The Signal
Yang's Counterexample Defeats a Quantum-Channel Optimality Conjecture
Jianting Yang’s paper, published in Quantum on October 28, 2026, gives a two-dimensional counterexample to an optimality test proposed by Bryan Coutts, Mark Girard and John Watrous. The example shows that a spectral-sign calculation can fai
Quantum.world Team · Written with AI · fact-checked ·
An optimal quantum channel can fail a test designed to certify its optimality. In a paper published in Quantum on October 28, 2026, Jianting Yang gives an explicit counterexample to a conjecture about when a spectral calculation can certify the best channel for a state-transformation task.
“By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture,” Jianting Yang writes. The result targets a particular proposed test, not the broader use of convex optimization or dual certificates in quantum information.
The gap appears when the error matrix is singular
The optimization problem asks which quantum channel best transforms an input state into a target state, with the mismatch measured by trace distance. In the 2021 paper “Certifying optimality for convex quantum channel optimization problems,” Bryan Coutts, Mark Girard and John Watrous set out conditions for certifying optimal channels and proposed a spectral test for this transformation problem.
The test takes the sign of the eigenvalues of the error matrix—the difference between the target state and the channel’s output. Yang focuses on the case where that matrix is rank-deficient, meaning at least one eigenvalue is zero. The earlier work established the test when the error matrix is full-rank; the unresolved case was whether the proposed condition continued to work when it was singular.
For a zero eigenvalue, the trace norm’s subdifferential—the set of operators that can supply a valid certificate—is no longer a single element. The conjecture held that assigning zero to the sign of each zero eigenvalue would still produce a necessary and sufficient optimality test. Yang’s example shows that this particular choice can fail, even though an optimal channel exists.
Symmetry reduces the search to two parameters
Yang constructs the counterexample by adding symmetry to the state-transformation problem. The reduction takes a nuclear-norm minimization over a four-by-four positive-semidefinite matrix and turns it into a lower-dimensional linear program. In the final check, feasible diagonal solutions are described by two real parameters, each ranging from zero to one.
The constructed solution reaches a lower bound on the objective and is therefore optimal, Yang writes. Yet its spectral-sign operator does not satisfy the conjectured certification conditions. The paper reports that numerical methods helped find the example, while the verification of its optimality is analytic.
One failed certificate does not remove the others
The counterexample narrows what can be inferred from the spectrum of a rank-deficient error matrix: that calculation alone does not always produce a valid dual certificate. Yang’s paper says a valid certificate may instead lie elsewhere in the subdifferential set. It does not show that optimal channels cannot be certified, or that the earlier full-rank result fails.
The practical consequence is specific but important for users of these tests: in singular cases, a spectral-sign calculation cannot by itself settle whether a candidate channel is optimal. The paper leaves open how to obtain an efficient certificate across the general rank-deficient case; its concrete result is the two-dimensional example that defeats the proposed test.