slug: iqbal-2025-pure-state-ecqc
bibkey: iqbal2025cqc
doi: 10.1007/s11128-026-05258-2
url: https://arxiv.org/abs/2509.08286v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Information/PureStateECQCRefutation.result

Pure-state ECQC fails in dimension five

Problem

H. Iqbal, On the CQC Conjecture: A sufficient condition and an extension, arXiv:2509.08286v2, Closing Remarks and Future Works, p. 24, asks:

So it would be interesting to develop a method to prove ECQC for pure states analytically, similar to the case of the original CQC conjecture.

The ECQC assertion is

$$I(A:B)\ge \underset{S\subseteq \{0,\dots ,d\},|S|=d}{\min }\sum _{i\in S}I(M_i^A:M_i^B).$$

The source says: “The elements of the set $\mathcal{S}$ are collections of $d$ mutual information resulting from $d+1$ bipartite MUB measurements.” Issue #12896 preregisters its pure-state specialization at prime dimension five, with Shannon and von Neumann entropy in bits and $0\log 0=0$. The arXiv PDF labels the conjecture 3.1, §3.1, p. 6, and the dimension-five bases §4.4.2, pp. 22–23; the preregistration identifies the same quoted content as Conjecture 2.1 and §4.5.2.

Motivation

D5/S3/Quantum/Information/PureStateECQCRefutation.result negates the universal pure-state claim using the source’s six bases. They are $M_0=F$, $M_1=I$, and $M_{k+1}=D^{k}F$ for $k=1,\dots ,4$, where $F_{xa}={\omega }^{xa}/\sqrt{5}$, $D_{xx}={\omega }^{x^2}$ and $\omega =\exp (2\pi i/5)$. Both parties use the same indexed column basis: $p_i(a,b)=|{\sum }_{x,y}\overline{m_{i,a}(x)}\overline{m_{i,b}(y)}\psi (x,y){|}^2$. This convention reproduces the source’s dimension-three isotropic calculation, which has zero mutual information in the two quadratic bases.

Gap

The checked arXiv v2 poses the pure-state question. Wang, Wang and Chen, arXiv:2608.03828v3, refute general ECQC with a rank-two classical–classical mixed state at dimension seven. Their pure-state sentence concerns CQC. Qian, arXiv:2608.14806v2, treats two-basis CQC; its ECQC occurrence is a bibliographic reference. Neither checked source supplies a pure-state ECQC settlement. This literature check has the scope recorded in #12896 and does not assert worldwide priority.

Route

Take the normalized pure vector $\psi =5^{-1/2}{\sum }_{x=0}^4|x⟩|2x\mathrm{mod}5⟩$. Since $1+2^2=0$ modulo five, the two quadratic phases cancel in every noncomputational joint amplitude. Additive-character orthogonality leaves $p_i(a,b)=\frac{1}{5}[b=2a]$ for all six bases. Both marginals are uniform, so each classical mutual information is ${\log }_{2}5$. Both partial-trace density matrices are $I/5$ and the global density is pure; its quantum mutual information is $2{\log }_{2}5$. The minimum sum over five measurements is $5{\log }_{2}5$, giving the strict violation $3{\log }_{2}5$.

Falsifier

The refutation requires normalization, the same-basis measurement law, phase cancellation in all six source generators, uniform marginals, the actual partial traces and a strict comparison. These facts occur on the live path of result. Replacing the second basis by its conjugate changes the assertion; this module does not settle that replacement.

Evidence

The canonical Lean module directly uses existing density-state, partial-trace, classical-information and von Neumann entropy declarations. Its public surface is bases, jointLaw, claim and result : ¬ claim. The sole theorem is honestly classified bind-only and admitted as the preregistered external open-problem-resolution; there are no private helper theorems. The mathematical truth is the kernel-checked declaration and its axiom closure, rather than the numerical computations below.

Triage

Tier 1 published quant-ph question, preregistered in #12896; resolution: Refuted at dimension five with the stated convention. Utility: certified-instance, typed refutes edge to claim. Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

• Proved in this module (PureStateECQCRefutation.result): a maximally entangled pure state can repeat the full classical ${\log }_{2}5$ correlation across all six source measurements. Purity and maximal local mixedness do not prevent the five-measurement sum from exceeding quantum information. The failure mechanism is cancellation of the quadratic phase under the pairing $x\mapsto 2x$, rather than insufficient normalization or a different measurement convention.
• Open for Lean formalization; analytic derivation: for an odd prime $d$ and $u^2=-1(\mathrm{mod}d)$, the vector ${\psi }_u=d^{-1/2}{\sum }_x|x⟩|ux⟩$ has quadratic-basis amplitude $d^{-3/2}\sum_x\omega^{-k(1+u^2)x^2-(a+ub)x} =d^{-1/2}[a+ub=0]$.Thecomputationallawis$d^{-1}[b=ua]$; the linear condition gives the same bijection because $u^2=-1$. Uniform marginals give $d{\log }_{2}d$ on the measurement side and $2{\log }_{2}d$ on the quantum side, a violation $(d-2){\log }_{2}d$. Among odd primes a root exists exactly when $d\equiv 1(\mathrm{mod}4)$. At $d=2$ a root also exists but this family gives equality. The general dimension statement and its number-theoretic condition are separate formalization candidates, not results of this module.
• Computed (python3 /Users/auric/.sshx/4d5197e63c823aead5c96fae/attempt-1/ecqc-numpy.py): exhaustively among the 500 dimension-five states $5^{-1/2}{\sum }_x{\omega }^{c{x}^2+ex}|x⟩|ux+v⟩$ ($u=1,2,3,4$; $v,c,e=0,\dots ,4$), exactly 50 are perfectly correlated in all six bases, all with $u\in \{2,3\}$. The Schmidt-rank-two state $(|00⟩+|11⟩)/\sqrt{2}$ violates by 0.1047442234152 bits under both same-basis and conjugate-basis conventions. The latter numerical observation is not a theorem of this module.
• Computed (same command, NumPy RNG seed 1, interleaved real/imaginary Gaussian draws, 20,000 normalized dimension-five vectors): zero sampled Haar violations; minimum quantum-minus-classical gap 0.3320889666305926 bits. This finite sample did not detect the displayed structured witness; it is not a universal statement about Haar-random states. The finite structured family has Haar measure zero; no zero-measure claim is made for the full violating set.
• Computed (python3 /Users/auric/.sshx/4d5197e63c823aead5c96fae/attempt-1/ecqc-perturb.py): the normalized family $\sqrt{1-t^2}\psi +t|0,1⟩$ still violates at $t=0.01,0.1,0.25,0.5$, respectively by 6.9591076771894596, 6.564445676791794, 5.137460533580348 and 1.9390574282495043 bits. Open for formalization: entropy continuity makes the strict comparison persist on a neighborhood and would imply positive Haar measure of the violating set; that neighborhood statement is not proved in this module.
• Computed / open (NumPy command above): in dimension three the 54 affine quadratic-phase states and 20,000 random maximally entangled states (seed 5) show no violation. Their minimum gaps are 1.584962500721156 and 1.2642168569856405 bits. The dimension-three universal pure-state question remains open. In dimension seven, 28,000 sampled aligned Schmidt states (seed 2; 4,000 per rank) include a rank-four violation of 1.7635455647231533 bits; its exact Lean settlement remains open.
• Nearest surviving statements and source dependencies: the numerical dimension-three observations remain finite-search results. The source’s pure-state two-measurement CQC result is a different assertion and is not refuted here. Any proposed general pure-state ECQC proof or consequence using ECQC needs an added hypothesis, a restricted dimension range or a weaker bound. This module does not invalidate independently proved sufficient conditions or classify all such replacements. The earlier mixed-state dimension-seven refutation settles general ECQC; this result separately answers the source’s pure-state question.

ASSUMED-UNVERIFIED

The journal full text is unverified; the formal statement is tied to arXiv v2. The finite numerical searches and general-dimension derivation are not kernel-verified theorems. Dimension three, sharp replacement bounds, classification of all violating states and the general-dimension/continuity formalization candidates remain open. The numerical programs are host-side experiments whose commands and parameters are given above; they do not enter Lean’s trust boundary.