Remote entanglement stabilization and concentration by quantum reservoir engineering

Nicolas Didier, Jérémie Guillaud, Shyam Shankar, and Mazyar Mirrahimi

Phys. Rev. A 98, 012329 – Published 27 July 2018

Abstract

Quantum information processing in a modular architecture requires the distribution, stabilization, and distillation of entanglement in a qubit network. We present autonomous entanglement stabilization protocols between two superconducting qubits that are coupled to distant cavities. The coupling between cavities is mediated and controlled via a three-wave mixing device that generates either a two-mode squeezed state or a delocalized mode between the remote cavities depending on the pump applied to the mixer. Local drives on the qubits and the cavities steer and maintain the system to the desired qubit Bell state. Most spectacularly, even a weakly squeezed state can stabilize a maximally entangled Bell state of two distant qubits through an autonomous entanglement concentration process. Moreover, we show that such reservoir-engineering-based protocols can stabilize entanglement in the presence of qubit-cavity asymmetries and losses.

Received 18 August 2017

DOI:https://doi.org/10.1103/PhysRevA.98.012329

Physics Subject Headings (PhySH)

Quantum entanglement Quantum information with solid state qubits

Superconducting qubits

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Nicolas Didier^1,*, Jérémie Guillaud¹, Shyam Shankar², and Mazyar Mirrahimi^1,3

¹QUANTIC Team, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France
²Departments of Applied Physics, Yale University, New Haven, Connecticut 06520, USA
³Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA

^*Present address: Rigetti Computing, 775 Heinz Avenue, Berkeley, California 94710, USA.

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Vol. 98, Iss. 1 — July 2018

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Figure 1
Autonomous remote entanglement stabilization protocols with three-wave mixers. (a) For a directional coupling in the amplification mode, the TWM is a source of two-mode squeezed light that steers the cavities to two-mode squeezed vacuum for the odd-qubit subspace and to a two-mode squeezed thermal state for the even-qubit subspace (dashed red arrows). Qubit drives couple the two subspaces (plain purple arrows). A coupling between the Bell states $| ϕ_{+} 〉$ and $| ϕ_{-} 〉$ is activated by the fluctuations of the photon-number imbalance present in the two-mode squeezed thermal state (light blue double-line arrow). Entanglement accumulation leads to the stabilization of the Bell state $| ϕ_{-} 〉$ in the two-mode squeezed vacuum with $100 %$ fidelity. The different notations of the figure are defined in Sec. 2. (b) For a bidirectional coupling in the conversion mode, the TWM generates delocalized modes between the two distant cavities. These delocalized modes are displaced by the cavity drives for even-qubit states (light blue double-line arrows). Qubit drives couple the even and odd subspaces (plain purple arrows). Cavity dissipation steers the cavities to vacuum for the odd-qubit subspace (dashed red arrow). For large enough coupling the physics becomes effectively single mode and the protocol of Ref. [10] can be applied. Notations are defined in Sec. 3b.
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Figure 2
Remote entanglement stabilization in the amplification mode and with the directional connection. The convergence rate is plotted versus amplification strength in decibels. For a fixed strength $Ω$ of the qubit drives, the convergence rate reaches a maximum for a finite value of the squeezing strength. However, this decrease in convergence rate, observed for strong squeezing powers, can be compensated by increasing the qubit drive strength (solid curve, corresponding to an optimal choice of $Ω$ ). Parameters are $χ_{1} / 2 π = χ_{2} / 2 π = 5$ MHz, $κ / 2 π = 1$ MHz. Inset: Temporal dynamics of the fidelity $F$ for different values of the squeezing strength.
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Figure 3
Effect of imperfections on remote entanglement stabilization in the amplification mode and directional connection. Using the same parameters as in Fig. 2, and $Ω = 1.6 κ$ , the complement of the steady-state concurrence is plotted against (a) transmission losses, (b) asymmetry in the cavity couplings to the transmission lines, and (c) qubit relaxation and dephasing (for $T_{1} = T_{2}$ ). One observes that applying a weak amount of squeezing provides more robustness to transmission losses and asymmetries but leads to a slower rate of convergence, harming the asymptotic concurrence in the presence of the qubit's relaxation and dephasing. In practice, one needs to choose an optimal value of squeezing to compromise between these two effects.
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Figure 4
Remote entanglement stabilization in the conversion mode and with the bidirectional connection. The single-cavity result (dashed and dotted lines) is recovered for a sufficiently large tunnel coupling $g$ (solid and dash-dotted lines). Results are obtained for different values of the damping rate of the long resonators, $κ_{ℓ} = 0$ in cyan and $κ_{ℓ} = 10 κ$ in black. The damping rate of the cavities and TWM modes is $κ / 2 π = 1$ MHz, the effective dispersive shift is equal to $χ_{eff} = 5 κ$ , the cavity drives are set to displace the mode ${\hat{d}}_{1}$ by an average of $\bar{n} = 3$ photons, and the qubit drive strength is $Ω = κ_{eff} / 2$ . For the simulations of the single-cavity case, we vary the effective damping rate $κ_{eff}$ similarly to the two-mode case. This explains the slight reduction of the fidelity for $κ_{ℓ} = 10 κ$ with increasing $g$ (and therefore $κ_{eff}$ ). Inset: Temporal dynamics of fidelity establishment and stabilization for different values of the tunnel coupling and $κ_{ℓ} = κ$ .
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Figure 5
Parameter $μ_{j_{1} j_{2}}$ for the four two-qubit states, obtained for $\bar{n} = 4$ . The dashed lines are obtained from Hamiltonian (13) with the effective parameters $E = g \frac{χ_{1} + χ_{2}}{\sqrt{χ_{1} χ_{2}}} {sin}^{2} φ$ and $χ_{eff} = \frac{χ_{1} χ_{2}}{χ_{1} + χ_{2}} {cos}^{2} φ$ .
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