What is the optimal way to prepare a Bell state using measurement and feedback?

Abstract

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.

Locally optimal control of continuous-variable entanglement

“…We will now assume the possibility of performing instantaneous, local symplectic transformations, of the form given by Eq. (6), which cannot generate quantum correlations but can, as we shall see, delay their decay during the interaction with the thermal bath described by the diffusive dynamics (15).…”

“…Since any manipulation of quantum systems requires them to be in contact with a noisy environment, a major question in advancing control upon them -arguably the main directive towards the development of functional quantum technologies -is how we can preserve this phenomenon for as long as it takes for an experiment to unfold. The design and application of quantum control techniques aimed at sustaining the entanglement of quantum systems has thus been a lively area of work over the last fifteen years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] that has seen the exploration of open-loop [2,7] as well as measurement-based [3][4][5][6][8][9][10] and quantum coherent feedback [13] strategies, applicable in principle to a wide variety of systems, although quantum optical scenarios seem to offer accurate enough control and low enough noise to facilitate such endeavours [1,11,16].…”

Continuous measurements for control of superconducting quantum circuits

“…We have jψð0Þi ¼ jψ 0 � and jψðt � 1=ΓÞi ¼ jψ þ i, indicating that we can deterministically prepare an entangled state from a separable state using feedback. A detailed derivation including a proof of global optimality of this resulting protocol is given in Ref [51]. Note that jψðtÞi only depends on time, and not on any function of the measurement record.…”

Manipulation of Multi‐Level Quantum Systems via Unsharp Measurements and Feedback Operations

“…For instance, even in the presence of random noise fields, quantum state still maintains its coherence in the presence of unsharp measurements and feedback operations . Indeed, measurement‐feedback control is invaluable in the manipulation of quantum systems . For instance, it can be used to fight against decoherence, manipulate number of photons, modulate frequency of Rabi oscillations, realize state tomography, enhance sideband cooling, and generate interaction potentials between remote particles …”