# What is the optimal way to prepare a Bell state using measurement and feedback?. (arXiv:1704.00332v2 [quant-ph] UPDATED)

Recent work has shown that use of quantum feedback can significantly enhance

both the speed and success rate of measurement-based remote entanglement

generation, but it is generally unknown what feedback protocols are optimal for

these tasks. Here we consider two common measurements that are capable of

projecting into pairwise entangled states, namely half- and full-parity

measurements of two qubits, and determine in each case a globally optimal

protocol for generation of entanglement. For the half-parity measurement, we

rederive a previously described protocol using more general methods and prove

that it is globally optimal for several figures of merit, including maximal

concurrence or fidelity, and minimal time to reach a specified concurrence or

fidelity. For the full-parity measurement, we derive a protocol for rapid

entanglement generation related to that of (C. Hill, J. Ralph, Phys. Rev. A 77,

014305 (2008)), and then map the dynamics of the concurrence of the state to

the Bloch vector length of an effective qubit. This mapping allows us to prove

several optimality results for feedback protocols with full-parity

measurements. We further show that our full-parity protocol transfers

entanglement optimally from one qubit to the other amongst all

measurement-based schemes. The methods developed here will be useful for

deriving feedback protocols and determining their optimality properties in many

other quantum systems subject to measurement and unitary operations.