Time-Continuous Bell Measurements

Abstract

We combine the concept of Bell measurements, in which two systems are projected into a maximally entangled state, with the concept of continuous measurements, which concerns the evolution of a continuously monitored quantum system. For such time-continuous Bell measurements we derive the corresponding stochastic Schrödinger equations, as well as the unconditional feedback master equations. Our results apply to a wide range of physical systems, and are easily adapted to describe an arbitrary number of systems and measurements. Time-continuous Bell measurements therefore provide a versatile tool for the control of complex quantum systems and networks. As examples we show show that (i) two two-level systems can be deterministically entangled via homodyne detection, tolerating photon loss up to 50%, and (ii) a quantum state of light can be continuously teleported to a mechanical oscillator, which works under the same conditions as are required for optomechanical ground state cooling.Introduction.-According to the basic rules of quantum mechanics a multipartite quantum system can be prepared in an entangled state by a strong projective measurement of joint properties of its subsystems. Measurements which project a system into a maximally entangled state are called Bell measurements and lie at the heart of fundamental quantum information processing protocols, such as quantum teleportation and entanglement swapping. In systems which are amenable to strong projective measurements (e. g., photons [1,2] and atoms [3,4]), Bell measurements constitute a well established, versatile tool for quantum control and state engineering. However, in many physical systems only weak, indirect, but time-continuous measurements are available. Over the last years a multitude of experiments have demonstrated quantum-limited time-continuous measurement and control in a range of physical systems, including single atoms [5][6][7], cavity modes [8,9], atomic ensembles [10][11][12], superconducting qubits [13,14], and massive mechanical oscillators [15][16][17][18]. Continuously monitored quantum dynamics are described through the formalism of stochastic Schrödinger and master equations [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], which in itself constitutes a cornerstone of quantum control. Surprisingly, no exhaustive connection between these important concepts-Bell measurements and time-continuous measurements-has been made so far.In this letter we establish this connection and introduce the notion of time-continuous Bell measurements [34], which are realized via continuous homodyne detection of electromagnetic fields, and can be applied to a great number of systems, including those which cannot be measured projectively. We derive the constitutive equations of motion-the conditional stochastic Schrödinger/master equation and the unconditional feedback master equation-of the monitored systems. * sebastian.hofer@univie.ac.at The systems may take the form of (c) a harmonic oscillator (e. g., an optomechanical cavity), or (d) a t...

Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits

“…Measurement-induced entanglement of spatially separated quantum systems is a startling prediction of quantum mechanics [1][2][3][4][5][6][7]. Recent experiments have demonstrated this effect via single-photon heralding [8][9][10][11], as well as via continuous measurement of photons interacting with qubits [12].…”

Remote Controlled Entanglement

“…We take advantage of the versatility of continuous measurement to monitor the dynamics of entanglement generation and demonstrate quantitative agreement to a theoretical model that captures the experimental details of the physical circuit [22]. Moreover, our characterization of the state of the joint system under continuous measurement suggests the feasibility of future continuous feedback stabilization of entanglement [30,31]. Further technical improvements in quantum efficiency, coherence times, and transmission characteristics hold the promise of on-demand, stabilized remote entanglement-a powerful resource for quantum information processing.…”

On-demand maximally entangled states with a parity meter and continuous feedback

“…While the readout of individual qubits is typically associated with the irreversible destruction of the given coherent state, it has been shown that a joint measurement of two qubits can serve as an effective mechanism to generate entanglement between two measured qubits initially in a product state, [1][2][3][4][5][6][7][8][9][10][11][12] and may be used for error correction schemes. 13,14 This measurement-based creation of entanglement is achieved by operating the detector as a parity meter.…”