Pure-state ECQC fails in dimension five

Abstract

A maximally entangled pure state in dimension five has log₂ 5 bits of mutual information in each of Iqbal’s six measurements, contradicting pure-state ECQC.

Definition 1.1 (The six measurement bases).

$$\forall i:\mathrm{Fin}\left(6\right),\forall a:\mathrm{ZMod}\left(5\right),\forall x:\mathrm{ZMod}\left(5\right),\mathrm{bases}\left(i,a,x\right)=\mathrm{ite}\left(i=1,Pi.single\left(a,1,x\right),\frac{\mathrm{stdAddChar}\left(castZMod5\left(\mathrm{ite}\left(i=0,0,\mathrm{natSub}\left(\mathrm{val}\left(i\right),1\right)\right)\right)\cdot x^2+a\cdot x\right)}{\mathrm{ofReal}\left(\mathrm{sqrt}\left(5\right)\right)}\right)$$

Formalization. D5/S3/Quantum/Information/PureStateECQCRefutation.bases (✓ std3).

Citation. H. Iqbal (2025). On the CQC Conjecture: A sufficient condition and an extension. DOI: 10.1007/s11128-026-05258-2. URL: https://arxiv.org/abs/2509.08286v2.

Commentary.

Iqbal, arXiv v2 pp. 22–23, Dimension 5: “These matrices are generated by the following special matrices, where note that, M_0 itself is the quantum Fourier matrix of order 5, and is one of the MUB bases in this dimension:” D = diag(1, ω, ω⁴, ω⁴, ω), and M_0 = (ω^(xa))/sqrt(5). “Based on these two matrices D and M_0 and remembering that identity matrix, which is denoted by M_1 in the following, is also part of the six MUBs found in this dimension, we generate the four remaining MUB bases:” M_1 = I, M_2 = DM_0, M_3 = D²M_0, M_4 = D³M_0, M_5 = D⁴M_0. The text continues: “where ω = e^(2πi/5) is the fifth root of unity.” The variable i indexes the source’s M_i, and a indexes its column. ZMod 5 is the residue field with representatives 0,…,4; stdAddChar(z) = exp(2πiz/5). castZMod5 is the natural-number cast; natSub is truncated natural subtraction. Pi.single(a,1,x) is the computational column, equal to 1 at x = a and 0 elsewhere. The complex denominator is the real square root embedded in ℂ.

Definition 1.2 (Same-basis joint probabilities).

$$\forall i:\mathrm{Fin}\left(6\right),\forall psi:(\mathrm{ZMod}\left(5\right)\times \mathrm{ZMod}\left(5\right))\to \mathbb{C},\forall ab:(\mathrm{ZMod}\left(5\right)\times \mathrm{ZMod}\left(5\right)),\mathrm{jointLaw}\left(i,psi,ab\right)=\mathrm{normSq}\left(\sum _{xy\in (\mathrm{ZMod}\left(5\right)\times \mathrm{ZMod}\left(5\right))}\mathrm{star}\left(\mathrm{bases}\left(i,\mathrm{fst}\left(ab\right),\mathrm{fst}\left(xy\right)\right)\right)\cdot \mathrm{star}\left(\mathrm{bases}\left(i,\mathrm{snd}\left(ab\right),\mathrm{snd}\left(xy\right)\right)\right)\cdot psi\left(xy\right)\right)$$

Formalization. D5/S3/Quantum/Information/PureStateECQCRefutation.jointLaw (✓ std3).

Citation. H. Iqbal (2025). On the CQC Conjecture: A sufficient condition and an extension. DOI: 10.1007/s11128-026-05258-2. URL: https://arxiv.org/abs/2509.08286v2.

Commentary.

Both parties measure M_i, using its columns without conjugating the second party’s basis. The amplitude conjugates both measured column vectors. normSq(z) = |z|². fst and snd select the coordinates of an ordered pair. This same-basis convention reproduces the paper’s dimension-3 isotropic probabilities p_(g_i g_j) = 1/9 and zero mutual information for the quadratic bases.

Definition 1.3 (The pure-state ECQC assertion).

$$claim\iff (\forall psi:(\mathrm{ZMod}\left(5\right)\times \mathrm{ZMod}\left(5\right))\to \mathbb{C},\forall hpsi:(\mathrm{dotProduct}\left(\mathrm{star}\left(psi\right),psi\right)=1),\mathrm{sInf}\left(\{r:\mathbb{R}|(\exists S:\mathrm{Finset}\left(\mathrm{Fin}\left(6\right)\right),(\mathrm{card}\left(S\right)=5)\wedge (r=\sum _{i\in S}\frac{\mathrm{mutualInformation}\left(\mathrm{jointLaw}\left(i,psi\right)\right)}{\log \left(2\right)}))\}\right)\le \frac{\mathrm{quantumMutualInformation}\left(\mathrm{pureDensityState}\left(psi,hpsi\right)\right)}{\log \left(2\right)})$$

Formalization. D5/S3/Quantum/Information/PureStateECQCRefutation.claim (✓ std3).

Citation. H. Iqbal (2025). On the CQC Conjecture: A sufficient condition and an extension. DOI: 10.1007/s11128-026-05258-2. URL: https://arxiv.org/abs/2509.08286v2.

Commentary.

Iqbal, Closing Remarks and Future Works, p. 24: “For example, the original CQC conjecture holds for all pure states, but the technique that was used to prove this turned out to be inapplicable for the ECQC conjecture. However, as our simulations for dimensions $3$ and $5$ show, there are no observed contradictions of the ECQC for pure states. 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 on p. 6 reads: “Let ${\rho }_{AB}$ be a bipartite state where each subsystem has prime dimension $d$. If both parties make $d+1$ measurements on their individual subsystems, then the mutual information of the resulting random variables will have the following upper bound: $I\left(A:B\right)\ge {\min }_{\mathcal{S}\in [d+1{]}^d}{\sum }_{i\in \mathcal{S}}I\left({\left(M_i\right)}^A:{\left(M_i\right)}^B\right)$.” Its counting convention is: “The elements of the set $\mathcal{S}$ are collections of $d$ mutual information resulting from $d+1$ bipartite MUB measurements.” The six generators are D = diag(1, ω, ω⁴, ω⁴, ω), M_0 = (ω^(xa))/sqrt(5), M_1 = I, and M_(k+1) = D^k M_0 for k = 1,…,4. These are the statement identified as Conjecture 2.1 and the bases identified as §4.5.2; the arXiv PDF labels them Conjecture 3.1 and §4.4.2. Here d = 5 and ψ ranges over all normalized pure amplitudes on ZMod 5 × ZMod 5. The proof hψ witnesses the normalization equation. pureDensityState is the existing positive trace-one rank-one projector; quantumMutualInformation uses its actual partial-trace marginals and von Neumann entropy. mutualInformation is the existing finite classical information, equal to the sum of the two marginal Shannon entropies minus joint entropy. Division by log(2) converts both existing natural-logarithm quantities to bits. Their zero-probability convention is 0 log 0 = 0. The set in sInf contains the sums over every five-element subset of Fin 6; it is finite and nonempty, so this infimum is the source’s minimum.

Theorem 1.4 (A maximally entangled counterexample).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/PureStateECQCRefutation.result (✓ std3). ∎

Resolves. Problems/iqbal-2025-pure-state-ecqc (refuted) by D5/S3/Quantum/Information/PureStateECQCRefutation.result.

Source. Repository-derived.

Acknowledgement. H. Iqbal (2025). On the CQC Conjecture: A sufficient condition and an extension. DOI: 10.1007/s11128-026-05258-2. URL: https://arxiv.org/abs/2509.08286v2.

Commentary.

Take ψ(x,y) = 1/sqrt(5) when y = 2x, and 0 otherwise, with arithmetic modulo 5. In every quadratic-phase basis, 1 + 2² = 0 cancels the quadratic phase; additive-character orthogonality gives amplitude 1/sqrt(5) precisely when a + 2b = 0. This is the same graph b = 2a as for the computational basis. Thus each joint law is uniform on a bijection graph and each classical mutual information is log₂ 5. Both reduced density matrices are I/5. The global rank-one projector has zero entropy, so its quantum mutual information is 2 log₂ 5. Every five-element measurement sum is 5 log₂ 5, which is strictly larger.

References

• Truth anchor: D5/S3/Quantum/Information/PureStateECQCRefutation.bases
• Truth anchor: D5/S3/Quantum/Information/PureStateECQCRefutation.claim
• Truth anchor: D5/S3/Quantum/Information/PureStateECQCRefutation.jointLaw
• Truth anchor: D5/S3/Quantum/Information/PureStateECQCRefutation.result
• Dependency: D5/S3/Entropy/MutualInformationEntropy
• Dependency: D5/S3/Quantum/Dynamics/EnergyEigenstateStationarity
• Dependency: D5/S3/Quantum/Information/InputInformationBalance