bibkey: iqbal2025cqc authors: H. Iqbal year: 2025 title: “On the CQC Conjecture: A sufficient condition and an extension” doi: 10.1007/s11128-026-05258-2 url: https://arxiv.org/abs/2509.08286v2 claim: “ECQC bounds the smallest sum of d classical mutual informations from d+1 bipartite MUB measurements by quantum mutual information; the closing remarks ask for an analytic pure-state proof.” strata_touched:

• D5/S3/Quantum/Information/PureStateECQCRefutation license: citation-only triage: anchor

Pure-state ECQC

H. Iqbal, On the CQC Conjecture: A sufficient condition and an extension, arXiv:2509.08286v2; Quantum Information Processing 25, 250 (2026).

The arXiv PDF, p. 6, Conjecture 3.1, §3.1, states:

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:

$$I(A:B)\ge \underset{\mathcal{S}\in [d+1{]}^d}{\min }\sum _{i\in \mathcal{S}}I(M_i^A:M_i^B).$$

The immediately following sentence fixes the counting convention:

The elements of the set $\mathcal{S}$ are collections of $d$ mutual information resulting from $d+1$ bipartite MUB measurements.

The Closing Remarks and Future Works, p. 24, ask:

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 dimension-5 construction, pp. 22–23, §4.4.2, reads:

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:

$$D=\mathrm{diag}(1,\omega ,{\omega }^4,{\omega }^4,\omega ),M_0=\frac{1}{\sqrt{5}}({\omega }^{xa}{)}_{x,a=0}^4.$$

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:

$$M_1=I,M_2=D{M}_0,M_3=D^2{M}_0,M_4=D^3{M}_0,M_5=D^4{M}_0.$$

where $\omega =e^{\frac{2\pi i}{5}}$ is the fifth root of unity.

The formal encoding uses these generators with indices in $\mathbb{Z}/5\mathbb{Z}$. The standard additive character is exactly $x\mapsto \exp (2\pi ix/5)$. Both parties use the same indexed basis, with joint probability $|{\sum }_{x,y}\overline{m_{i,a}(x)}\overline{m_{i,b}(y)}\psi (x,y){|}^2$. This convention agrees with the paper’s dimension-3 isotropic calculation: the quadratic bases give $p_{g_i{g}_j}=1/9$ and zero mutual information. The existing quantum and classical information definitions use natural logarithms; division by $\log 2$ expresses both quantities in bits, including the zero-probability convention.

Verified locator

• URL: https://arxiv.org/abs/2509.08286v2. The PDF locators are Conjecture 3.1, p. 6; the dimension-5 generators in §4.4.2, pp. 22–23; and the closing question, p. 24. The source statement is also identified as Conjecture 2.1 and the bases as §4.5.2 in the problem’s preregistration; the quoted content is the same.
• DOI: https://doi.org/10.1007/s11128-026-05258-2. The journal full text is unverified; the formal claim concerns the arXiv v2 statement and generators.