Remote Entanglement by Coherent Multiplication of Concurrent Quantum Signals

Concurrent remote entanglement of distant, noninteracting quantum entities is a crucial function for quantum information processing. In contrast with the existing protocols which employ the addition of signals to generate entanglement between two remote qubits, the continuous variable protocol we present is based on the multiplication of signals. This protocol can be straightforwardly implemented by a novel Josephson junction mixing circuit. Our scheme would be able to generate provable entanglement even in the presence of practical imperfections: finite quantum efficiency of detectors and undesired photon loss in current state-of-the-art devices.

Figure 1

Remote entanglement protocol schematic. The first step of the protocol (I) consists of entangling two stationary qubits, Alice (in dark red) and Bob (in dark green), with two propagating modes (indicated by thicker lines), ancilla signal (a) and ancilla signal (b), respectively, each initially prepared in a coherent state. This is achieved by first applying a $\pi /2$ rotation ($Y^{1/2}$) on Alice (Bob), followed by a conditional displacement gate (CD) on Alice (Bob) and ancilla signal (a) [ancilla signal (b)]. In the next step (II), a nonlinear interaction of the ancilla signal (a), ancilla signal (b) and parity signal (c) [implemented by the Josephson Parametric Multiplier (JPM), see below], followed by a homodyne detection of the parity signal mode, effectively realizes a joint two-qubit parity measurement. The resulting qubit-photon state has either even or odd joint qubit parity conditioned on the integrated homodyne current being in either lobe of the distribution (indicated by dashed lines). The last step (III), which disentangles the qubits from the photon states, comprises of homodyne measurements of the $\mathbf{a}$ and $\mathbf{b}$ modes, denoted by ${\mathrm{HD}}_{\mathbf{a}}$ and ${\mathrm{HD}}_{\mathbf{b}}$. Conditioned on the measurement outcome in (II) and (III), the qubits are projected onto the even or odd Bell manifold, with a relative phase that depends on the measurement outcome in (III).

Figure 2

Schematic of the JPM of Fig. 1. The modes $\mathbf{a}({\omega }_a),\mathbf{b}({\omega }_b)$ and $\mathbf{c}({\omega }_c)$ when pumped at ${\omega }_p={\omega }_c-{\omega }_a-{\omega }_b$ (orange) participate in a nonlinear, three-wave interaction ${\mathbf{H}}_{\mathrm{int}}/\hslash =ig{e}^{-i{\omega }_{p}t}\mathbf{a}\mathbf{b}{\mathbf{c}}^{†}+\mathrm{H}.\mathrm{c}.$ This nonlinear mode mixing arises out of the Josephson Four Wave Mixer (JFWM). (Inset) The JFWM has four nominally identical Josephson junctions connected electrically as shown. The coupled system has four mutually orthogonal normal (electrical) modes [24], which couple nonlinearly, corresponding to the cavity modes $\mathbf{a},\mathbf{b},\mathbf{c}$ and the pump mode.

Figure 3

Probability distribution of outcomes, overlap ($F$) with Bell-state $|{\mathrm{\Phi }}^{+}⟩=(|ee⟩+|gg⟩)/\sqrt{2}$, concurrence ($C$) of the joint two-qubit system ${\rho }_q$ and gradient of overlap are plotted when the qubit-photon state is projected onto one with even two-qubit parity after the second step of the protocol. The top (bottom) row panels correspond to homodyne detection of the $X$ ($Y$) quadratures of modes $\mathbf{a},\mathbf{b}$. We choose $\alpha =\beta =0.75$ and assume perfect quantum efficiency and zero spurious photon loss. Panel (a) shows the probability of outcomes $P(x_a,x_b)$ for $X$ measurements, corresponding to which, we see that the two-qubit state is projected on to $|{\mathrm{\Phi }}^{+}⟩$ for values around the diagonal $x_b=-x_a$ [panel (b)]. For events occurring in the quadrant $x_a,x_b>0(<0)$, the two-qubit state is projected on to $|ee⟩(|gg⟩)$. Panel (c) shows the concurrence $\mathcal{C}(x_a,x_b)$ of ${\rho }_q$ for the different outcomes, which varies from 0 (for ${\rho }_q$ being a separable state $|ee⟩$ or $|gg⟩$) to 1 (in case of maximum entanglement). Panel (d) shows the gradient of the overlap as a function of $x_a,x_b$, which is zero (in green) for a narrow region around the line $x_a=-x_b$ where an entangled state is obtained. It changes rapidly on moving away from the line $x_a=-x_b$ and goes back to zero when the qubit state is projected on to $|ee⟩$ or $|gg⟩$. Panel (e) shows the probability of outcomes $P(y_a,y_b)$ for $Y$ measurements. For events around the line $y_a=-y_b$, the two-qubit state is once again projected on to $|{\mathrm{\Phi }}^{+}⟩$. However, for $Y$ measurements, the phase of the generated Bell state varies continuously, depending on the particular outcome $(y_a,y_b)$, indicated by the existence of alternating bright and dark fringes in fidelity [panel (f)]. For instance, along the line $y_a=y_b$, the two-qubit state oscillates continuously between $|{\mathrm{\Phi }}^{+}⟩$ and $|{\mathrm{\Phi }}^{-}⟩$. Panel (g) shows the concurrence $\mathcal{C}(y_a,y_b)$ of ${\rho }_q$, which is $\approx 1$ for all measurement outcomes, indicating generation of maximal entanglement for all outcomes $(y_a,y_b)$. The small regions of low entanglement are artifacts of the analytic approximation which replaces stochastic evolution of the system with a deterministic Lindblad evolution (see further discussion in the text). Panel (h) shows the gradient of overlap which is zero where the two-qubit state is projected onto $|{\mathrm{\Phi }}^{\pm }⟩$ and changes rapidly as the two-qubit state oscillates between $|{\mathrm{\Phi }}^{+}⟩$ and $|{\mathrm{\Phi }}^{-}⟩$.

Figure 4

Maximum overlap with Bell state $|{\mathrm{\Phi }}^{+}⟩$ is shown upon variation of incident state amplitude $\alpha$ and efficiency parameter $\eta$ for measurement of $Y$ quadratures of $\mathbf{a}$ and $\mathbf{b}$. A sample of 500 trajectories were simulated for each data point in the $(\alpha ,\eta )$ space and maximum fidelity was noted. For perfect efficiency ($\eta =1$), it is always possible to generate $|{\mathrm{\Phi }}^{+}⟩$. The maximum target fidelity goes down as $\eta$ is lowered. Higher values of $\alpha$ are more susceptible to finite efficiency. However, even for $\eta =0.7$ and $\alpha =0.5$, we obtain fidelities in excess of 80%, indicating the robustness of the scheme to photon loss and finite quantum efficiency.