A Counterexample to the Optimality Conjecture in Convex Quantum Channel Optimization

This paper presents a counterexample to the optimality conjecture in convex quantum channel optimization proposed by Coutts et al. The conjecture posits that for nuclear norm minimization problems in quantum channel optimization, the dual certificate of an optimal solution can be uniquely determined via the spectral calculus of the Choi matrix. By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture.

💡 Research Summary

The paper addresses a conjecture put forward by Coutts, Girard, and Watrous (2021) concerning optimality certification in convex quantum channel optimization. The conjecture claims that for the trace‑distance‑based optimal state‑transformation problem, the dual certificate of an optimal completely positive map can be uniquely obtained by applying the spectral sign function to the error matrix (the difference between the target state and the transformed source state). The authors prove that this claim holds when the error matrix is full‑rank, but they leave the rank‑deficient case open.

To resolve the open case, the authors exploit symmetry in the problem. They consider three two‑dimensional Hilbert spaces (X, Y, Z) and introduce a maximally mixed reference state (\rho_{0}= \frac12\sum_{i,j}|i\rangle\langle j|\otimes|i\rangle\langle j|). Using the Choi–Jamiołkowski isomorphism they show that for any channel (\Phi) the map ((\operatorname{Id}Y\otimes\Psi{\rho_0})(J(\Phi))) reduces to a simple scalar multiplication (\frac12 J(\Phi)). Consequently the original nuclear‑norm minimization problem (2) is equivalent to a much simpler SDP (11):
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