Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits

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.

Popular Summary

Of all of the predictions of quantum measurement theory, the ability to entangle by measurement is surely one of the most dramatic and exciting. This process can be thought of as dynamical and random given the continuous measurement of photons interacting with multiple spatially separated quantum systems. Many questions about the entanglement creation process are outstanding, such as the following: (i) What is the complete characterization of the dynamics of entanglement creation as a continuous trajectory? (ii) What is the statistical distribution of the entanglement at any time during the process? (iii) What is the most-likely way entanglement is created? Here, we provide systematic answers to these questions and others by analyzing experimentally entangled quantum trajectories of jointly measured superconducting quantum circuits.

We track the quantum trajectories of two transmon qubits housed in separate superconducting cavities designed to minimize signal loss. We observe how the qubits evolve under joint continuous measurement by microwave frequency radiation, demonstrating a comprehensive understanding of dynamical entanglement generation by the measurement process. This entanglement generation is stochastic, and we present a detailed theoretical analysis of the statistics of the process, finding the full statistical distribution of created entanglement at any point during the process, as well as the most-likely paths in the quantum-state space to reach entangled states, and the associated times. In particular, we show that the two-qubit state collapses to one of three subspaces, and we determine the experimentally most-likely paths. We show excellent agreement between experimental data and our theory.

Our results present new ways to use joint measurement as a control mechanism to entangle remote systems for quantum information processing purposes.

Figure 1

Experimental setup and transmon qubit trajectories. Panel (a) shows a simplified illustration of the experimental setup. Two transmon qubits in two remote cavities are linked via a single microwave circulator in a bounce-bounce geometry; the output of the circulator is routed to a high-efficiency amplification chain (not shown). The upper inset represents the distribution of the microwave field in the ($X_1$, $X_2$) quadrature plane at the output of the amplifier. The three possible outcomes correspond to three different subspaces: $|00⟩$, $|11⟩$, and the odd-parity subspace (spanned by the $|01⟩$ and $|10⟩$ states). Panel (b) shows the evolution in time of the measurement outcome probability distribution $p(V_t)$. Panel (c) shows the agreement between the experimentally generated conditional tomography $x_i(V_t)$ (symbols and shaded error bars) and the theoretical Bayesian reconstruction (dashed lines), for a single time $t=0.48\mu \mathrm{s}$. In panel (d), we use such reconstructions to predict and verify the trajectory of a single iteration of the experiment, showing both density matrix elements $x_i$ and concurrence (inset).

Figure 2

Concurrence distribution for the qubits under the half-parity measurement. In panel (a) we plot the concurrence probability density function, Eq. (10), for different values of time. The values of time for the presented curves are chosen so as to see their unique features as they develop. The grey dotted curve joining the high-concurrence peaks shows the concurrence upper bound, Eq. (11). The inset shows an example of how the concurrence (at time $t=1\mu \mathrm{s}$) varies as a function of the readout $V_t$. Panels (b) and (c) are the histograms of the concurrence at any time points from $t=0$ to $T=1.6\mu \mathrm{s}$ (with a step size $0.01\mu \mathrm{s}$), comparing theory and experimental data. For the theory plot, we coarse-grain the distribution $p_{\mathcal{C},t}(c)$ in Eq. (10) by integrating it with a pixel size $\delta c\sim 0.015$, which is the bin size of the experimental histogram. For presentation purposes, a histogram at any time $t$ is normalized by its maximum element.

Figure 3

The most-likely paths from the theoretical prediction and the experimental data. The first, second, and third columns are from three different data sets, measuring the same qubits and cavities with three different measurement-readout powers (indicated by the characteristic measurement time): ${\tau }_m=2.10\mu \mathrm{s}$, $0.60\mu \mathrm{s}$, and $0.36\mu \mathrm{s}$, respectively. Panels (a)–(c) show the concurrence of the multiple most-likely paths: the path with high concurrence (solid purple line), and the two paths with low concurrence projecting onto $|00⟩$ (dashed orange line) and $|11⟩$ (dotted green line) subspaces. The vertical lines (labeled as $A_{1,2}$, $B$, $C_{1,2}$, $D$) represent the times at which the concurrence is maximum for each of these paths. The insets show the log-likelihood density as functions of optimal measurement readouts, and the grey-scale histograms in the background show experimental concurrence histograms similar to the one in Fig. 2. Panels (d)–(f) present the evolution of the density matrix elements of the high-concurrence most-likely paths: The solid curves indicate theoretical solutions, whereas the data points indicate experimental most-likely paths postselected with the most entangled state at the final time $T=1.6\mu \mathrm{s}$. Examples of the postselected trajectories are shown as fluctuating curves.

Figure 4

Histograms of the time to maximum concurrence for individual trajectories for three measurement strengths indicated by the values of ${\tau }_m$. For the two cases with strong readout powers (shown in red and green histograms), there exists two peaks corresponding to two most-likely times to reach their most entangled states. The theoretical predictions of these times are shown as vertical dashed lines labeled as $A_{1,2}$, $B$, $C_{1,2}$, $D$ [same as in Figs. 3].