Quantum information researchers have proposed a new route toward solving one of the field’s most persistent open problems: the CQC conjecture, a mathematical statement about how much information two quantum systems can retain after being measured in incompatible ways. In a newly published study, Hasan Iqbal of the University of Wyoming identifies a sufficient condition that guarantees the conjecture is correct for a broader class of quantum states. The work also proposes an extension involving many mutually unbiased bases and higher-dimensional systems, and reports numerical tests showing no contradiction in thousands of randomly generated examples. Although the conjecture itself remains unproven in full generality, the results offer new mathematical tools for studying quantum correlations, uncertainty, entanglement detection and the security of quantum communication.
The CQC conjecture concerns a tension at the heart of quantum mechanics. Two parties, commonly labelled Alice and Bob, may share a quantum state containing correlations that cannot be described entirely as ordinary classical information. When both measure their systems, however, the outcomes become classical data. The conjecture states that if Alice and Bob measure their systems in two mutually unbiased bases, the sum of the classical mutual information obtained from those two experiments cannot exceed the original quantum mutual information shared by the systems. In symbols, the conjecture is written as (I(Z^A:Z^B)+I(X^A:X^B)leq I(A:B)). Here, (Z) and (X) represent incompatible measurements, while (I(A:B)) quantifies the total correlations in the original quantum state. The challenge is that measurement can reveal different aspects of a quantum state, raising the possibility that correlations extracted in separate experiments might collectively appear larger than the correlations present before measurement.
Mutually unbiased bases are central to the problem because they represent maximally complementary measurement choices. If a particle is prepared in one basis, a measurement in a mutually unbiased basis produces outcomes with equal probability. In a (d)-dimensional system, the computational basis can be paired with a Fourier basis, whose vectors are coherent superpositions of all computational states. Measuring in one basis can make the outcome of the other completely unpredictable. This complementarity underlies entropic uncertainty relations, which place lower bounds on the combined uncertainty associated with incompatible measurements. Unlike simple uncertainty statements about position and momentum, the CQC conjecture tracks mutual information between two systems, making it sensitive to both local randomness and shared correlations.
The conjecture was introduced more than a decade ago by researchers studying uncertainty relations for mutual information. It has already been established for several important families of states, including pure states, states with one maximally mixed subsystem and situations in which one of the measurements is minimally disturbing. It also has potential practical consequences. If the conjecture is correct, unusually large classical correlations observed in two incompatible measurement settings can serve as evidence of entanglement. The result may also strengthen uncertainty relations involving quantum memories and constrain how much information an eavesdropper can obtain in quantum key distribution. In cryptographic settings, Alice and Bob can use correlations between their measurement outcomes to establish a secret key, while the conjecture would help bound the information available to an adversary.
Iqbal’s first contribution is a sufficient condition derived from an information-exclusion result developed by researchers Patrick Coles and Marco Piani. That earlier result limits the combined mutual information Alice can obtain about Bob’s quantum system when she measures in two mutually unbiased bases. In simplified form, it states that (I(Z^A:B)+I(X^A:B)) cannot exceed (log d-H(A|B)), where (d) is the system dimension and (H(A|B)) is the quantum conditional entropy. The new condition compares the loss of information caused by measuring Bob’s quantum system with the loss caused by converting it into classical outcomes. If the decrease from quantum memory to classical measurement is sufficiently large, the original CQC inequality follows automatically. This does not prove the conjecture for every state, but it identifies a measurable structural feature that guarantees its validity.
The condition is especially interesting because it applies beyond the examples already known to satisfy CQC. Iqbal reports numerical demonstrations using random mixed, separable two-qubit states that are neither pure nor equipped with a maximally mixed subsystem. These states were selected to satisfy the new mathematical criterion. For each state, the researchers compared the original quantum mutual information with the sum of the two classical mutual informations produced by incompatible measurements. The difference remained non-negative in the simulations, meaning the classical information extracted from the two measurement settings never exceeded the total quantum correlation. Such numerical evidence cannot replace an analytical proof, but it illustrates how the sufficient condition can identify previously inaccessible regions of the space of quantum states.
The study then advances a broader proposal called the extended CQC, or ECQC, conjecture. Instead of using only two mutually unbiased bases, the extension considers all (d+1) mutually unbiased bases available in prime dimensions and asks whether the original quantum mutual information is at least as large as the sum of the (d) smallest classical mutual informations generated by those measurements. Excluding the largest term is essential to the proposed formulation: summing every available basis can produce a quantity that is too strong to be universally plausible, whereas selecting all but the most informative measurement creates a more balanced comparison. Prime dimensions such as three and five are particularly useful because complete sets of (d+1) mutually unbiased bases are known to exist there.
The author also derives a sufficient condition for ECQC using a multiple-measurement entropic uncertainty relation. This relation connects the sum of conditional entropies from (d+1) measurements to both the dimension of the system and the quantum conditional entropy (H(A|B)). The resulting criterion compares the total information available when Alice measures while Bob retains a quantum memory with the information remaining after Bob also measures. If the reduction is large enough, the extended conjecture follows. The analysis further produces a bipartite generalization of the Maassen–Uffink uncertainty relation, placing a lower bound on the combined joint entropies of Alice’s and Bob’s measurement outcomes. In physical terms, the more complementary measurements the parties perform, the more uncertainty must appear in their combined classical records.
One of the most detailed tests involves isotropic states, a family that mixes a maximally entangled state with completely mixed noise. These states are described by (rho{AB}=p|Psi^+ranglelanglePsi^+|+(1-p)mathbb{I}{AB}/d^2), where (p) controls the weight of the entangled component. Their individual subsystems remain maximally mixed for the full allowed range of (p), while their shared quantum mutual information changes continuously with the noise level. The calculations show that, in the chosen complete sets of mutually unbiased bases, two particular measurements retain nonzero classical mutual information, while the other measurements produce uniformly distributed joint outcomes and therefore zero mutual information. For prime dimensions, the author derives an explicit expression for the two nonzero contributions and shows that their sum remains below the original quantum mutual information. This establishes ECQC for isotropic states across the examined prime-dimensional family, including states that are entangled and states that are separable.
The numerical investigation extends beyond isotropic states. For dimension three, the researchers tested 100,000 random pure bipartite states and 100,000 random mixed bipartite states using four mutually unbiased bases, then compared the quantum mutual information with the sum of the three smallest classical mutual informations. No violation was observed. Additional tests examined the isotropic family across its entire physical parameter range. In dimension five, a more computationally demanding study used 10,000 random pure states and 10,000 random mixed states with six mutually unbiased bases. Again, removing the largest classical mutual information and summing the remaining five produced no value greater than the original quantum mutual information. These experiments are best understood as evidence supporting the conjecture rather than proof: random sampling cannot rule out rare counterexamples, and the structure of the chosen bases may influence the numerical outcome.
The findings arrive with important limitations and an open invitation to the quantum information community. The original CQC conjecture remains unresolved for arbitrary mixed states, and the extended version is even less established. The proposed sufficient conditions cover only states satisfying specific inequalities, while the numerical simulations explore finite samples and selected prime dimensions. Composite dimensions pose an additional obstacle because the maximum number of mutually unbiased bases is not known in general, and some available collections can behave differently. The next major goal is an analytical proof of ECQC for all pure states, followed by a formulation that works in both prime and composite dimensions. If those challenges can be overcome, the conjecture could become a powerful bridge between quantum correlations, uncertainty and communication security, turning incompatible measurements into a practical diagnostic for the hidden information structure of quantum matter.
Subject of Research: Quantum mutual information, entropic uncertainty relations, mutually unbiased bases, quantum correlations and entanglement
Article Title: On the CQC conjecture: a sufficient condition and an extension
Article References: Iqbal, H. “On the CQC conjecture: a sufficient condition and an extension.” Quantum Information Processing 25, Article 250 (2026). Foundational references include Schneeloch, Broadbent and Howell, “Uncertainty relation for mutual information,” Physical Review A 90, 062119 (2014); Coles and Piani, “Improved entropic uncertainty relations and information exclusion relations,” Physical Review A 89, 022112 (2014); and Berta et al., “The uncertainty principle in the presence of quantum memory,” Nature Physics 6, 659–662 (2010).
Image Credits: AI Generated
DOI: 10.1007/s11128-026-05258-2
Keywords: quantum mutual information, CQC conjecture, ECQC conjecture, entropic uncertainty relations, mutually unbiased bases, quantum entanglement, quantum correlations, isotropic states, quantum information theory, quantum cryptography
Tags: CQC conjecturehigher-dimensional quantum systemsmathematical conditions for quantum correlationsmutually unbiased basesquantum communication securityquantum correlationsquantum entanglement detectionquantum information theoryquantum measurement incompatibilityquantum mutual informationquantum state measurementquantum uncertainty
