Quantum Entanglement with Geometric Measures

Xuanran Zhu

REVIEW 3 major objections 5 minor 2 references

Quantum Entanglement with Geometric Measures

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A ladder of geometric entanglement measures splits entanglement dimensionality exactly and is computable by gradient descent.

desk verdict Solid k-GME framework with some nice analytic results, but the flagship border-rank claim for the matrix multiplication tensor is internally inconsistent as printed. read the letter →

arxiv 2506.11453 v1 pith:GO5XAAGB submitted 2025-06-13 quant-ph

Xuanran Zhu This is my paper · ORCID

classification quant-ph MSC 81P4081P45 PACS 03.67.Mn03.67.-a

keywords geometricmeasureofentanglementk-GMEmonotonesSchmidtnumberborderranktensorconvexroofconstructionmanifoldtrivializationgradientdescent

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 thesis sets out to show that the geometric measure of entanglement, the squared-overlap distance from a state to the product states, can be promoted into a complete ladder of monotones: the $k$-th member $E_G^{(k)}$ measures the distance to all states of entanglement dimensionality below $k$. For bipartite mixed states the ladder exactly separates Schmidt numbers, and for multipartite pure states it exactly separates tensor-border ranks, provided the quantities can be evaluated. The thesis further argues they can be evaluated: the defining overlap maximization is rewritten as an unconstrained gradient-descent problem on a trivialized manifold, and the same scheme supplies upper bounds for subspace and mixed-state monotones alongside semidefinite-programming lower bounds. If the claims hold, one computational framework covers pure, subspace, and mixed entanglement quantification, including high-dimensional entanglement.

What carries the argument

The central object is the $k$-GME ladder. For each $k$, $E_G^{(k)}$ is the squared-overlap deficit from the given state to all states of Schmidt or tensor rank below $k$; for subspaces it is minimized over the subspace and reduces to the smallest expectation of the orthogonal-complement projector, and for mixed states it is the convex roof, which the thesis identifies with a fidelity distance to states of Schmidt number at most $k-1$. The computational engine is manifold trivialization: states of bounded rank are parameterized by unconstrained real parameters through the SoftPlus map for positive coefficients and normalization maps for local vectors, converting each non-convex overlap maximization into unconstrained gradient descent with automatic differentiation. The same parameterization yields upper bounds, while PPT and generalized-reduction relaxations provide semidefinite-programming lower bounds.

What would settle it

For the reported transition $E_G^{(6)}(|\Phi_2\rangle)\approx 10^{-14}$, replace the heuristic gradient-descent search by a certified global maximization over states of tensor rank below 6; a certified positive value would falsify the border-rank claim $\mathrm{BR}(|\Phi_2\rangle)=7$, while a certified zero would confirm it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a one-parameter extension of the geometric measure, $E_G^{(k)}(|\psi\rangle)=1-\max_{\mathrm{SR}(|\phi\rangle)<k}|\langle\phi|\psi\rangle|^2$, carries exactly the information that distinguishes entanglement dimensionalities. For a mixed state $\rho$, the convex-roof extension is simultaneously the fidelity distance $1-F(\rho,\sigma)$ to the set of states with Schmidt number at most $k-1$, and the thesis establishes that $\mathrm{SN}(\rho)=k$ exactly when $E_G^{(k)}(\rho)>0$ and $E_G^{(k+1)}(\rho)=0$. For multipartite pure states the same construction with tensor rank gives $\mathrm{BR}(|\psi\rangle)=k$ exactly when $E_G^{(k)}(|\psi\rangle)>0$ and $E_G^{(k+1)}(|\psi\rangle)=0$. The thesis demonstrates the ladder on Haar-random bipartite states, subspace examples, Werner and isotropic states, Dicke states, and the $2\times 2$ matrix-multiplication tensor, where the transition of $E_G^{(k)}$ at $k=6$ to a near-zero value is read as border rank $7$.

Load-bearing premise

The border-rank and Schmidt-number conclusions stand on the assumption that the non-convex gradient descent over the trivialized parameter space actually finds the global maximum overlap with states of bounded tensor rank, so that reported near-zero values such as $10^{-14}$ are genuine zeros rather than optimization failures.

Editorial extensions

If this is right

Computing the $k$-GME ladder of a bipartite mixed state reads off its Schmidt number, because the first $k$ with $E_G^{(k)}(\rho)>0$ and $E_G^{(k+1)}(\rho)=0$ is exactly $\mathrm{SN}(\rho)$.
For multipartite pure states the same ladder determines border rank, giving a numerical probe of tensor border rank, a quantity connected to algebraic complexity.
Because the mixed-state $k$-GME is a fidelity distance, it can be sandwiched between gradient-descent upper bounds and semidefinite-programming lower bounds, so reported values can carry two-sided error estimates.
The subspace version certifies entanglement and its dimensionality in cases where PPT-based relaxations are blind, including completely entangled subspaces and high-dimensional entangled subspaces.
For Haar-random $d\otimes d$ pure states the top-level value $E_G^{(d)}$ has the closed-form distribution $d(d^2-1)(1-dx)^{d^2-2}$, so high-dimensional entanglement is exponentially rare under uniform sampling and the optimal distillation probability of the maximally entangled state follows the same law.

Reading between the lines

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

If the ladder characterization is stable under noise, a single fidelity measurement $|\langle\phi|\psi\rangle|^2$ with an optimized low-rank $\phi$ could certify an entanglement-dimensionality lower bound without full tomography; the thesis does not develop this experimental witness interpretation.
The trivialized gradient-descent method is not specific to entanglement: the same parameterization could be pointed at tensor-rank and border-rank questions elsewhere in algebraic complexity, where exact answers are known for only a few tensors.
A testable extension is to apply the $k$-GME ladder to bound-entangled families with high Schmidt number and compare the predicted transitions with independent rigorous bounds from $k$-positive maps; agreement would strengthen the numerical optimality assumption.
Because the set of bounded-tensor-rank states is not closed, the max in the multipartite definition is formally a supremum; the numerical transition criteria presuppose that the trivialized search reaches that supremum, something the thesis does not prove.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the k-GME generalization is a natural and mostly coherent extension of the geometric measure of entanglement, and the random-state analytic distributions are worth a look. But the central numerical demonstration—the border rank of the 2x2 matrix multiplication tensor—is broken by the paper's own definitions. Since E^(k) = 1 - max_{TR|^2, the allowed set grows with k, so E^(k) must be nonincreasing in k. The thesis reports E^(7)(Phi2) ~ 1/8 and E^(6)(Phi2) ~ 10^-14, which violates that necessary inequality. If E^(6) is really ~10^-14, then E^(7) must also be ~0, and the claimed BR=7 conclusion cannot follow—the numbers would suggest BR<=5. A typo or off-by-one in the k-1 parameterization is the likely explanation, but as printed the flagship numerical claim doesn't hold up.

What is actually new and good: the k-GME family is a clean extension to rank-bounded tensor rank; Theorem 6.1.4 gives an explicit analytic distribution for the smallest Schmidt eigenvalue and the associated distillation success probability, and the numerics agree with that distribution for 4x4 Haar random states. The subspace and mixed-state formulations are coherent, although the key distance-based equivalence is deferred to the author's own Ref. [217]. Numerical checks on Dicke states match known border ranks, and the manifold-trivialization approach scales better than SDP on the tested cases. That is genuine work.

Soft spots beyond the Phi2 problem: the optimization is non-convex gradient descent, which the thesis itself says provides only upper bounds; there is no code or data release, no convergence diagnostics, and no error bars. The border-rank inferences therefore rest on an unverified global-optimality assumption. The internal contradiction is more decisive, but the reproducibility gaps are substantial for a thesis making numerical certification claims.

This is for quantum information readers interested in entanglement dimensionality measures and Haar random state distributions. The analytic parts are useful; the numerical border-rank claims need correction before anyone should rely on them.

Recommendation: yes, send it to peer review—the framework and analytic results deserve referee time—but a referee should demand a fix for the monotonicity violation, proper convergence diagnostics, and code release. I would not cite the Phi2 result until it is corrected.

Referee Report

3 major / 5 minor

Summary. The thesis proposes a family of geometric-measure-based entanglement monotones E_G^(k), defined for pure states, subspaces, and mixed states in both bipartite and multipartite settings, with the goal of characterizing entanglement dimensionality through Schmidt rank, Schmidt number, tensor rank, and border rank. It derives analytic distributions for k-GME and optimal entanglement-distillation probabilities for Haar-random bipartite pure states, and it develops a manifold-trivialization gradient-descent framework, complemented by SDP relaxations, for numerical computation. The central applications are k-GME transitions for Dicke states and for the 2x2 matrix-multiplication tensor |Φ2>, from which border-rank values are inferred.

Significance. If the central claims are corrected and properly supported, the k-GME family would provide a unified quantitative tool for entanglement dimensionality across pure, subspace, and mixed-state settings, with analytic predictions such as Eq. (6.11) that are independently checkable and numerically verified. The hybrid non-convex/SDP computational strategy is practically valuable and substantially cheaper than hierarchical methods in the subspace examples. However, as printed, the headline border-rank demonstration for |Φ2> is internally inconsistent, so the significance presently hinges on a load-bearing correction rather than on the results as stated.

major comments (3)

[Sec. 6.1.2, Eq. (6.21) and the paragraph after Eq. (6.30)] The reported values for |Φ2>_ABC violate the monotonicity that is immediate from Definition 18. Since the admissible set {|φ>: TR(|φ>)<k} is nested, E_G^(k)(|ψ>) must be nonincreasing in k. The text states E_G^(7) ≈ 1/8 and E_G^(6) ≈ 10^-14, which would require E_G^(6) ≥ E_G^(7) and is therefore impossible. Taken literally, E_G^(6) ≈ 10^-14 would imply BR(|Φ2>) ≤ 5, contradicting both the known border rank 7 and the thesis's own conclusion. This appears to be an off-by-one or typographical error, with E_G^(8) ≈ 10^-14 the plausible intended value, but as printed the central numerical demonstration is not supported. The same monotonicity check should be applied to the Dicke-state transitions reported in Fig. 6.3.
[Sec. 5.2.5 and Eq. (6.24)] The gradient-descent method returns an upper bound on E_G^(k), not a two-sided approximation; the thesis states in Sec. 5.2.5 that the non-convex method does not guarantee global convergence. For the border-rank inference, the positivity condition E_G^(k) > 0 requires a lower bound, whereas a positive numerical value such as E_G^(7) ≈ 1/8 only gives E_G^(7)_true ≤ 1/8 and does not certify that the true value is positive. Without an independent certificate (for example, an algebraic border-rank lower bound or an SDP/dual-witness lower bound), the conclusion BR(|Φ2>) = 7 is not established by the reported numerics. Conversely, the near-zero value for E_G^(6), being an upper bound, would certify E_G^(6)_true ≈ 0 and hence BR(|Φ2>) ≤ 5, further reinforcing that the data need correction rather than reinterpretation.
[Definition 18 and Eq. (6.24)] The set of states with tensor rank strictly less than k is not closed, and the maximum in Eq. (6.21) need not be attained; the W state is a standard example where the supremum over rank-2 states is 1 but is not achieved. The definition should therefore use a supremum, and the numerical parameterization of Eq. (6.24) must be justified as approximating that supremum. As written, the text silently replaces the supremum by a maximum over a parameterized family, which is a formal gap in the definition of the central monotone and also affects the claimed equivalence between BR(|ψ>) = k and the pair of conditions E_G^(k)(|ψ>) > 0, E_G^(k+1)(|ψ>) = 0.

minor comments (5)

[Fig. 6.3] The Dicke-state transitions would be easier to assess if the numerical values of E_G^(k) and the number of random restarts or other hyperparameters were reported; the caption currently only states that transitions occur.
[Fig. 6.6 caption] The word 'istropic' should be 'isotropic' in the caption of Fig. 6.6.
[Sec. 6.3.1, Theorem 6.3.1] The proof of Theorem 6.3.1 is deferred to 'Appendix A, Theorem 2, in Ref. [217]'; since the manuscript is a thesis, including the proof or a self-contained sketch would improve verifiability.
[Sec. 6.1.1, Eq. (6.10)] The marginal eigenvalue distribution in Eq. (6.10) depends on an unspecified normalization constant N and on polynomials A_j^{(d,d)}(x) that are only partly displayed; the formulas are difficult to reproduce without the cited reference, and the displayed 4x4 example would benefit from a consistency check against the normalization condition.
[Table 6.5] The summary table would be clearer if each row explicitly stated the domain of the monotone (pure state, subspace, or mixed state) and the exact rank parameter being bounded.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: k-GME is defined by rank thresholds, the Schmidt/border-rank equivalences are explicit definitional consequences, and numerical results are externally benchmarked; the reported Phi2 numbers are internally inconsistent but that is a correctness issue.

full rationale

The derivation chain is self-contained in the relevant sense. Definitions 17 and 18 define E_G^(k) directly as 1 minus the maximal overlap with states of bounded Schmidt/tensor rank; therefore the characterizations SR(|ψ⟩)=k iff E_G^(k)>0 and E_G^(k+1)=0 and BR(|ψ⟩)=k iff E_G^(k)>0 and E_G^(k+1)=0 are immediate consequences of the definitions and the closure meaning of border rank, not circular predictions. The paper explicitly attributes the border-rank equivalence to the definition of border rank. The numerical claims for Haar-random states are compared with analytical marginal distributions (Eqs. 6.10-6.15), and the Dicke and 2x2 matrix-multiplication border-rank examples are benchmarked against known results [179, 182]. The mixed-state distance representation (Theorem 6.3.1) is cited to the author's own prior work (Ref. [217]) rather than proved in the thesis; this is a self-citation, but the Schmidt-number characterization does not depend on it, since the convex-roof definition plus the pure-state threshold already gives SN(ρ)=k iff E_G^(k)>0 and E_G^(k+1)=0. Hence it is not load-bearing circularity. The concrete risk is instead empirical/consistency: Sec. 6.1.2 reports E_G^(6)(Φ2)≈10^-14 and E_G^(7)(Φ2)≈1/8, which violates the monotonicity E_G^(k)≥E_G^(k+1) required by Definition 18, and Sec. 5.2.5 states the non-convex method only gives upper bounds; these undermine the printed border-rank inference but are correctness issues rather than circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim introduces no new physical entities such as particles or forces. It depends on standard quantum information mathematics (Schmidt decomposition, convex roof, positive maps) and on the author's prior theorems for the distance-based k-GME. The main hidden assumption is that the non-convex numerical optimization reaches global optima, so that zero/nonzero transitions are real.

free parameters (2)

Number of random restarts for gradient descent
Sec. 5.2.5 says the objective is minimized multiple times with randomly initialized parameters, treating the number of trials as a hyperparameter; no values are given. The numerical results in Tables 6.1, 6.2 and Figs. 6.3, 6.4 depend on this.
Optimizer hyperparameters (learning rate, L-BFGS settings)
The thesis states L-BFGS is the default optimizer, but does not specify learning rates, line search tolerances, or iteration counts. These affect the reported numerical values.

assumptions (3)

ad hoc to paper The set of states with tensor rank < k can be approximated by the unnormalized parameterization with SoftPlus coefficients and sphere-normalized local states (Eqs. 6.23-6.24), so that the supremum over the non-closed rank-bounded set is attainable in the limit.
Introduced in Sec. 6.1.2 to reformulate k-GME as an unconstrained optimization; no proof is given that the optimization converges to the supremum, and the non-closed nature of the rank set is not formally addressed.
domain assumption The distance-based representation E_G^(k)(ρ) = min_{σ∈S_{k-1}} (1 - F(ρ,σ)) holds for mixed states, as stated in Theorem 6.3.1.
The theorem is stated without proof in the thesis and cited to Ref. [217], the author's own prior paper. If this theorem were false, the claim that SN(ρ) = k iff E_G^(k)(ρ) > 0 and E_G^(k+1)(ρ) = 0 would fail.
standard math The joint eigenvalue distribution for Haar random pure states (Eq. 6.9) and the marginal distributions from Ref. [169] are correct.
These results are imported from the random matrix theory literature and used to derive the analytic distribution of E_G^(d).

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Cite this review

Pith. "Pith review of Quantum Entanglement with Geometric Measures." pith.science (2026). https://pith.science/paper/GO5XAAGB

@misc{pith2026250611453,
  author       = {Pith},
  title        = {Pith review of: Quantum Entanglement with Geometric Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GO5XAAGB}},
  note         = {Machine review of arXiv:2506.11453}
}

read the original abstract

Quantifying quantum entanglement is a pivotal challenge in quantum information science, particularly for high-dimensional systems, due to its computational complexity. This thesis extends the geometric measure of entanglement (GME) to introduce and investigate a suite of GME-based entanglement monotones tailored for diverse quantum contexts, including pure states, subspaces, and mixed states. These monotones are applicable to both bipartite and multipartite systems, offering a unified framework for characterizing entanglement across various scenarios. Notably, the proposed monotones are adept at identifying entanglement with varying entanglement dimensionalities, making them particularly effective for detecting high-dimensional entanglement. To support practical computation, we develop a non-convex optimization framework that yields accurate upper bounds, complemented by semidefinite programming techniques to establish robust lower bounds. Together, these approaches provide a consistent and efficient computational methodology. This work advances both the theoretical understanding and algorithmic tools for entanglement quantification, contributing to the study of complex quantum correlations in entangled systems.

Figures

Figures reproduced from arXiv: 2506.11453 by the authors.

**Figure 4.** Geometric representation of entanglement witnesses. The quantum state [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 6.** Entanglement distillation analysis for random pure states: (a) Optimal [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

Figures from the paper (23 more)

**Figure 2.1.** Figure 2.1: Graphical representation of basic tensors: (a, b) vectors, (c) a linear operator, and [PITH_FULL_IMAGE:figures/full_fig_p025_2_1.png]

**Figure 3.1.** Figure 3.1: Illustration of the 𝑑 ⊗ 𝑑 (𝑑 > 2) quantum state space, partitioned according to Schmidt numbers. The set 𝑆𝑘 comprises states with Schmidt number SN ≤ 𝑘, forming a convex subset of the quantum state space and satisfying the hierarchy 𝑆1 ⊂ 𝑆2 ⊂ ⋯ ⊂ 𝑆𝑑 . The set of sepa…

**Figure 3.2.** Figure 3.2: The entanglement structure of tripartite mixed states [PITH_FULL_IMAGE:figures/full_fig_p037_3_2.png]

**Figure 3.3.** Figure 3.3: Illustration of entanglement distillation and formation protocols. In the distillation [PITH_FULL_IMAGE:figures/full_fig_p042_3_3.png]

**Figure 4.1.** Figure 4.1: Geometric representation of entanglement witnesses. The quantum state space is [PITH_FULL_IMAGE:figures/full_fig_p050_4_1.png]

**Figure 4.2.** Figure 4.2: The convex roof entanglement measure 𝐸(𝜌) is calculated by minimizing a weighted sum of the entanglement measure for pure states 𝐸(|𝜓𝑖 ⟩) over all possible pure-state decompositions 𝜌 = ∑𝑖 𝑝𝑖 |𝜓𝑖 ⟩⟨𝜓𝑖 |. For a pure state |𝜓𝐴𝐵⟩, the EoF, denoted as 𝐸𝑓 (|𝜓𝐴𝐵⟩), is equ…

**Figure 4.3.** Figure 4.3: Visualization of a distance-based entanglement measure [PITH_FULL_IMAGE:figures/full_fig_p060_4_3.png]

**Figure 5.1.** Figure 5.1: Visualization of the interior point method. The algorithm typically starts with an [PITH_FULL_IMAGE:figures/full_fig_p066_5_1.png]

**Figure 5.2.** Figure 5.2: Illustration of the PPT relaxation for bipartite separable states. Here, SEP represents [PITH_FULL_IMAGE:figures/full_fig_p069_5_2.png]

**Figure 5.3.** Figure 5.3: Geometric interpretation of gradient descent. Beginning from an arbitrary initial [PITH_FULL_IMAGE:figures/full_fig_p075_5_3.png]

**Figure 5.4.** Figure 5.4: The computational graph of the function 𝑓(𝑥1 , 𝑥2 ) = ln 𝑥1 + 𝑥1𝑥2 − sin(𝑥2). 𝑣1 , 𝑣2 , … , 𝑣5 are used to record the intermediate variables. The solid blue lines represent the forward mode, while the dashed orange lines denote the reverse mode. In the forward mode, …

**Figure 5.5.** Figure 5.5: The Bloch sphere representing single-qubit pure states. Points on the surface cor [PITH_FULL_IMAGE:figures/full_fig_p082_5_5.png]

**Figure 5.6.** Figure 5.6: Schematic diagram of parameterized quantum circuit [PITH_FULL_IMAGE:figures/full_fig_p087_5_6.png]

**Figure 5.7.** Figure 5.7: Flowchart of the manifold optimization framework for solving quantum information [PITH_FULL_IMAGE:figures/full_fig_p090_5_7.png]

**Figure 6.1.** Figure 6.1: 𝑘-GME distribution for Haar random pure states in a 4 ⊗ 4 system. The green, orange, and blue areas represent the numerical probability distribution function (P.D.F.) of 𝐸 (𝑘) 𝐺 for 𝑘 = 2, 3, 4, respectively. For comparison, the red curve denotes the analytical resul…

**Figure 6.2.** Figure 6.2: Entanglement distillation analysis for random pure states: (a) Optimal success prob [PITH_FULL_IMAGE:figures/full_fig_p097_6_2.png]

**Figure 6.3.** Figure 6.3: Results of 𝑘-GME for Dicke states |𝐷𝑚 𝑛 ⟩ with different 𝑛 and 𝑚. The transition of 𝐸 (𝑘) 𝐺 at 𝑘 = 𝑚 + 1 and 𝑘 = 𝑚 + 2 implies the border rank of Dicke states. We compute the 𝑘-GME 𝐸 (𝑘) 𝐺 (|𝐷𝑚 𝑛 ⟩) for various values of 𝑛 and 𝑚, as shown in [PITH_FULL_IMAGE:figures…

**Figure 6.4.** Figure 6.4: Results for 2 ⊗ 𝑑 entangled subspace 𝒮 𝜃 2⊗𝑑: (a) Numerical and analytical results for 𝐸 (2) 𝐺 (𝒮 𝜃 2⊗𝑑) with respect to 𝜃. Here, GD denotes the method based on the gradient descent and PPT represents the results from the PPT relaxation. (b) Minimum values of 𝐸 (2) 𝐺…

**Figure 6.5.** Figure 6.5: The 𝑘-GME results for 4 ⊗ 4 Werner states. The solid lines represent numerical results obtained via gradient descent (GD), while the dots indicate analytical results. Werner states are categorized into two regions based on their Schmidt number. For the 𝑑 ⊗ 𝑑 isotropi…

**Figure 6.6.** Figure 6.6: The 𝑘-GME results for isotropic states: (a) Results for 4 ⊗ 4 istropic state 𝜌𝐼 (𝐹 ). (b) Results for two copies of 2 ⊗ 2 isotropic states 𝜌𝐼 (𝐹 ) ⊗ 𝜌𝐼 (𝐹 ) in 4 ⊗ 4 system. Here, “GD” represents the gradient descent method, “Reduction” means the generalized reductio…

**Figure 6.7.** Figure 6.7: Geometric measure of entanglement for Horodecki states [PITH_FULL_IMAGE:figures/full_fig_p124_6_7.png]

**Figure 6.8.** Figure 6.8: The 𝑘-GME results for the edge state under a depolarizing channel. The noise level of the depolarizing channel is controlled by 𝑝. A sharp transition indicates the disappearance of entanglement with a certain Schmidt number. dimensional entanglement within these PPT …

**Figure 6.9.** Figure 6.9: The geometric measure of entanglement (GME) for the mixed state [PITH_FULL_IMAGE:figures/full_fig_p128_6_9.png]

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Works this paper leans on

2 extracted references · 2 linked inside Pith

[1990]

Simpler semidefinite programs for completely bounded norms

(cit. on p. 52). 142I. Bengtsson et al., Geometry of quantum states: an introduction to quantum entanglement (Cambridge university press, 2017) (cit. on p. 53). 143J. Watrous, “Simpler semidefinite programs for completely bounded norms”, arXiv preprint arXiv:1207.5726 (2012) (cit. on p. 55). 144C. Lemaréchal, “Cauchy and the gradient method”, Doc Math Ext...

arXiv 2012
[2010]

Conditions for a class of entanglement transformations

(cit. on pp. 25, 55, 101, 102). 79M. A. Nielsen, “Conditions for a class of entanglement transformations”, Phys. Rev. Lett.83, 436–439 (1999) (cit. on pp. 26, 27, 80). 80R. Bhatia, Matrix analysis, Vol. 169 (Springer Science & Business Media, 2013) (cit. on p. 26). 81G. Vidal, “Entanglement of pure states for a single copy”, Phys. Rev. Lett. 83, 1046–1049...

arXiv 1999

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1990]

Simpler semidefinite programs for completely bounded norms

(cit. on p. 52). 142I. Bengtsson et al., Geometry of quantum states: an introduction to quantum entanglement (Cambridge university press, 2017) (cit. on p. 53). 143J. Watrous, “Simpler semidefinite programs for completely bounded norms”, arXiv preprint arXiv:1207.5726 (2012) (cit. on p. 55). 144C. Lemaréchal, “Cauchy and the gradient method”, Doc Math Ext...

arXiv 2012

[2] [2010]

Conditions for a class of entanglement transformations

(cit. on pp. 25, 55, 101, 102). 79M. A. Nielsen, “Conditions for a class of entanglement transformations”, Phys. Rev. Lett.83, 436–439 (1999) (cit. on pp. 26, 27, 80). 80R. Bhatia, Matrix analysis, Vol. 169 (Springer Science & Business Media, 2013) (cit. on p. 26). 81G. Vidal, “Entanglement of pure states for a single copy”, Phys. Rev. Lett. 83, 1046–1049...

arXiv 1999