CPDNN quantum channels with qubit output are CPCP

TL;DR

This paper proves that for quantum channels with qubit output, the class of doubly nonnegative channels exactly coincides with the class of CPCP channels, resolving an open question in quantum resource theory.

Contribution

It establishes that all CPDNN quantum channels with qubit output are necessarily CPCP, confirming the exactness of the doubly nonnegative relaxation for these channels.

Findings

Every CPDNN quantum channel to a qubit is CPCP.

No qutrit-to-qubit CPDNN channel is outside CPCP.

Doubly nonnegative relaxation is exact for qubit-output channels.

Abstract

The resource theory for nonnegativity of quantum amplitudes distinguishes completely positive completely positive (CPCP) quantum channels from the larger and more tractable class of completely positive doubly nonnegative (CPDNN) quantum channels. It was left open whether there exists a qutrit-to-qubit quantum channel \(\Phi:M_3\to M_2\) that is CPDNN but not CPCP. We answer this question in the negative and prove the stronger statement that every CPDNN quantum channel \(\Phi:M_n\to M_2\) is CPCP for every \(n\in\mathbb N\). Equivalently, for qubit-output quantum channels the doubly nonnegative relaxation is exact.