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arXiv:2608.14806

Separable Counterexamples to Complementary Quantum Correlations, and Why Random Search Missed Them

Lilong Qian

quant-phSubmitted Aug 14, 2026• Announced Aug 18, 2026• 0 citations
v1 · Aug 14, 2026v2 · Aug 19, 2026
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Abstract

The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair \(m\times n\) with \(m,n\ge3\), with closed-form excess at least \(1/(8m^2n^2)\) nats, and in every qubit--qudit pair \(2\times n\) with \(n\ge3\) except \(n=3,5\); every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound \(S\le8/3\) to \(S\le3.8265583\ldots\); the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity \(X=1-4\operatorname{Var}(c)\) turns the prime-Fourier full-spark barrier into a variance bound. A \(128\)-bit interval cover then closes that family at \(2\times3\) with gap at least \(0.012021\) nats, and at \(2\times5\) an exact saturator attaining \(S=(14+2\sqrt5)/5\) refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at \(2\times2\). Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies \(7.3\) standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees (\(\sim10^{-21}\) of frames), and extrapolating the sample minimum demands \(10^{10}\) to \(10^{18}\) samples against the \(10^7\) ever run.

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The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair (m\times n) with (m,n\ge3), with closed-form excess at least (1/(8m^2n^2)) nats, and in every qubit--qudit pair (2\times n) with (n\ge3) except (n=3,5); every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound (S\le8/3) to (S\le3.8265583\ldots); the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity (X=1-4\operatorname{Var}(c)) turns the prime-Fourier full-spark barrier into a variance bound. A (128)-bit interval cover then closes that family at (2\times3) with gap at least (0.012021) nats, and at (2\times5) an exact saturator attaining (S=(14+2\sqrt5)/5) refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at (2\times2). Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies (7.3) standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees ((\sim10^{-21}) of frames), and extrapolating the sample minimum demands (10^{10}) to (10^{18}) samples against the (10^7) ever run.
Mutually unbiased measurements are commonly expected to expose independent facets of a quantum state: a correlation that is classical in one basis should disappear in a complementary basis. In higher dimensions, however, this intuition becomes particularly subtle because correlations recovered in different settings need not represent different information. To expose this loophole, we propose a two-branch classical null test: before the setting is chosen, a shared bit selects one of two orthogonal product preparations, producing a rank-two classical--classical state, and the candidate protocol then runs unchanged. Different settings can read the same bit through different outcome patterns. This two-branch classical architecture disproves the complementary-quantum correlation (CQC) conjecture in every dimension d≥3d\geq3d≥3. A distinct rank-two classical--classical state disproves its complete-basis extension (ECQC) at d=7d=7d=7, with an overrun that grows without bound along prime dimensions. Its qutrit CQC instance also gives classical false positives for a proposed quantum-correlation measure and a proposed one-sided semi-device-independent steering criterion, and refutes a conditional-probability conjecture. The failures identify the missing requirement: information read in different settings must be nonredundant. In experiments and applications, the same low-overhead architecture can serve as a calibration test before a multibasis score is assigned quantum meaning.
Extracting total correlations from a quantum system usually requires reconstructing its state, whereas many experiments access only a few measurement settings. A possible shortcut is to add the mutual informations obtained from complementary measurements; in dimensions above two, however, this procedure can count the same classical correlation twice. We establish that qubits are protected from such overcounting. For every two-qubit state, the correlations observed in two complementary local bases are bounded by the premeasurement quantum mutual information. The proof traces this protection to binary-entropy curvature on the Bloch ball and combines a qubit information-exclusion tradeoff with data processing under local dephasing. Consequently, two correlation tables give a tomography-free lower bound on total correlation. A score above one bit also certifies a quantitative one-way entanglement-distillation rate; when applied to the Choi state of a qubit channel, the same data lower bound its quantum capacity. The theorem therefore identifies both an operational use of complementarity and the trusted two-dimensional setting in which its correlation accounting is valid.
Wu et al. introduced a symmetric measure QsQ^{s}Qs in terms of mutually unbiased bases (MUBs) that quantifies residual quantum correlations (RQC). The definition of this measure involves determining two complementary local bases via an optimization of the quantum mutual information. One defines the first local bases, the optimal computational ones, by maximizing the classical correlations measure. Next, one defines a new set of local bases that are complementary to the first ones. In the latter, one also calculates the corresponding measure of quantum correlations by maximization. Considering alternative optimizations of the mentioned measures, one can define other quantifiers. In particular, Mundarain and Ladrón de Guevara introduced local available quantum correlations (LAQC, L\mathcal{L}L) in terms of the complementary local bases to the ones defining the least classically correlated state. That is, they characterize LAQC by defining the optimal bases as those that minimize the classical correlations measure. This type of quantum correlation is closely related to an entanglement activation protocol proposed by Piani et al. In previous articles, we derived an exact solution for the LAQC measure for 2-qubit X states. Based on these previous results and methodology, we determine the corresponding exact analytical expression for QsQ^sQs of 2-qubit X states. We then analyze the behavior of these two measures for two non-Markovian quantum dephasing channels: Random Telegraph (RT) and Modified Ornstein-Uhlenbeck (MOU) noises. We derive general conditions for sudden death and revival of these measures in X states and illustrate these results with four families of bipartite qubit states: Werner states, Maximally Nonlocal Mixed States (MNMS), Maximally Entangled Mixed States (MEMS), and a mixed symmetric separable state.
The CQC conjecture by Schneeloch et al. (Physical Review A 90.6, 2014) asserts that the sum of classical mutual information between two parties obtained by measuring individual systems in two mutually unbiased bases cannot exceed their shared quantum mutual information. This remarkable conjecture, which deserves more work, still remains open. In this article, using a result by Coles and Piani (Physical Review A 89.2, 2014), we derive a sufficient condition for which the CQC conjecture holds. We show that this condition validates the conjecture for a wider class of states compared to the states that were mentioned in the original work. Furthermore, we extend this conjecture to a higher number of mutually unbiased bases and prime dimensions. We prove this extended CQC conjecture on isotropic states of arbitrary prime dimensions and simulate it extensively for dimensions 3 and 5 for random bipartite states, observing no contradiction.
The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension m∗=2⌊n/2⌋m_*=2^{\lfloor n/2 \rfloor}m∗​=2⌊n/2⌋ (equivalently ⌊n/2⌋\lfloor n/2 \rfloor⌊n/2⌋ copies of maximally entangled two-qubit pairs), together with nnn numbers of anti-commuting observables on one side. To this end, we employ an nnn-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having 2n−12^{n-1}2n−1 and nnn number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least 2⌊n/2⌋2^{\lfloor n/2 \rfloor}2⌊n/2⌋ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with nnn mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.
We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.