State-independent entanglement extraction: exact exponents and Gaussian threshold — analytic preprint v1

Nine-page analytic preprint by Yongxian Zhang: State-independent entanglement extraction: exact exponents, reliability and a Gaussian threshold.

Precise task and universal quantifiers

For every fixed pair 2 <= m <= n, construct one family of global unitaries on the original k-copy system followed by deterministic local product channels. The family is indexed only by m,n,k and requested target dimension K; it is independent of the input spectrum, eigenbasis, rank and entropy, and works simultaneously for every fixed density matrix. There is no preliminary tomography, measurement, additional pre-unitary ancilla, state-dependent random choice, or postselection. The initial global entangling operation is an explicit resource.

The family attains the same all-rate target-fidelity exponent as the state-aware completely-PPT optimum. For A=log m, L=log(mn), c_R=L-2 min(R,A):

E_p(R) = (R-A)_+ + max_{1<=alpha<=2} [(alpha-1)/alpha] (S_alpha(p)-c_R).

The faithful capacity is C=min{A,(L-H(p))/2}. Eventual-exact extraction has capacity C0=min{A,(L-log rank(rho))/2}. The fidelity convention is overlap with the normalized maximally entangled projector, not its square root.

New strict-direct and Gaussian laws

For C0 < R < C, set c=L-2R. The exact error-decay exponent is

lim -(1/k) log(1-F_k) = min_{H(q)>=c} D(q||p),

both for the universal local construction and for the state-aware completely-PPT optimum. Relative entropy is on the positive support. This strict range is essential; exact zero-error transition behavior is not obtained by continuity.

If 0 < C < log m and V(p)=Var_p(-log p)>0, then

log K_k = k C + z sqrt(k) + o(sqrt(k))
  implies F_k -> Phi(-2z/sqrt(V(p))).

This gives the same nondegenerate Gaussian threshold without any state knowledge. The protocol does not need C or V: it takes the requested integer K as its parameter.

Mathematical mechanism and attribution

A single simultaneous Schur-Weyl packing retains the entire unknown representation factor and a known fraction of its identity multiplicity factor in every block. It does not select a block after seeing the input. Large blocks retain enough mass for the optimal fidelity exponent, while small blocks are extracted exactly and preserve the reliability exponent. A universal dominating state reduces the Gaussian statement to ordinary iid self-information, without assuming a new Schur-measure central limit theorem.

Standard Schur-Weyl duality, Weyl/hook dimensions and highest-weight character bounds are explicit external mathematical inputs. Universal compression, pure-state concentration, universal ordinary mixed-state LOCC distillation, source-coding asymptotics, and state-agnostic work extraction are credited. Rains's fidelity-effect constraints and Lami's entropy converse are not claimed as new discoveries. The ordinary CLT is used for the Gaussian conclusion.

Verification and source binding

This is a written analytic argument, not Lean certification or external peer review. Independent ancillary CI run 38034511091 passed at 45c677c89cef268033b6d6c68c3e44eeb20a0b1e. It checks exact combinatorial and rational identities, noncommuting inputs, coherent code dyads, multiplicity truncation and packing. Two unchanged checker runs pass, including assertions disabled; nine modified programs are rejected for specified mathematical diagnostics. Finite regression counts do not certify the full unbounded argument.

Publication commit: f465a11c55101fd38a10046daed65feeef007eeb.
Repository directory: research/universal-schur-extraction.
CI: https://github.com/mxym/math/actions/runs/38034511091

From the extracted source archive:

python3 verify_sources.py
python3 check.py
python3 -O test_checker.py

The checks use only Python's standard library. Compile main.tex twice with pdfLaTeX, or rebuild from PROOF.md using build_manuscript.py with Pandoc and pdfLaTeX. The archive contains the full proof, manuscript, programs, source hashes and literal CI records. PREPRINT_BINDING.json and SHA256SUMS bind the assets to the tested and published sources.

Limits

This does not solve ordinary fixed-input LOCC distillation or its universal reliability questions: free global entangling preprocessing changes the task. No claim is made for uniform finite-block convergence over varying states, exact zero-error-boundary reliability, cap-limited/zero-varentropy Gaussian laws, efficient circuits, APPT=AS, or the finite-copy higher-dimensional APPT purity conjecture. No arXiv submission, new DOI, external peer review or exhaustive priority certification is asserted. The earlier immutable qutrit, negativity and extraction releases are unchanged. Funding and AI-assistance disclosures appear in the manuscript.