Optimal pure state cloning and transposition are complementary channels

Abstract

State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. We first determine the optimal fidelity for transforming $N$ qudits into $K$ copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding $N \to K$ transposition map is the complementary channel of the optimal universal symmetric $N \to N + K$ quantum cloning machine on pure states. We then present an explicit quantum circuit that realises $N \to K$ transposition and $N \to N + K$ cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state $N \to 1$ qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.