On-demand maximally entangled states with a parity meter and continuous feedback

Generating on-demand maximally entangled states is one of the cornerstones for quantum information processing. Parity measurements can serve to create Bell states and have been implemented via an electronic Mach-Zehnder interferometer among others. However, the entanglement generation is necessarily harmed by measurement-induced dephasing processes in one of the two parity subspaces. In this work, we propose two different schemes of continuous feedback for a parity measurement. They enable us to avoid both the measurement-induced dephasing process and the experimentally unavoidable dephasing, e.g., due to fluctuations of the gate voltages controlling the initialization of the qubits. We show that we can generate maximally entangled steady states in both parity subspaces. Importantly, the measurement scheme we propose is valid for implementation of parity measurements with feedback loops in various solid-state environments.

Figure 1

Parity measurement through a MZI and induced dephasing. (a) Sketch of the MZI operated as a parity meter for two charge qubits, $Q1$ and $Q2$, including the feedback circuit. The feedback is implemented via a gate voltage, which controls the bias energies of the qubits. Electrons are injected in lead $|1\rangle$ and the current is measured in lead $|3\rangle$. The input $|2\rangle$ and output $|4\rangle$ are not used in this proposal. (b) Signal at the detector's output. The Aharonov-Bohm flux $\Phi$ can be tuned to have indistinguishable signals for even and odd states, $I_e$ and $I_o$ respectively. (c) Probabilities for the two qubits to be in each of the four maximally entangled states as a function of time without feedback ($f=0$, dashed lines) and with optimal feedback ($f=\text{opt}$, solid lines).

Figure 2

Generation of entanglement through the parity measurement with feedback. (a) Amplitude of the oscillations between the populations of the even Bell states $|{\Psi }_e^{\pm }\rangle$ as a function of time for different values of the feedback parameter. For $f=1$, the amplitude does not decay as expected for the optimal feedback when considering a QND measurement. (b) Conditional concurrences plotted as a function of time in units of the measurement time set by the dephasing rate ${\Gamma }_{eo}$. The optimal value for the feedback parameter $f=1$ leads to steady maximally-entangled states in both parity subspaces: $C_e=C_o$ do not decay. For comparison, conditional concurrences are shown for different values of non optimal feedback $f$: $f=0,0.5,0.75$.

Figure 3

Deterministic generation of entangled states in the presence of external sources of noise. Top: Conditional concurrence $C_e$ as a function of the measurement strength ${\Gamma }_x$ normalized by ${\Gamma }_{eo}$ for different noise strengths ${\Gamma }_{\xi }/{\Gamma }_{eo}$. In the absence of external noise, the Markovian feedback that compensates the intrinsic dephasing is enough to produce Bell states, which do not decohere in time: $C_e=1$ (red crosses). For nonzero values of noise, $C_e$ reaches a saturated value, which decreases when the noise strength becomes large. Bottom: Conditional concurrence $C_o$ follows the same evolution as $C_e$: here the behavior of $C_o$ as a function of the noise strength ${\Gamma }_{\xi }/{\Gamma }_{eo}$ is plotted.