Optimal quantum-enhanced interferometry

We analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer's initial 50:50 beam splitter $B$ is a product state of the two input modes. Requiring a product state is a natural restriction: If one were allowed to input an arbitrary, entangled two-mode state $|\Xi \rangle$ to the beam splitter, one could generally just as easily input the state $B|\Xi \rangle$ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number.

Figure 1

Two modes $a$ and $b$ are incident on a 50:50 beam splitter. After the beam splitter, phase shifts ${\phi }_1$ and ${\phi }_2$ are imposed in the two arms. A measurement is then made to detect the differential phase shift ${\varphi }_d={\phi }_1-{\phi }_2$. When the measurement is pushed beyond a second 50:50 beam splitter, i.e., when the DETECTION box includes a 50:50 beam splitter before measurement, the result is a Mach-Zehnder interferometer, which is sensitive only to ${\varphi }_d$.