slug: saleem-et-al-2025-noisy-curvature-of-entanglement
bibkey: saleem2025curvature
doi: 10.48550/arXiv.2504.13729
url: https://arxiv.org/abs/2504.13729
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Information/SaleemNoisyCurvatureRefutation.result

Noisy entanglement curvature can exceed quantum Fisher information

Problem

Z. H. Saleem, A. Shaji, A. M. Babu, D.-W. Luo, Q. Langfitt, T. Yu and S. K. Gray, Quantum Fisher Information and the Curvature of Entanglement, arXiv:2504.13729v4, Section IV.B, p. 9:

For the case with decay also there are more general examples we can consider but in all cases we conjecture that CoE ≤ QFI with equality at specific points.

The source note locates Section II, Eq. (7), the flip-flop Hamiltonian (21), the open evolution (41), the particular examples (42)–(43), and Appendix B. Preregistration #17681 fixes the literal conventions and the unrestricted-initial-density-matrix reading. Section III explicitly introduces losses and mixed probe states after treating pure lossless evolution. The resolution is Refuted.

Statement echo

Motivation

A bound on entanglement curvature by QFI would connect the sharpness of dynamically generated entanglement to metrological sensitivity. A diagonal, separable mixed probe already separates these quantities under the source’s symmetric amplitude damping.

Gap

The source versions v1–v4 and the QUANTUM 2026 abstract retain the coincidence theme; the abstract reports persistence under amplitude damping and gives no violation. The literature readings supplied with the preregistration and research seat screen Chatterjee arXiv:2508.12497, Saleem arXiv:2606.05696 and 2606.15484, Saleem–Saleem arXiv:2606.18415, and Saleem et al. arXiv:2609.31853 without locating a refutation for evolution (41). These follow-up screens are seat-reported. The bounded search conclusion is not-found-in-searched-scope; literature completeness is ASSUMED-UNVERIFIED. A coincidence established for particular probes does not supply a universal inequality.

Route

Take the parameter-independent state ${\rho }_0=\mathrm{diag}(1,11,3,1)/16$ and evolve it by (41). At $\kappa =\log 2$, $t=1$, $g_{\ast }=\pi /4$, the state is

$${\rho }_{\ast }=\frac{1}{64}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 15& 8i& 0\\ 0& -8i& 15& 0\\ 0& 0& 0& 33\end{array}\right).$$

In a neighbourhood of $g_{\ast }$ its concurrence is $C(g)=\sin (2g)/4-\sqrt{33}/32$, so $-C^{\prime \prime }(g_{\ast })=1$. The Hermitian SLD $L=\mathrm{diag}(0,-16/15,16/15,0)$ gives $F_Q=8/15$. Thus the universal inequality would require $1\le 8/15$.

Falsifier

The interpretation would fail if the conjecture excluded mixed initial probes or used a different lowering convention. Section III’s explicit mixed-state scope, Eq. (41) and Appendix B fix those choices. A nonsmooth concurrence point would not refute the smooth-point formal claim; concurrence_smooth proves the required regularity here. The pure lossless theorem of Section III.B has different assumptions.

Evidence

The settling declaration is D5/S3/Quantum/Information/SaleemNoisyCurvatureRefutation.result : ¬ claim. The consumed private theorems rho0_density, trajectory, concurrence_smooth, curvature, sld_point and qfi_point establish the initial density matrix, exact trajectory, regularity, curvature and SLD-QFI used by result. The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). No other public theorem is introduced.

Experiment

The pinned experiment directory contains check.py. Command: python3 /Users/auric/trureturing-experiments/docs/reports/saleem-et-al-2025-curvature-of-entanglement-noisy/check.py; exit 0; ALL_OK; script SHA-256 205e590442177c267051f09ef129df7f7fead0f3be102714075a1a9dccada488. The checked blob belongs to commit 75dbfafee01a748b83764044ebe12042bcdb48e2. The exact symbolic checks cover the trajectory, trace, spin-flip polynomial and point spectrum, concurrence, SLD equation and family QFI/curvature identities. The numerical checks compare the mixed witness at finite-difference steps $10^{-3},10^{-4}$, the pure product witness at $10^{-3},3\cdot 10^{-4},10^{-4}$, and the source’s $|01⟩$ formulas at three specified parameter triples. The zero-damping control samples 3000 random pure states with seed 0, of which 20 pass the script’s concurrence and finite-difference stability filters; their maximum gap is $-12.1389$. This control is not a prevalence test at positive damping.

Triage

Tier 1; resolution Refuted; admission_basis: open-problem-resolution (#17681; Refuted). Utility is kind=certified-instance; basis=refutes: the instance contradicts the literal universal claim. Every theorem is classified bind-only after expansion; the external named settlement supplies the admission basis.

What the settlement shows

1. Mechanism at the point — kernel-checked. concurrence_smooth, curvature, sld_point, qfi_point and result prove that smooth CoE is $1$ while QFI is $8/15$ at the stated positive-damping point. These statements concern the explicit witness, not a uniform family inequality.

2. Counterfamily — computed identities, with positivity proved in prose. The experiment verifies exactly $\mathrm{CoE}=2q{t}^2$, $F_Q=8q{t}^2/(8-q)$ and $\mathrm{CoE}-F_Q=2q{t}^2(4-q)/(8-q)$ for the displayed family, where $q=e^{-\kappa t}$. For $\kappa ,t>0$, $0<q<1$, so all factors in this last quotient are positive. The same sign condition makes the positive concurrence branch smooth near $g_{\ast }=\pi /(4t)$. The block explanation is that curvature is linear in the imbalance $\Delta$, while QFI is quadratic in $\Delta$ divided by block weight $K$; $\mathrm{CoE}/F_Q=K/\Delta =2-q/4$ for $\Delta =1/2$, $K=1-q/8$. This is an explanation of the symbolic identities, not an additional Lean theorem.

3. Pure product probe — computed. The experiment’s stable finite differences for $(|00⟩+|10⟩)/\sqrt{2}$ at $\kappa =0.3$, $t=3$, $g=2.9$ give gap $3.3708$ and concurrence $0.0505$. This supports a noisy pure-state violation numerically; it is not kernel evidence for a pure-state refutation or a proof of exact differentiability there. Random pure-state prevalence — open. The supplied research reading says about $7\%$ of sampled pure initial states violate the bound at smooth points. The pinned checker does not test that statistic; the population, sample size, positive-damping sampling rule and smoothness filter are unverified here. The estimate is ASSUMED-UNVERIFIED, not a reproduced experimental result.

4. Consequences — proved in prose from the refutation and source scope. Section III.B’s pure lossless theorem cannot be extended to all initial states under (41): such an extension would imply claim, contradicted by result. The two noisy examples’ coincidence properties remain statements about those probes; their validity does not imply an inequality for every probe. No lossless theorem or specific noisy coincidence formula is refuted. Any subsequent argument using the universal noisy bound needs an additional restriction or a different inequality.

5. Neighbouring questions — open. Classify initial states for which $\mathrm{CoE}\le F_Q$ holds under (41); find a corrected inequality such as $\mathrm{CoE}\le c{F}_Q$ with explicit admissible $c$ or a block-weight term; and determine what holds for other noise models. The nearest surviving statements are the source’s pure lossless theorem and its particular noisy examples, each with its own original hypotheses.

6. Literature — open completeness. The source versions and conference abstract above give the context; the named follow-up screens locate no noisy violation in the searched scope. This is a literature reading with completeness ASSUMED-UNVERIFIED, not a proof that no prior refutation exists.

ASSUMED-UNVERIFIED

Literature completeness, the reported $7\%$ prevalence and the unformalized pure-state numerical extension remain unverified. Full-rank and separability properties are not separate Lean predicates in this module: the initial diagonal entries are positive and form a convex mixture of computational-basis product projectors, which gives those properties by an elementary paper argument. The evolved-point eigenvalues are $1/64,7/64,23/64,33/64$, all positive. The exact family and numerical observations have the experiment’s stated scopes. The kernel settlement is the closed negation of the preregistered universal claim at one smooth positive-damping witness.