Covariant feedback Control of arbitrary qudit state against Depolarizing Noise is impossible

TL;DR

This paper proves that covariant quantum feedback control cannot improve the fidelity of arbitrary qudit states affected by depolarizing noise, highlighting fundamental limitations in quantum error correction.

Contribution

It establishes the impossibility of covariant feedback control for correcting depolarizing noise on unknown qudit states, extending to bipartite entangled states.

Findings

No covariant CPTP maps outperform identity in fidelity

Proof applies to arbitrary qudit states and bipartite entangled states

Fundamental limit on covariant quantum feedback control

Abstract

In this paper, we prove that it is impossible to construct a convariant quantum feedback control instruments which aim to correct the channel noise imposed on an unknown qubit. The proof is based on the searching for the optimal quantum control protocol and it turns out there exist no better %\textit{Complete Positive Trace %Preserving}(CPTP) maps complete positive and covariant quantum operations which provide a higher fidelity than the trivial Identity operators. The generalization of the investigation to bipartite entangled pure state is also included.