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arXiv:1603.09623 · Physical Review X

Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits

Areeya Chantasri, Mollie E. Schwartz, Nicolas Roch, Irfan Siddiqi, Andrew N. Jordan

Published Dec 14, 2016• 48 citations
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Abstract

We experimentally and theoretically investigate the quantum trajectories of jointly monitored transmon qubits embedded in spatially separated microwave cavities.Using nearly quantum-noise-limited superconducting amplifiers and an optimized setup to reduce signal loss between cavities, we can efficiently track measurement-induced entanglement generation as a continuous process for single realizations of the experiment.The quantum trajectories of transmon qubits naturally split into low and high entanglement classes.The distribution of concurrence is found at any given time, and we explore the dynamics of entanglement creation in the state space.The distribution exhibits a sharp cutoff in the high concurrence limit, defining a maximal concurrence boundary.The most-likely paths of the qubits' trajectories are also investigated, resulting in three probable paths, gradually projecting the system to two even subspaces and an odd subspace, conforming to a "half-parity" measurement.We also investigate the most-likely time for the individual trajectories to reach their most entangled state, and we find that there are two solutions for the local maximum, corresponding to the low and high entanglement routes.The theoretical predictions show excellent agreement with the experimental entangled-qubit trajectory data.

DOI
10.1103/physrevx.6.041052

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We experimentally and theoretically investigate the quantum trajectories of jointly monitored transmon qubits embedded in spatially separated microwave cavities.Using nearly quantum-noise-limited superconducting amplifiers and an optimized setup to reduce signal loss between cavities, we can efficiently track measurement-induced entanglement generation as a continuous process for single realizations of the experiment.The quantum trajectories of transmon qubits naturally split into low and high entanglement classes.The distribution of concurrence is found at any given time, and we explore the dynamics of entanglement creation in the state space.The distribution exhibits a sharp cutoff in the high concurrence limit, defining a maximal concurrence boundary.The most-likely paths of the qubits' trajectories are also investigated, resulting in three probable paths, gradually projecting the system to two even subspaces and an odd subspace, conforming to a "half-parity" measurement.We also investigate the most-likely time for the individual trajectories to reach their most entangled state, and we find that there are two solutions for the local maximum, corresponding to the low and high entanglement routes.The theoretical predictions show excellent agreement with the experimental entangled-qubit trajectory data.
The creation of a quantum network requires the distribution of coherent information across macroscopic distances. We demonstrate the entanglement of two superconducting qubits, separated by more than a meter of coaxial cable, by designing a joint measurement that probabilistically projects onto an entangled state. By using a continuous measurement scheme, we are further able to observe single quantum trajectories of the joint two-qubit state, confirming the validity of the quantum Bayesian formalism for a cascaded system. Our results allow us to resolve the dynamics of continuous projection onto the entangled manifold, in quantitative agreement with theory.
Measurement has traditionally been viewed as a means to restore classical behavior to a quantum system: a coherent superposition, once observed, transforms into a single classical state. However, it is possible to design a measurement that instead projects into an entangled state, thereby purifying, rather than destroying, quantum correlations. We use continuous measurement to generate entanglement between two superconducting qubits that are separated by more than a meter of cable, demonstrating that quantum information can be transferred over the metallic wires that comprise a low-loss channel for microwave photon propagation. We further generate a faithful, time-resolved record of single quantum trajectories. Studying the statistics of these trajectories and of the ensemble of measurements provides insight into the dynamics of measurement-induced entanglement in an extended quantum network.
An open challenge in quantum information processing with superconducting circuits is to entangle arbitrary pairs of qubits, not simply nearest neighbors. This can be accomplished by first entangling the qubits with flying microwave oscillators in the form of traveling pulses and then performing joint manipulation and measurement on a pair of such flying oscillators at a distant site. This method has the advantage of avoiding the overhead required to move information over large distances via only nearest neighbor coupling while also possessing an extremely high on/off ratio of coupling. Remarkably, such a process is naturally embedded in the act of phase-preserving amplification, which transforms two input modes, termed signal and idler, into a two-mode squeezed output state. For an ideal system, this operation generates heralded, perfectly entangled states between remote qubits with a fifty percent success rate. For an inefficient system, the partial loss of information from the flying states degrades the purity of the entanglement. Data will be shown on such a protocol involving two transmon qubits embedded in superconducting cavities connected to the signal and idler inputs of a Josephson Parametric Converter (JPC) operated as a nearly-quantum limited phase-preserving amplifier. I will also discuss new devices which will allow us to improve the process and achieve high success rates with imperfect hardware. Light refreshments will be served.
We present a first-principles theoretical analysis of the entanglement of two superconducting qubits in spatially separated microwave cavities by a sequential (cascaded) probe of the two cavities with a coherent mode, that provides a full characterization of both the continuous measurement induced dynamics and the entanglement generation. We use the SLH formalism to derive the full quantum master equation for the coupled qubits and cavities system, within the rotating wave and dispersive approximations, and conditioned equations for the cavity fields. We then develop effective stochastic master equations for the dynamics of the qubit system in both a polaronic reference frame and a reduced representation within the laboratory frame. We compare simulations with and analyze tradeoffs between these two representations, including the onset of a non-Markovian regime for simulations in the reduced representation. We provide conditions for ensuring persistence of entanglement and show that using shaped pulses enables these conditions to be met at all times under general experimental conditions. The resulting entanglement is shown to be robust with respect to measurement imperfections and loss channels. We also study the effects of qubit driving and relaxation dynamics during a weak measurement, as a prelude to modeling measurement-based feedback control in this cascaded system.
On-demand creation of entanglement between distant qubits is a necessary ingredient for distributed quantum computation. We propose an entanglement scheme that allows for single-shot deterministic entanglement creation by detecting a single photon passing through a Mach–Zehnder interferometer with one transmon qubit in each arm. The entanglement production essentially relies on the fact that superconducting microwave structures allow one to achieve strong coupling between the qubit and the photon. By detecting the photon via a photon counter, a parity measurement is implemented and the wave function of the two qubits is projected onto a maximally entangled state. Most importantly, the entanglement generation is heralded such that our protocol is not susceptible to photon loss due to the indivisible nature of single photons.
We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two noncommuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with solutions obtained using the Fokker-Planck equation. For a more general case with decoherence effects and inefficiency, we use a diagrammatic approach to find the correlators perturbatively in the quantum efficiency. We also calculate the state correlators for the quantum trajectories which are extracted from readout signals measured in a transmon qubit experiment, by means of the quantum Bayesian state update. We find an excellent agreement between the correlators based on the experimental data and those obtained from our analytical and numerical results.