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arXiv:2003.02952 · Physical Review A

Entanglement-preserving limit cycles from sequential quantum measurements and feedback

Philippe Lewalle, Cyril Elouard, Andrew N. Jordan

Published Dec 28, 2020• 11 citations
arXivHTMLPDF

Abstract

Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local $\ensuremath{\pi}$ pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e., using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e., using diffusive quantum trajectories) [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. Our method provides a pathway towards generating and protecting entangled states, complementing others in the literature with similar aims, such that two-qubit entanglement may be used in various applications on demand.

DOI
10.1103/physreva.102.062219

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Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e., using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e., using diffusive quantum trajectories) [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. Our method provides a pathway towards generating and protecting entangled states, complementing others in the literature with similar aims, such that two-qubit entanglement may be used in various applications on demand.
We discuss recent developments in measurement protocols that generate quantum entanglement between two remote qubits, focusing on the theory of joint continuous detection of their spontaneous emission. We consider a device geometry similar to that used in well-known Bell state measurements, which we analyze using a conceptually transparent model of stochastic quantum trajectories; we use this to review photodetection, the most straightforward case, and then generalize to the diffusive trajectories from homodyne and heterodyne detection as well. Such quadrature measurement schemes are a realistic two-qubit extension of existing circuit QED experiments, which obtain quantum trajectories by homodyning or heterodyning a superconducting qubit’s spontaneous emission, or an adaptation of existing optical measurement schemes to obtain jump trajectories from emitters. We mention key results, presented from within a single theoretical framework, and draw connections to concepts in the wider literature on entanglement generation by measurement (such as path information erasure and entanglement swapping). The photon which-path information acquisition, and therefore the two-qubit entanglement yield, is tunable under the homodyne detection scheme we discuss, at best generating equivalent average entanglement dynamics as in the comparable photodetection case. In addition to deriving this known equivalence, we extend past analyses in our characterization of the measurement dynamics: we include derivations of bounds on the fastest possible evolution toward a Bell state under joint homodyne measurement dynamics and characterize the maximal entanglement yield possible using inefficient (lossy) measurements.
We consider protocols to generate quantum entanglement between two remote qubits, through joint time-continuous detection of their spontaneous emission. We demonstrate that schemes based on homodyne detection, leading to diffusive quantum trajectories, lead to identical average entanglement yield as comparable photodetection strategies; this is despite substantial differences in the two-qubit state dynamics between these schemes, which we explore in detail. The ability to use different measurements to achieve the same ends may be of practical significance; the less-well-known diffusive scheme appears far more feasible on superconducting qubit platforms in the near term.
Generating on-demand maximally entangled states is one of the cornerstones for quantum information processing. Parity measurements can serve to create Bell states and have been implemented via an electronic Mach-Zehnder interferometer among others. However, the entanglement generation is necessarily harmed by measurement-induced dephasing processes in one of the two parity subspaces. In this work, we propose two different schemes of continuous feedback for a parity measurement. They enable us to avoid both the measurement-induced dephasing process and the experimentally unavoidable dephasing, e.g., due to fluctuations of the gate voltages controlling the initialization of the qubits. We show that we can generate maximally entangled steady states in both parity subspaces. Importantly, the measurement scheme we propose is valid for implementation of parity measurements with feedback loops in various solid-state environments.
The quantum resources of entanglement and single photons are key to a range of quantum applications. These resources are challenging to produce cleanly and efficiently. This thesis investigates and improves methods of producing entanglement and single photons using measurement and control. To robustly produce entanglement, this thesis extends work by Carvalho et al., who developed a method for creating an entangled state of two atoms. This is done by coupling the atoms to a damped cavity, and using measurement of the output of the cavity to trigger a feedback pulse on the atoms. The robustness of this scheme against imperfect localisation of the two atoms is tested in this thesis, and limits are placed on the size of a trap that will still allow an entangled state to be produced. Additionally, using three-level (Lambda configuration) atoms slows the rate at which the system evolves, allowing more time for measurement and feedback to be applied, though it does not counteract the influence of spontaneous emission. In extending this work to multiple atoms we found that the system rapidly becomes more complex, and we developed a strategy for choosing a feedback pulse that stabilises a specific, highly entangled steady state of multiple particles. In addition to this general strategy, we developed a local and separable feedback for generating entanglement in a four-partite system. This thesis also investigates the production of single photons through the rephased amplified spontaneous emission (RASE) protocol. This protocol offers the promise of a stream of precisely shaped single photons, though it is plagued by efficiency issues. We develop a model of RASE and show that it overcomes the trade-off between efficiency and photon spacing by using different optical depths for the amplified spontaneous emission (ASE) preparation of the ensemble, and the RASE emission of single photons. By tailoring the density profile of the ensemble to mode-match the RASE emission of a single photon, we also able to eliminate reabsorption of the emitted photon, which would otherwise have limited the efficiency to 70%.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.
We present a general approach to measurement-based quantum feedback that employs proportional and quantum state--(PaQS-) based feedback components to obtain locally optimal protocols. To demonstrate the power of the method, we first show that it reproduces many known feedback protocols and then apply it to generation of multipartite entanglement with an emphasis on remote entanglement, which requires spatially local feedback Hamiltonians. The symmetry of both measurement and feedback operators is found to be essential for construction of effective protocols. We show that under perfect measurement efficiency, entangled states can be reached with fidelity approaching unity under non-Markovian feedback control protocols, while Markovian protocols resulting from optimizing the feedback unitaries on ensemble-averaged states still yield fidelities above 94%. Application of the PaQS approach to generation of NNN-qubit WWW, general Dicke, and Greenberger-Horne-Zeilinger states shows that such entangled states can be efficiently generated with high fidelity, for up to N=100N=100N=100 in some cases.