A Counterexample to the Optimality Conjecture in Convex Quantum Channel Optimization

TL;DR

This paper constructs a specific counterexample in 2D quantum channels that disproves a conjecture claiming the dual certificate in nuclear norm minimization can always be uniquely determined by spectral calculus of the Choi matrix.

Contribution

It provides the first explicit counterexample to the conjecture, challenging assumptions in convex quantum channel optimization theory.

Findings

Counterexample disproves the conjecture in 2D cases

Dual certificates are not always uniquely determined by spectral calculus

Challenges existing beliefs in quantum channel optimization

Abstract

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.