Observation of Measurement-Induced Entanglement and Quantum Trajectories of Remote Superconducting Qubits

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.

Figure 1

Experimental setup. (a) Simplified representation of the experimental setup. (b) and (c) Schematic of the phase shift acquired by a coherent state sequentially measuring first qubit 1 (b) and then qubit 2 (c) in reflection. (d) Picture of the base-temperature setup.

Figure 2

Demonstration of indistinguishability between $|01⟩$ and $|10⟩$ computational states during measurement. (a) Example of the temporal evolution of the measurement signal $V_m$. The inset shows the associated instantaneous voltage $V(t)$. (b) Histogram of $V_m$ for each of the four computational states $|00⟩$, $|01⟩$, $|10⟩$, and $|11⟩$. The range of data postselected for tomographic reconstruction at $t_m=0.65\mu \mathrm{s}$ is represented as a shaded grey area.

Figure 3

Generation and verification of entanglement between two spatially separated superconducting qubits. (a) Concurrence of the entangled state as a function of $t_m$. The inset displays the evolution of the basis state populations (${\rho }_{00,00}$, etc.) and odd-parity coherence (${\rho }_{01,10}$). The shaded region represents the standard deviation centered about the average (circles). Dashed lines are theoretical simulations based on a Bayesian approach, and solid lines are calculated using a rigorous master equation; in both cases, no fitting parameters are used [22]. (b) Full-density matrices of the postselected entangled subspace for increasing $t_m$.

Figure 4

Resolving single quantum trajectories for cascaded quantum systems. (a) Absolute value of the density matrix elements conditioned on the measured voltage $V_m$ for $t_m=0.65\mu \mathrm{s}$ and ${\overline{n}}_1=1.2$, presenting an instantaneous mapping $V_m\mapsto \rho (V_m)$. The shaded region represents the standard deviation about the average (circles); dashed lines (respectively, solid lines) are theoretical simulations based on a Bayesian approach (respectively, on a full master equation) without fitting parameter [22]. (b)–(d) Examples of reconstructed quantum trajectories for diagonal and principal off-diagonal density matrix elements. The dots represent tomographic reconstruction based on the mapping $V_m\mapsto \rho (V_m)$ for every $t_m$. The dashed lines are Bayesian estimations based on the measured $V_m(t)$ (insets). The solid lines for the full master equation were obtained by running 100000 instances of the stochastic differential equation with 1-ns resolution and averaging the obtained populations conditioned on $V_m$ at $t_m$[22].