Parity detection and entanglement with a Mach-Zehnder interferometer

A parity meter projects the state of two qubits onto two subspaces with different parities, the states in each parity class being indistinguishable. It has application in quantum information for its entanglement properties. In our work we consider the electronic Mach-Zehnder interferometer (MZI) coupled capacitively to two double quantum dots (DQDs), one on each arm of the MZI. These charge qubits couple linearly to the charge in the arms of the MZI. A key advantage of an MZI is that the qubits are well separated in distance so that mutual interaction between them is avoided. Assuming equal coupling between both DQDs and the arms and the same bias for each DQD, this setup usually detects three different currents, one for the odd states and two for each even state. Controlling the magnetic flux of the MZI, we can operate the MZI as a parity meter: only two currents are measured at the output, one for each parity class. In this configuration, the MZI acts as an ideal detector, its Heisenberg efficiency being maximal. Initially unentangled DQDs become entangled through the parity measurement process with probability one and for a class of initial states our parity meter deterministically generates Bell states.

Figure 1

(Color online) MZI coupled capacitively to two qubits. ${\chi }_1\left({\chi }_2\right)$ is the phase accumulated by the electrons passing through the upper (lower) arm of the MZI over a length $L$. These phases depend on the state of the qubits. $C_1$, $C_2$, and $C_i$ are the capacitances characterizing the DQDs.

Figure 2

Current measured at lead 3 as a function of the magnetic flux between the arms of the MZI. (a) At an arbitrary flux three currents $I_o$, $I_{\uparrow \uparrow }$, and $I_{\downarrow \downarrow }$ are distinguishable even if the incremental phases are identical: $\Delta {\chi }_1=\Delta {\chi }_2=\Delta \chi$. (b) To measure only two currents corresponding to the parity classes, $I_o$ and $I_e$, we set the flux to $2\pi \Phi /{\Phi }_0=0\text{modulo}\pi$. A maximum of the current is reached at $2\pi \Phi /{\Phi }_0=2\pi$.