Non-tightness of the antidistinguishability overlap bound

Johnston, Russo and Sikora, “Tight bounds for antidistinguishability and circulant sets of pure quantum states”, Quantum 9, 1622, arXiv:2311.17047v2, Section 6, asks:

We still do not know whether or not Corollary 5.4 is tight when n ≥ 5.

Corollary 5.4 guarantees antidistinguishability when every distinct-state overlap is at most $c_n=\sqrt{(n-2)/(2(n-1))}$. Issue #15573 specifies the finite-dimensional unit-state convention and the squared-overlap margin. The settlement proves non-tightness for every $n\ge10$.

Statement echo

| Source condition | Formal counterpart | Scope | | --- | --- | --- | | A set of $n$ pure states | $\psi:\operatorname{Fin}(n)\to\mathbb C^d$ | Every finite ambient dimension $d$ | | Normalized states | $|\psi_i|=1$ for every $i$ | No dimension or distinctness restriction | | POVM with zero probability for the corresponding state | IsAntidistinguishable | Positive semidefinite effects, identity sum, zero Born quadratic form | | Tightness of the overlap guarantee | IsTight | Every positive squared-overlap margin admits a non-antidistinguishable family | | Positive source cap | $c_n>0$ for $n\ge10$ | Squaring reparametrizes positive margins: $(c_n+\varepsilon)^2-c_n^2>0$ and $\sqrt{c_n^2+\delta}-c_n>0$ | | Non-tightness | claim and result | Every $n\ge10$; cases $5\le n\le9$ remain open |

The positive margin $\delta_n$ is additional proved information, rather than a hypothesis strengthening the source question. The displayed finite-dimensional POVM definition is literal. The Gram construction is reused as Matrix.gram, and the squared Frobenius norm is reused as RHLinalg.frobSq.

D5/S3/Quantum/Measurement/AntidistinguishabilityOverlapNonTight.result proves the universal non-tightness claim. Its quantitative theorem gives a uniform guarantee for arbitrary unit-state ensembles, including non-circulant Gram matrices.

The literature screen recorded in #15573 covers the source, its later restatement and citing works. It found no settlement within that scope. The revised Hillery–Andersson–Puthoor published body is ASSUMED-UNVERIFIED: its accepted manuscript is embargoed until 2026-10-17. The previously available arXiv body does not settle the arbitrary-ensemble guarantee.

The proof constructs a POVM from an incoherent positive semidefinite Gram decomposition (AntidistinguishabilityGram.incoherent_gram_antidistinguishable). Closedness and separation of that cone (IncoherentCone.incoherent_closed, incoherent_separation) combine with coordinate-hyperplane positivity and its reciprocal spectral budget (LocallyPositiveSpectrum.local_psd_kernel_eq_zero, local_psd_negIndex, local_psd_reciprocal_budget). The source spectral sufficient condition is AntidistinguishabilitySpectrum.spectral_sufficient_antidistinguishable.

Failure of the spectral condition constrains the fourth eigenvalue and rank-three residual (AntidistinguishabilityOverlapNonTight.spectral_failure_fourth_bound, topThree_failure_distance). Squaring the entries of the rank-three approximation gives rank at most nine (topThree_square_rank). A full-rank equal-overlap matrix has a positive distance from every such matrix (OverlapLowRankDistance.low_rank_distance_lower, equalOverlap_low_rank_distance). The quantitative spectral condition then follows by contradiction.

A unit-state family with $n\ge10$, squared overlaps at most $c_n^2+\delta_n$, and no corresponding POVM would contradict quantitative_antidistinguishability. A source clause not represented by the statement echo would invalidate the claimed correspondence. A prior publication establishing this exact arbitrary-ensemble improvement would invalidate a priority claim; the bounded literature screen does not establish priority beyond its checked scope.

Kernel-checked declarations

The five supporting modules and AntidistinguishabilityOverlapNonTight carry the formal proof. quantitative_antidistinguishability has the complete unit-norm and overlap hypotheses. result proves claim. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$.

Experiment

The numerical check is pinned to commit d3fce0cdf5066a77d2084a1ab9b8ec68018f674e. Command: python3 check.py, exit 0, final reading bad= 0. Script SHA-256: 96e3a911084ab29d6c727a6842e1d734be97b120845bbec1cf10ef6079830610.

Computed scope: the Lemma A1 identity on 60 random rational spectra; the final inequality's closed form and strict sign for $10\le n\le400$; rank exactly nine for one random rank-three complex Gram matrix at $n=20$; and 747 admissible Gram matrices at $n=10,12,16$, with zero spectral-condition violations. The experiments establish those readings only; the uniform theorem is supplied by Lean.

What the settlement shows

Mechanism — kernel-checked. Write $\Delta=|G|_F^2-n^2/2$. Under spectral failure, AntidistinguishabilityOverlapNonTight.topThree_failure_distance gives $|G-G_3|_F\le\sqrt{(n-3)\Delta}/2$. norm_square_frob_distance_le gives $||G|^2-|G_3|^2|_F\le\sqrt{(n-3)\Delta}$, and topThree_square_rank gives rank at most nine for $|G_3|^2$. OverlapLowRankDistance.equalOverlap_low_rank_distance gives its distance lower bound from $H_0=(1-c_n^2)I+c_n^2J$. The coefficient satisfies $1-c_n^2>1/2$ by cSq_lt_half; this is the lower spectral bound used in the distance argument. These estimates force the spectral sufficient condition under the improved overlap cap.

Quantitative statement — kernel-checked. AntidistinguishabilityGram.delta defines

$$\delta_n=\frac{n-9}{16n(n-1)(n-3)}.$$

OverlapLowRankDistance.delta_pos proves positivity for $n\ge10$. AntidistinguishabilityOverlapNonTight.quantitative_antidistinguishability proves antidistinguishability whenever $|\langle\psi_i,\psi_j\rangle|^2\le c_n^2+\delta_n$. result therefore proves non-tightness for every $n\ge10$.

Remaining finite cases — open. The source question for $n=5,6,7,8,9$ is not settled by these declarations.

Asymptotic threshold — proved in prose only. Define $T_n$ as the infimum of the largest absolute overlap among non-antidistinguishable unit-state families. For any fixed $0<\varepsilon<1$, suppose all pairwise absolute overlaps are at most $1-\varepsilon$. If the family is non-antidistinguishable, the contrapositive of the spectral sufficient condition gives $\sum_k\sqrt{\lambda_k}<2\sqrt n$. Choose an integer $r\ge64/\varepsilon$. For $n>r$, put $s_k=\sqrt{\lambda_k}$ in decreasing order. Then $s_r\le(\sum_k s_k)/r$ and

$$\sum_{k\ge r}\lambda_k\le s_r\sum_{k\ge r}s_k<\frac{4n}{r}.$$

The singular-value decomposition of the state synthesis map identifies its squared singular values with the Gram eigenvalues; zero padding covers ambient dimension below $r$. Projection onto the top $r$ state-space spectral directions has total squared residual less than $4n/r$. At least half the states have squared residual at most $8/r\le\varepsilon/8$. Any two original states have Euclidean distance at least $\sqrt{2\varepsilon}$ because the real part of their inner product is at most its absolute value. Their projections therefore have distance at least $\sqrt{2\varepsilon}-2\sqrt{\varepsilon/8}=\sqrt{\varepsilon/2}$. They lie in the unit ball of $\mathbb R^{2r}$. Disjoint balls of radius $\sqrt{\varepsilon/2}/2$ centered at those projections lie in the ball with radius $1+\sqrt{\varepsilon/2}/2$; comparison of Euclidean volumes bounds their number by $(1+2\sqrt{2/\varepsilon})^{2r}$. Thus $n/2$ is bounded by this number. For larger $n$ every such family is antidistinguishable, so $T_n\ge1-\varepsilon$. Identical unit states are non-antidistinguishable and have maximum overlap one, so $T_n\le1$. Hence $T_n\to1$. This argument is a prose derivation; no Lean declaration in this delivery states the limit.

Literature boundary — ASSUMED-UNVERIFIED. The revised Hillery–Andersson–Puthoor body remains unchecked for the stated embargo reason. The literature scope and this boundary are recorded in #15573.

Numerical readings — computed. The pinned experiment entry in Evidence supplies exactly the finite scopes listed there, including bad= 0. Its rank-nine reading is for one sampled matrix; the universal rank-at-most-nine bound is the separate kernel-checked theorem topThree_square_rank.

The source's circulant necessity statement and its sharp Frobenius bound are unaffected mathematical results. The non-tightness proof supplies an improved sufficient overlap guarantee for arbitrary ensembles; it does not claim a full classification of non-antidistinguishable sets.

Escape audit

The escape audit is unfinished under #15730. Dimension-dependent matrices, proof-dependent spectral maps, guarded inequalities and existential POVM decompositions still require faithful source-family reconstructions, lawful template readouts, selection coordinates, whole-family variation, slot sensitivity and observational-dependence evidence. The settling Reg mirror is a typed candidate; compilation alone does not establish a complete registration. The settling candidate decodes as state=declared_unresolved with IE-C050 ClosedTruthReadout, diagnostic unclassified_form, rule source.residual_requires_open; it has no source-bound escape certificate. The concrete gap is the exact dependent source-family reconstruction, lawful template readout, selection coordinates, whole-family variation, slot sensitivity and observational dependence evidence for the theorem family. The unfinished audit is tracked in #15730.

The revised Hillery–Andersson–Puthoor published body is not checked. The asymptotic threshold derivation is proved in prose, with Lean formalization open. Escape registration evidence is incomplete as specified in #15730. No exhaustive literature-priority assertion is made.