Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities

We describe a method to project photonic two-qubit states onto the symmetric and antisymmetric subspaces of their Hilbert space. This device utilizes an ancillary coherent state, together with a weak cross-Kerr nonlinearity, generated, for example, by electromagnetically induced transparency. The symmetry analyzer is nondestructive, and works for small values of the cross-Kerr coupling. Furthermore, this device can be used to construct a nondestructive Bell-state detector.

Figure 1

Schematic circuit showing how two relatively small cross-Kerr nonlinearities can be used to discriminate between $(\mid 2,0⟩\pm \mid 0,2⟩)∕\sqrt{2}$ and ∣1, 1⟩ nondestructively. The boxes labeled by $\theta$ and $-\theta$ represent the cross-Kerr nonlinearities that induce a phase shift in the coherent state $\mid {\alpha }_c⟩$ proportional to the number of photons in the corresponding signal mode. A homodyne measurement of the $X$ quadrature with outcome $x$ will then discriminate between the two input states. Depending on this outcome, a relative phase shift $2\varphi \left(x\right)$ is needed to restore the $\mid 2,0⟩\pm \mid 0,2⟩$ state.

Figure 2

(a) Schematic phase space illustration of the state $\mid {\psi }_1⟩=d_1\mid 1,1⟩\mid {\alpha }_c⟩+(d_2∕\sqrt{2})(\mid 2,0⟩\mid {\alpha }_c{e}^{2i\theta }⟩\pm \mid 0,2⟩\mid {\alpha }_c{e}^{-2i\theta }⟩)$. For the ∣1, 1⟩ component of the state, the probe mode receives zero total phase shift, whereas for the ∣2, 0⟩ and ∣0, 2⟩ components, the phase shifts are $+2\theta$ and $-2\theta$, respectively. (b) The corresponding probability distribution for the outcome of the $X$ quadrature measurement of the probe beam. The two peaks in this distribution are associated with the states ∣1, 1⟩ and $\mid 2,0⟩\pm \mid 2,0⟩$.

Figure 3

The symmetry analyzer: photonic two-qubit input states interfere on a 50:50 beam splitter (BS). The polarizing beam splitters (PBS) will transform the beam splitter output into four modes with equal polarization. Using the methodology of Fig. 1, one can distinguish between the resulting bunched and balanced states. This results in a measurement of the symmetry of the photonic two-qubit input state. The PBS and BS after the homodyne measurement return the two modes to the two-qubit space.

Figure 4

The Bell-state detector: repeated application of the symmetry analyzer (SA) and local unitary rotations in $\mathcal{H}$ subsequently project onto different Bell states. The local unitaries are simply performed with suitable optical plates for polarization qubits.