slug: jmpp-2022-absolute-two-incoherence-purity
bibkey: johnstonmoeinpereiraplosker2022incoherence
doi: 10.1103/PhysRevA.106.052417
url: https://arxiv.org/abs/2205.05110v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Measurement/AbsoluteTwoIncoherencePurityRefutation.result

Absolute two-incoherence beyond the purity bound

Problem

Johnston, Moein, Pereira and Plosker, Absolutely k-incoherent quantum states and spectral inequalities for factor width of a matrix, Phys. Rev. A 106, 052417, arXiv:2205.05110v2, Section 5, printed page 17:

Theorem 7 gives a sufficient condition for absolute 2-incoherence that is also necessary in dimension n = 3. It is unclear whether or not it is necessary in dimension n = 4 or higher.

The condition is $\mathrm{Tr}({\rho }^2)\le 1/(n-1)$. The universal necessity assertion for all $n\ge 4$ is refuted by one density matrix in dimension five. The source and literal definitions are in the Library note; the external question is preregistered in #17863.

Statement echo

FactorWidthTwoCone.IsLocally2 is the Hermitian support-two quadratic-form condition in Definition 1(b). It is an auxiliary dual-cone predicate, not an extra hypothesis in claim or result.

Motivation

The purity ball in Theorem 7 is sufficient for absolute two-incoherence. Its boundary need not be the full boundary of that set. The settlement identifies a state outside the ball whose entire unitary orbit remains in the factor-width-two cone.

Gap

The missing necessity fails because the orbit of a particular trace-one, unit-Frobenius-norm Hermitian matrix is separated from the trace-one locally two-positive matrices. The separation leaves room for an affine perturbation beyond the purity threshold. The assertion here concerns the source’s universal necessity question; neither necessity at $n=4$ nor a uniform formal family over all $n\ge 5$ is settled.

Route

Put $r=\sqrt{6}$, $a=(1+r)/5$, $b=(1-2r/3)/5$, and

$$B=\mathrm{diag}(a,a,b,b,b),{\rho }_t=\frac{1+t}{4}(I-B)-\frac{t}{5}I.$$

The settling proof uses $t=10^{-18}$. The private lemmas rho_density and rho_purity establish density and

$$\mathrm{Tr}({\rho }_t^2)=\frac{1}{5}+\frac{(1+t{)}^2}{20}.$$

For $t>0$ this is strictly greater than $1/4$. With $\epsilon =10^{-8}$, gap_n5 proves a uniform trace-pairing gap ${\epsilon }^2/2$ against every trace-one locally two-positive matrix and every unitary conjugate of $B$.

A hypothetical smaller gap forces a squared-modulus Gram matrix $H$ of rank at most four close to the full-rank equal-overlap matrix with off-diagonal entry $3/8$. H_distance_lower gives squared Frobenius distance at least $25/64$, whereas H_distance_upper gives at most $10^{-6}$. Their contradiction proves the gap. FactorWidthTwoCone.fw2_duality, whose proof uses compactness of finite support decompositions, then converts the nonnegative dual pairings into sparse outer-product sums for every unitary conjugate of ${\rho }_t$.

Falsifier

The exact result would fail if the displayed state had a unitary conjugate outside the factor-width-two cone or did not exceed purity $1/4$. The kernel checks both required properties through witness_absolute, rho_purity, and their use in result. Numerical optimization supplies no premise of these proofs.

Evidence

The settling GID is D5/S3/Quantum/Measurement/AbsoluteTwoIncoherencePurityRefutation.result, with admission_basis: open-problem-resolution (#17863; Refuted). The supporting module is D5/S3/Quantum/Measurement/FactorWidthTwoCone, with admission_basis: escape-witness supplied by block_closed on the proof path to fw2_duality. Existing owner statements are reused at their original declarations in IncoherentCone and AntidistinguishabilityOverlapNonTight.

The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9).

The six supporting public theorems and eight published owner theorems have an unfinished four-slot escape audit in #18980. No validated registration is claimed without compiled registration evidence and its required-CI reading.

Experiment

Numerical checker and results, pinned to cb4521985393e13d9517583014c2d21a6e82873f.

The checked program is check.py, SHA-256 5cd6f44dd762913ea53d6073e94f2571525a6cf0a2bbe04cb6d36293dce7588a. The executed command is /Users/auric/.sshx/6f7f08e6d50b6f8f47ab13e8/attempt-1/experiment-venv/bin/python -u /tmp/op-jmpp/check.py; its input is exactly the linked commit’s check.py, and its exit code is 0 (ALL_OK). The environment has NumPy 2.5.4, CVXPY 1.9.3 and Clarabel 0.11.1, with seed 20261017.

Numerical optimization is finite sampling and local search, not a universal orbit certificate. The solver emits a possible-inaccuracy warning; the displayed optimization readings are approximate. They do not certify necessity or non-necessity at $n=4$, an optimal gap, or an all-dimension family. The exact formal witness uses $t=10^{-18}$, independently of this experiment.

Triage

What the settlement shows

ASSUMED-UNVERIFIED

Literature completeness, worldwide priority, and overlap with unpublished frame or boundary-spectrum work are not established. The follow-up papers named in #17863 are seat-reported literature readings. The numerical checker’s local optima and sampled decompositions prove no statement about untested unitary matrices or dimensions. Dimension four, a formal all-$n\ge 5$ family, and optimal perturbation constants remain open in this delivery. The four-slot escape audit remains unfinished under #18980.