03.65.Ud04.62.+v
keywords quantumentanglementwitnessnonlineardetectionmulti-copyWernerstateGHZWconcentration
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 proposes a nonlinear entanglement detection strategy: take existing entanglement witnesses and apply them to multiple copies of a state with the local subsystems crossed between copies, rather than aligned. The central claim is that this reordering can produce negative expectation values for entangled states on which every witness alone, and the aligned two-copy construction, gives nonnegative values. That is demonstrated for the Bell state $|\psi^+\rangle$, for Werner states with mixing parameter $w<0.206$ using three copies, for a Werner-like family using a witness combined with a positive semidefinite operator, and for three-qubit GHZ and W states using two-copy bipartite witnesses. The same crossed layout yields an entanglement concentration protocol. A sympathetic reader would care because the strategy extracts new detection power from already-known witnesses without constructing new ones.

What carries the argument

The central object is the crossed-copy witness tensor product: for two copies of a bipartite state, instead of the aligned product $W_{AA'}\otimes V_{BB'}$, one measures $W_{AB'}\otimes V_{BA'}$, and for $k$ copies one uses orderings such as $W_{1,A_1B_2}\otimes W_{2,A_2B_3}\otimes W_{3,B_1A_3}$. The factor order is chosen so that each witness's negative eigenspace is aligned with correlations of the entangled state across copies. The expectation value is automatically nonnegative on separable states because each factor is a witness or a positive semidefinite operator, so negativity of the expectation value is a valid entanglement certificate. The accompanying proofs that the constructed operators are witnesses use positive-partial-transpose type inequalities plus the negative-eigenvalue criterion for witnesses.

What would settle it

Compute the minimum of $\mathrm{Tr}(W\rho)$ and $\mathrm{Tr}(W_1\rho)$ over all separable two-qubit states, e.g. by minimizing over product Bloch vectors; if either minimum is negative, the operator is not a witness and the corresponding example collapses, whereas if both are nonnegative the crossed-copy negative expectations stand.

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Extended reading notes

Core claim

The paper's core discovery is that the tensor product of two entanglement witnesses is not invariant under swapping which subsystems are paired across copies, and this asymmetry can be used for detection. In Observation 1, for $\rho=|\psi^+\rangle\langle\psi^+|$ and witnesses $W=I-X\otimes X+Z\otimes Z$ and $V=2|\phi^+\rangle\langle\phi^+|_{\tau_2}$, the single-copy expectations are $\mathrm{Tr}(W\rho)=\mathrm{Tr}(V\rho)=1$, yet the crossed product $W_{AB'}\otimes V_{BA'}$ has expectation $-1/2$ on $\rho^{\otimes 2}$. Observation 2 extends the phenomenon to three copies: for the Werner family $\rho_w=w I/4+(1-w)|\psi^+\rangle\langle\psi^+|$ with $w<0.206$, every two-copy crossed pairing of the three witnesses $W_1,W_2,W_3$ is nonnegative, but the three-copy ordering $W_{1,A_1B_2}\otimes W_{2,A_2B_3}\otimes W_{3,B_1A_3}$ has negative expectation. The paper also shows that one factor in the product may be replaced by any positive semidefinite operator and that the construction detects tripartite entanglement using bipartite witnesses.

Load-bearing premise

Everything rests on each constructed operator being a true entanglement witness, never negative on a separable state, and for two of the operators the paper's written proof of that property contains an unjustified step, so as written the witness status of those operators is the load-bearing premise.

Editorial extensions

If this is right

  • Existing witnesses can be reused on few copies of an unknown state to detect entanglement that linear detection misses, as in the Bell state example where both single-copy expectations are $+1$ but the crossed two-copy expectation is $-1/2$.
  • For Werner states, more copies can activate detection: three-copy crossed ordering detects entanglement for $w<0.206$ even though every two-copy crossed pairing of the same witnesses fails.
  • A witness combined with a positive semidefinite operator under the crossed ordering detects entanglement for a family of Werner-like states, and tuning the operator extends the detection region toward the PPT boundary, approaching $a>1/\sqrt{3}$.
  • Two-qubit witnesses applied to two copies of three-qubit states detect genuine tripartite entanglement of noisy W states for $c<0.406$, a wider range than the standard tripartite W-state witness's $c<0.38$ threshold.
  • The crossed-copy layout also realizes entanglement concentration: a suitable measurement on the crossed subsystem $BA'$ projects two copies of a full-Schmidt-rank pure state onto the same state, or onto the maximally entangled state, with positive probability.

Reading between the lines

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

  • Our inference: the power of the crossed ordering likely comes from aligning the negative eigenvectors of different witnesses against different tensor factors of the state; a general criterion for when reordering enlarges the detected set could be expressed in terms of the support projections of the witnesses' negative eigenspaces.
  • Our inference: because nonnegativity on separable states is automatic for any factor permutation, the detection gain is purely about entangled states; one could test whether crossed orderings of a single witness $W\otimes W$, rather than two different witnesses, ever detect states that $W$ alone misses.
  • Our inference: the gap in the written proof that $W$ and $W_1$ are witnesses can be settled by a direct numerical minimization over product-state Bloch vectors; if the minima are nonnegative, Observation 1 and the Werner three-copy example survive independently of the flawed inequality chain.
  • Our inference: a natural experimental extension is to implement the crossed two-copy measurement on photonic or trapped-ion Bell states and record the negative expectation value, which would give a few-copy entanglement certificate without full tomography.
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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 / 4 minor

Summary. The paper proposes a nonlinear entanglement detection strategy in which, given two copies of a bipartite state, one measures a witness W on AB' and another witness V on BA' (a 'crossed' ordering), rather than the same-copy pairing used in [47]. It claims (Observation 1) that there exist witnesses W, V and a state ρ such that Tr(Wρ) ≥ 0 and Tr(Vρ) ≥ 0, but Tr((W_AB' ⊗ V_BA')ρ^⊗2) < 0. It further claims (Observation 2) that when the two-copy crossed strategy fails, a three-copy crossed ordering can succeed, and gives a Werner-state example with a negativity region w < 0.206. Observations 3 and 4 extend the idea to combining witnesses with positive semidefinite operators and to detecting tripartite entanglement with bipartite witnesses. An entanglement concentration protocol is also sketched.

Significance. If the examples are correct, the paper gives a simple and explicit demonstration that reordering tensor factors of existing witnesses can increase detection power, complementing the trace-polynomial approach of [47]. The two- and three-copy trace calculations in Examples 1–3 are explicit and largely checkable, and the central existence claims are credible. However, the written proof that the operator W in Example 1 is an entanglement witness contains an invalid inference, and the formula in Appendix E is numerically wrong. Because these points are load-bearing for the stated observations, the manuscript requires revision before its claims are fully supported.

major comments (3)
  1. [Section II, Example 1, Eq. (1)] The proof that W = I − X⊗X + Z⊗Z is an entanglement witness is invalid as written. From Tr(Wρ) < 0 the text obtains |ρ14| + |ρ23| > ρ11 + ρ44 ≥ 2√(ρ11ρ44), and then concludes |ρ23|^2 > ρ11ρ44. This conclusion does not follow: the displayed inequality can hold with |ρ14| large and |ρ23| = 0, in which case |ρ23|^2 = 0. The subsequent statement that W is a witness because it has a negative eigenvalue is also not a valid witness criterion. Since Observation 1 and Example 2 rely on W being a genuine witness, this gap must be closed. The claim is repairable: for product states, ⟨a|⊗⟨b|W|a⟩⊗|b⟩ = 1 − x_a x_b + z_a z_b ≥ 0 by the Bloch-vector bound |x_a x_b| + |z_a z_b| ≤ 1, so a correct proof can be supplied.
  2. [Appendix E, Example 5] The displayed expression for Tr((W4,AB' ⊗ W3,BC' ⊗ W3,CA')ρ_c^⊗2), namely 2(c/8)^2 + 12(c/8)(8−5c)/24 + 4((1−c)/3)^2, is positive for every c ∈ [0,1], so it cannot imply the stated negativity for c < 0.406. A direct six-qubit trace calculation gives (50c − 25c^2 − 16)/36, which is negative for c < 0.4. The numerical threshold should be corrected, and the formula in Appendix E should be replaced, although the qualitative claim that the crossed ordering detects ρ_c for a range of c remains valid.
  3. [Section II, Example 3] The assertion that Tr(Wi,A1B2 ⊗ Wj,B1A2 ρ_w^⊗2) ≥ 0 for all i,j ∈ {1,2,3} is stated without proof or calculation. This is the 'two-copy failure' half of Observation 2 and is essential for the claimed hierarchy two-copy failure / three-copy success. Please provide the verification, or at least summarize the computation in an appendix.
minor comments (4)
  1. [Throughout] There are numerous typos, including 'quanutm' in the abstract, 'whi ch' in the introduction, 'sperable' in Figure 4, 'Muliticoy' in reference [46], 'Wotters' in reference [2], and 'Lioyd' in reference [7].
  2. [Appendix C] The opening sentence writes W2,B1A3 where the main text and the subsequent calculation use W3,B1A3; please fix this notation.
  3. [Observation 4 and Example 5] Observation 4 is phrased with witnesses Wi for i=1,2,3, but Example 5 uses W3 and W4. Please clarify the indexing so that the statement and the example match.
  4. [Section II, multipartite discussion] The definition of W3 in the multipartite discussion omits the factor 2 that appears in the earlier definition (W3 = 2|ψ+⟩⟨ψ+|τ2). Please make the conventions consistent.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the detection claims are explicit analytic witness traces, and all self-citations are background or comparison only.

full rationale

The central observations are supported by direct finite-dimensional operator calculations rather than by fitted parameters or by conclusions imported from the authors' prior work. For example, Example 1 computes Tr(Wρ)=Tr(Vρ)=1 and Tr((W_AB'⊗V_BA')ρ^⊗2)=−1/2 for ρ=|ψ+><ψ+|, and Example 3 gives a closed expression for the three-copy expectation from which the w<0.206 region is read off. Example 4 derives Tr(W_bρ_a^⊗2)=[(1−6b)a^2+2b+1]/(16b); the parameter b is a free parameter of a positive-semidefinite operator family and is optimized in the limit b→∞, so it is not fitted to the target state's entanglement. The self-citations ([28], [51], [66]) appear only in background remarks and in comparisons to existing criteria, and the conclusions do not rest on any unverified uniqueness theorem or on a load-bearing self-citation. The witness-validity proof for W in Example 1 contains an invalid inference, from |ρ14|+|ρ23|>2√(ρ11ρ44) to |ρ23|^2>ρ11ρ44, but this is a correctness gap rather than a circular reduction: the operator's claimed witness status is checked against separable states separately, not defined by the target state's negativity. Accordingly, the paper's derivation chain is not circular, and the score is low.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard definitions of separability and witnesses, the PPT criterion for qubit systems, and explicit finite-dimensional trace calculations. No data are fitted and no new physical entities are introduced. The only hand-tuned quantity is the PSD-operator parameter b in Example 4, which is an optimization parameter rather than a fitted prediction.

free parameters (1)
  • b = b >= 1, taken to infinity in the threshold limit
    Parameter in the positive semidefinite operator P_b in Example 4; chosen by hand to improve the detection threshold from sqrt(3/5) toward sqrt(1/3). It is not fitted to data.
assumptions (4)
  • domain assumption An entanglement witness has nonnegative expectation on all separable states.
    Used to argue that the crossed multi-copy operator remains nonnegative on separable states; invoked before Observation 1 in Section II.
  • standard math For 2x2 systems, the positive partial transpose criterion is necessary and sufficient for separability.
    Used in Examples 1 and 3 to argue that the constructed operators are witnesses.
  • domain assumption When ρ is separable, the k-copy state ρ^{⊗k} factorizes with respect to the crossed pairing of subsystems.
    Underlies the claim that the crossed witness expectation is nonnegative for all separable states; stated informally in Section II and used throughout.
  • domain assumption The bipartite state used in the concentration protocol has full Schmidt rank so that the matrix Ψ is invertible.
    Needed in Appendix F for the definition of the measurement vectors involving (Ψ*)^{-1}.

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Cite this review

Pith. "Pith review of Witness based nonlinear detection of quantum entanglement." pith.science (2026). https://pith.science/paper/OKZB5LNF

@misc{pith2026250202868,
  author       = {Pith},
  title        = {Pith review of: Witness based nonlinear detection of quantum entanglement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKZB5LNF}},
  note         = {Machine review of arXiv:2502.02868}
}
read the original abstract

We present a nonlinear entanglement detection strategy which detects entanglement that the linear detection strategy fails. We show that when the nonlinear entanglement detection strategy fails to detect the entanglement of an entangled state with two copies, it may succeed with three or more copies. Based on our strategy, a witness combined with a suitable quanutm mechanical observable may detect the entanglement that can not be detected by the witness alone. Moreover, our strategy can also be applied to detect multipartite entanglement by using the witnesses for bipartite systems, as well as to entanglement concentrations.

Figures

Figures reproduced from arXiv: 2502.02868 by the authors.

Figure 1
Figure 1. FIG. 1: The expected value of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Tr( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The left (right) figure shows strategy of measuring [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5: The relations between nonlinear detection and linea [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Works this paper leans on

71 extracted references · 64 canonical work pages

  1. [47]

    ⊗Wn such that Tr( Wρ⊗ k ent)< 0 for some entangled states ρent, while Tr( Wρ⊗ k sep) ≥ 0 for all separable states, where k depends on the local dimension and n

    the authors introduced the nonlinear entanglement witness by using tensor product of linear counterparts, W =W1 ⊗W2 ⊗... ⊗Wn such that Tr( Wρ⊗ k ent)< 0 for some entangled states ρent, while Tr( Wρ⊗ k sep) ≥ 0 for all separable states, where k depends on the local dimension and n. It is shown that there are two bipartite witnesses W and V and a bipar- tit...

    work page

  2. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys 81, 865 (2009)

    work page 2009

  3. [2]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres and W. K. Wotters, Teleporting an unknown quan- tum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993)

    work page 1993

  4. [3]

    206. FIG. 3: The left (right) figure shows strategy of measuring witnesses on three (four) copies of state ρ. Observation 3. For some entangled states ρ and wit- nesses W such that Tr (Wρ ) ≥ 0, there exists pos- itive semidefinite operators P such that Tr ((PAB′ ⊗ WBA′ )ρ⊗ 2)< 0. Example 4 . Let us consider the another Werner state, ρa =a|ψ − ⟩⟨ψ − | + 1 −...

    work page

  5. [4]

    Gigena and R

    N. Gigena and R. Rossignoli, Bipartite entanglement in fermion systems, Phys. Rev. A 95, 062320 (2017)

    work page 2017

  6. [5]

    Ekert and R

    A. Ekert and R. Jozsa, Quantum algorithms: Entangle- ment enhanced information processing, Philos. Trans. R. Soc. A 356, 1769 (1998)

    work page 1998

  7. [6]

    Cleve and H

    R. Cleve and H. Buhrman, Substituting quantum en- tanglement for communication, Phys. Rev. A 56, 1201 (1997)

    work page 1997

  8. [7]

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

    work page 2000

Show all 71 references
  1. [8]

    Lioyd, Universal quantum simulators, Science 273, 1073 (1996)

    S. Lioyd, Universal quantum simulators, Science 273, 1073 (1996)

    1996

  2. [9]

    Datta, S

    A. Datta, S. T. Flammia and C. M. Caves, Entanglement and the power of one qubit, Phys. Rev. A 72, 042316 (2005)

    2005

  3. [10]

    Gisin, G

    N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Quan- tum cryptography, Rev. Mod. Phys. 74, 145 (2002)

    2002

  4. [11]

    Masanes, Universally composable privacy amplifica tion from causality constraints, Phys

    L. Masanes, Universally composable privacy amplifica tion from causality constraints, Phys. Rev. Lett. 102, 140501 (2009)

    2009

  5. [12]

    A. K. Ekert, Quantum cryptography based on Bell’s the- orem, Phys. Rev. Lett. 67, 661 (1991)

    1991

  6. [13]

    T´ oth and O

    G. T´ oth and O. G¨ uhne, Detecting genuine multipartite entanglement with two local measurements, Phys. Rev. Lett. 94, 060501 (2005)

    2005

  7. [14]

    Since 1 4 (ρ11 +ρ44)2 ≥ρ11ρ44 ≥ |ρ14|2 ≥Re(ρ14)2 > 1 16, we have ρ11 +ρ44> 1 2 , and then 2 √ρ22ρ33 ≤ρ22 +ρ33 < 1/ 2

    = 1 + 4Re(ρ14)< 0. Since 1 4 (ρ11 +ρ44)2 ≥ρ11ρ44 ≥ |ρ14|2 ≥Re(ρ14)2 > 1 16, we have ρ11 +ρ44> 1 2 , and then 2 √ρ22ρ33 ≤ρ22 +ρ33 < 1/ 2. This implies that ρ22ρ33< 1 16 < |ρ14|2 and thus ρ is entangled based on PPT criterion. Hence Tr( W1ρ) ≥ 0 for all separable states ρ. Accor...

  8. [15]

    Gurvits, Classical deterministic complexity of Edmonds’ problem and quantum entanglement, arXiv:quant-ph/0303055

    L. Gurvits, Classical deterministic complexity of Edmonds’ problem and quantum entanglement, arXiv:quant-ph/0303055

    arXiv

  9. [16]

    K. Wang, Z. Song, X. Zhao, Z. Wang and X. Wang, De- tecting and quantifying entanglement on near-term quan- tum devices, npj Quantum Information 8, 52 (2022)

    2022

  10. [17]

    Morelli, M

    S. Morelli, M. Huber and A. Tavakoli, Resource-efficient high-dimensional entanglement detection via symmetric projections, Phys. Rev. Lett. 131, 170201 (2023)

    2023

  11. [18]

    G¨ uhne and G

    O. G¨ uhne and G. T´ oth, Entanglement detection, Physics Reports 474, 1 (2009)

    2009

  12. [19]

    Hou and X.F

    J.C. Hou and X.F. Qi, Constructing entanglement wit- nesses for infinite-dimensional systems, Phys. Rev. A 81, 062351 (2010)

    2010

  13. [20]

    Morelli, H

    S. Morelli, H. Yamasaki, M. Huber and A. Tavakoli, Entanglement detection with imprecise measurements, Phys. Rev. Lett. 128, 250501 (2022)

    2022

  14. [21]

    Jurkowski and D

    J. Jurkowski and D. Chruscinski, Estimating concurren ce via entanglement witnesses, Phys. Rev. A 81, 052308 (2010)

    2010

  15. [22]

    Branciard, D

    C. Branciard, D. Rosset, Y.C. Liang and N. Gisin, Measurement-Device-Independent entanglement witnesses for all entangled quantum states, Phys. Rev. Lett. 110, 060405 (2013)

    2013

  16. [23]

    Facchi, S

    P. Facchi, S. Pascazio, V. Vedral and K. Yuasa, Quan- tumness and entanglement witnesses, J. Phys. A: Math. Theor. 45, 105302 (2012)

    2012

  17. [24]

    Debarba, T

    T. Debarba, T. O. Maciel and R. O. Vianna, Witnessed entanglement and the geometric measure of quantum dis- cord, Phys. Rev. A 86, 024302 (2012)

    2012

  18. [25]

    Reusch, J

    A. Reusch, J. Sperling and W. Vogel, Entanglement wit- nesses for indistinguishable particles, Phys. Rev. A 91, 042324 (2015)

    2015

  19. [26]

    Chru´ sci´ nski and G

    D. Chru´ sci´ nski and G. Sarbicki, Entanglement witnesses: construction, analysis and classification, J. Phys. A: Math. Theor. 47, 483001 (2014)

    2014

  20. [27]

    Huber and R

    M. Huber and R. Sengupta, Witnessing genuine mul- tipartite entanglement with positive maps, Phys. Rev. Lett. 113, 100501 (2014)

    2014

  21. [28]

    Shen, T.R

    S.Q. Shen, T.R. Xu, S.M. Fei, X.Q. Li-Jost and M. Li, Optimization of ultrafine entanglement witnesses, Phys. Rev. A 97, 032343 (2018)

    2018

  22. [29]

    Yuan, Q.X

    X. Yuan, Q.X. Mei, S. Zhou and X.F. Ma, Reliable and robust entanglement witness, Phys. Rev. A 93, 042317 (2016)

    2016

  23. [30]

    Shahandeh, M

    F. Shahandeh, M. Ringbauer, J. C. Loredo and T. C. Ralph, Ultrafine entanglement witnessing, Phys. Rev. Lett. 118, 110502 (2017)

    2017

  24. [31]

    K. Sen, C. Srivastava, S. Mal, A. S. De and U. Sen, De- tection loophole in measurement-device-independent en- tanglement witness, Phys. Rev. A 103, 032415 (2021)

    2021

  25. [32]

    A. Aloy, J. Tura, F. Baccari, A. Ac ´ ın, M. Lewenstein and R. Augusiak, Device-independent witnesses of entangle- ment depth from two-body correlators, Phys. Rev. Lett. 123, 100507 (2019)

    2019

  26. [33]

    Pandya, O

    P. Pandya, O. Sakarya and M. Wie´ sniak, Hilbert- Schmidt distance and entanglement witnessing, Phys. Rev. A 102, 012409 (2020)

    2020

  27. [34]

    Jannessary, F

    V. Jannessary, F. Rezazadeh, S. Raeisi and V. Karim- ipour, Witnessing entanglement of remote particles with incomplete teleportation, Phys. Rev. A 108, 042421 (2023)

    2023

  28. [35]

    Zhang, W.H

    C. Zhang, W.H. Zhang, P. Sekatski, J. D. Bancal, M. Zwerger, P. Yin, G.C. Li, X.X. Peng, L. Chen, Y.J. Han, J.S. Xu, Y.F. Huang, G. Chen, C.F. Li and G.C. Guo, Certification of genuine multipartite entanglement with general and robust device-independent witnesses, Phys. Rev. Le...

    2022

  29. [36]

    Ha and J

    D. Ha and J. S. Kim, Entanglement witness and multi- partite quantum state discrimination, J. Phys. A: Math. Theor. 56, 205303 (2023)

    2023

  30. [37]

    M.S. Li, Y.L. Wang, Sequentially witnessing entangle- ment by independent observer pairs, Phys. Lett. A 508, 129500 (2024)

    2024

  31. [38]

    Griffet, T

    C. Griffet, T. Haas and N. J. Cerf, Accessing continuous- variable entanglement witnesses with multimode spin ob- servables, Phys. Rev. A 108, 022421 (2023)

    2023

  32. [39]

    Yang, J.J

    N. Yang, J.J. Wu, X.Y. Dong, L.Y. Xiao, J. Wang and Ming Li, Entanglement measure based on optimal entan- glement witness, Quantum Inf. Process. 22, 296 (2023). 7

    2023

  33. [40]

    Ha and J

    D. Ha and J. S. Kim, Entanglement witness and non- locality in confidence of measurement from multipartite quantum state discrimination, arXiv:2405.20586

    arXiv

  34. [41]

    Shi, Lower bounds of entanglement quantifiers based on entanglement witnesses, arXiv:2312.17620

    X. Shi, Lower bounds of entanglement quantifiers based on entanglement witnesses, arXiv:2312.17620

    arXiv

  35. [42]

    Dai and N

    S. Dai and N. Bao, Entanglement Detection by Approx- imate Entanglement Witnesses, arXiv:2402.14755

    arXiv

  36. [43]

    Saggio and P

    V. Saggio and P. Walther, A perspective on few-copy en- tanglement detection in experiments, arXiv:2201.02641

    arXiv

  37. [44]

    Nicky K.H. Li, M. Huber and N. Friis, High-dimensional entanglement witnessed by correlations in arbitrary bases, arXiv:2406.04395

    arXiv

  38. [45]

    L.L. Sun, X. Zhou, A. Tavakoli, Z.P. Xu and S.X Yu, Bounding the amount of entanglement from witness op- erators, Phys. Rev. Lett. 132, 110204 (2024)

    2024

  39. [46]

    Tr´ enyi, ´A

    R. Tr´ enyi, ´A. Luk´ acs, P. Horodecki, R. Horodecki, T. V´ ertesi and G. T´ oth, Muliticoy metrology with many- particle quantum states, arxiv (2022), 2203.05538

    2022 arXiv

  40. [48]

    P. Liu, Z. Liu and X. Ma, Fundamental limitation on the detectability of entanglement, Phys. Rev. Lett. 129, 230503 (2022)

    2022

  41. [49]

    Yamasaki, S

    H. Yamasaki, S. Morelli, M. Miethlinger, J. Bavaresco, N. Friis and M. Huber, Activation of genuine multipar- tite entanglement: Beyond the single-copy paradigm of entanglement characterisation, Quantum 6, 695 (2022)

    2022

  42. [50]

    Rico and F

    A. Rico and F. Huber, Entanglement detection with trace polynomials, Phys. Rev. Lett. 132, 070202 (2024)

    2024

  43. [51]

    G¨ uhne and N

    O. G¨ uhne and N. L¨ utkenhaus, Nonlinear entanglement witnesses, Phys. Rev. Lett. 96, 170502 (2006)

    2006

  44. [52]

    Kotowski, M

    M. Kotowski, M. Kotowski and M. Ku´ s, Universal non- linear entanglement witnesses, Phys. Rev. A 81, 062318 (2010)

    2010

  45. [53]

    J. M. Arrazola, O. Gittsovich and N. L¨ utkenhaus, Ac- cessible nonlinear entanglement witnesses, Phys. Rev. A 85, 062327 (2012)

    2012

  46. [54]

    Shen, J.M

    S.Q. Shen, J.M. Liang, M. Li, J. Yu and S.M. Fei, Nonlin- ear improvement of qubit-qudit entanglement witnesses, Phys. Rev. A 101, 012312 (2020)

    2020

  47. [55]

    Hyllus, O

    P. Hyllus, O. G¨ uehne, D. Bruss and M. Lewenstein, Re- lations between entanglement witnesses and bell inequal- ities, Phys. Rev. A 72, 012321 (2005)

    2005

  48. [56]

    Peres, Separability criterion for density matrices , Phys

    A. Peres, Separability criterion for density matrices , Phys. Rev. Lett. 77, 1413 (1996)

    1996

  49. [57]

    Kraus, M

    B. Kraus, M. Lewenstein and J. I. Cirac, Characteriza- tion of distillable and activable states using entanglemen t witnesses, Phys. Rev. A 65, 042327 (2002)

    2002

  50. [58]

    M.C. Chen, Y. Li, R.Z. Liu, D. Wu, Z.E. Su, X.L. Wang, L. Li, N.L. Liu, C.Y. Lu and J.W. Pan, Directly Measur- ing a Multiparticle Quantum Wave Function via Quan- tum Teleportation, Phys. Rev. Lett. 127, 030402 (2021)

    2021

  51. [59]

    Xu and J.H

    Z.J. Xu and J.H. An, Noise mitigation in quantum tele- portation, Phys. Rev. A 110, 012442 (2024)

    2024

  52. [60]

    Horodecki and M

    M. Horodecki and M. Piani, On quantum advantage in dense coding, J. Phys. A: Math. Theor. 45, 105306 (2012)

    2012

  53. [61]

    Guo, B.H

    Y. Guo, B.H. Liu, C.F. Li and G.C. Guo, Advances in quantum dense coding, Adv. Quant. Technol 2, 1900011 (2019)

    2019

  54. [62]

    Hickey and G

    A. Hickey and G. Gour, Quantifying the imaginarity of quantum mechanics, J. Phys. Rev. A: Math. Theor. 51, 414009 (2018)

    2018

  55. [63]

    Werner, Quantum states with EPR correlations ad- mitting a hidden variable model, Phys

    R.F. Werner, Quantum states with EPR correlations ad- mitting a hidden variable model, Phys. Rev. A 40, 4277 (1989)

    1989

  56. [64]

    X.B. Zou, K. Pahlke and W. Mathis, Generation of an entangled four-photon W state, Phys. Rev. A 66, 044302 (2002)

    2002

  57. [65]

    V. N. Gorbachev, A. A. Rodichkina and A. I. Trubilko, On preparation of the W-states from atomic ensembles, Phys. Lett. A 310, 339 (2003)

    2003

  58. [66]

    Tashima, T

    T. Tashima, T. Wakatsuki, S. K. Ozdemir, T. Yamamoto, M. Koashi and N. Imoto, Local transformation of two EPR photon pairs into a three-photon W state, Phys. Rev. Lett. 102, 130502 (2009)

    2009

  59. [67]

    X.Y. Wu, Y. Cai, T. H. Yang, H. N. Le, J.-D. Bancal and V. Scarani, Robust self testing of the 3-qubit W state, Phys. Rev. A 90, 042339 (2014)

    2014

  60. [68]

    Bourennane, M

    M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Guehne, P. Hyllus, D. Bruss, M. Lewen- stein and A. Sanpera, Witnessing multipartite entangle- ment, Phys. Rev. Lett. 92, 087902 (2004)

    2004

  61. [69]

    Zhao, T.G

    M.J. Zhao, T.G. Zhang, X.Q. Li-Jost and S.M. Fei, Iden- tification of three-qubit entanglement, Phys. Rev. A 87, 012316 (2013). 8 Appendix A: Example 1 Utilizing our strategy, we write down each term of ρ⊗ 2 AB after measuring the operators WAB′ and VBA′ . Without causing confu...

    2013

  62. [70]

    i 2 ) = − 1 4 − 1 4 = − 1 2

    −i 2 ) − ( −i 2 . i 2 ) = − 1 4 − 1 4 = − 1 2 . Appendix C: Example 3 We calculate the value of Tr(W1,A 1B2 ⊗W2,A 2B3 ⊗W2,B 1A3ρ⊗ 3 w ). For simplicity, we only list the terms that contribute to the trace, |000000⟩⟨000000| W1,A1 B2 − − − − − − − − − − − − → W2,A2 B3 ,W 3,B1A3 ...

  63. [71]

    ( 1 −c 3 )2, where Tr((W4,AB ′ ⊗W3,BC ′ ⊗W3,CA ′ )ρ⊗ 2 c )< 0 when c< 0

    8 − 5c 24 + 4. ( 1 −c 3 )2, where Tr((W4,AB ′ ⊗W3,BC ′ ⊗W3,CA ′ )ρ⊗ 2 c )< 0 when c< 0. 406. Appendix F: Entanglement concentration We compute the reduced state associated with the subsystem AB ′ after measurement. For clarity and intuitiveness, we adopt the representation met...

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