Entanglement Thresholds of Quantum Networks containing Doubly-Parametric Microwave-Optical Transducers

Networking superconducting quantum processors over optical links will require transducers capable of entangling microwave and optical modes. However, decoherence from thermal noise, losses, and limited cooperativities pose significant challenges for designing networks which incorporate microwave-optical transducers. Thus, understanding the tolerable decoherence that the transducer can introduce into a network is essential to knowing when it can or cannot be used for quantum communication tasks. We find explicit expressions for when the quantum channel of a doubly-parametric transducer is separable, when it is PPT-preserving, and when it can create distillable optical-microwave entanglement. We then examine network topologies using two transducers and characterize their thresholds for entangling remote microwave modes over an optical link. The resources that are allowed, such as measurements and entangled states, directly affects the tolerable transducer performance. Conversely, the achievable transducer performance dictates the resources and network topology required to establish entanglement. Thus, having two transducers individually capable of quantum operation is not sufficient for demonstrating remote microwave entanglement due to the resource constraints of the network.