Measuring the parity of $N$ distant atoms with linear optics

It is known that parity measurement, together with single-qubit rotation, is sufficient for implementing scalable quantum computation. In this Brief Report, we propose a scheme for a projective measurement of the parity operator $P_z={\otimes }_{i=1}^N{\sigma }_{i,z}$ of $N$ distant atoms trapped in spatially separated cavities. Instead of direct interaction between the atoms, quantum interference of polarized photons decaying from the optical cavities is used to realize expected measurement without resorting to a sequence of single- and two-qubit operations. It is shown that parity measurement can be implemented repeatedly until success without destroying the qubits at any stage of the operation.

Figure 1

Relevant level structure and atomic transitions of the trapped atom.

Figure 2

Schematic setup for a parity measurement of two distant atoms. It includes three polarization beam splitters (PBS), which transmit the horizontal polarization and reflect vertical polarization, four half-wave plates (HWP), which implement transformation $|H\rangle \to \left(\right|H\rangle +|V\rangle )/\sqrt{2}$, $|V\rangle \to \left(\right|H\rangle -|V\rangle )/\sqrt{2}$, and four photon detectors $D$.

Figure 3

Schematic setup for a parity measurement of $N$ atoms trapped in distant cavities.