Local and Global Distinguishability in Quantum Interferometry

A statistical distinguishability based on relative entropy characterizes the fitness of quantum states for phase estimation. This criterion is employed in the context of a Mach-Zehnder interferometer and used to interpolate between two regimes of local and global phase distinguishability. The scaling of distinguishability in these regimes with photon number is explored for various quantum states. It emerges that local distinguishability is dependent on a discrepancy between quantum and classical rotational energy. Our analysis demonstrates that the Heisenberg limit is the true upper limit for local phase sensitivity. Only the “NOON“ states share this bound, but other states exhibit a better trade-off when comparing local and global phase regimes.

Figure 1

Mach-Zehnder interferometer with 2-mode input state $|{\psi }_0⟩$. Angle $\theta$ is the relative phase between the two arms. Beam splitters perform $\pm \frac{\pi }{2}$ rotations about the $x$ axis. Detectors at the outputs count photon numbers $n_a$ and $n_b$.

Figure 2

Comparing distinguishability $\mathcal{D}(\chi ,\Delta )$ in relatively local ($\Delta ={10}^{-3}$) and global ($\Delta =\pi$) regimes for various inputs $|{\psi }_0⟩$. Each scatter point corresponds to a photon number $2j=n\in [5,50]$, shaded according to $\chi \in \{\frac{\pi }{2},\frac{3\pi }{4},\pi \}$. Pentagons depict NOON states ($\zeta =0$), triangles $|j,0{⟩}_z$, squares $|j,+j{⟩}_z$, and stars denote phase states of Eq. (15) with $\gamma =\pi /2$. NOON states exhibit the best local distinguishability, yet in the $\Delta =\pi$ domain their $\mathcal{D}$ is small and does not improve with photon number. In contrast, $|j,+j{⟩}_z$ (corresponding to only one illuminated input port) has strong global characteristics but poor local distinguishability. The phase states seem to offer the best compromise performance, as the trail of (black) stars travels the furthest into the top right corner of the graph. Notice that the choice of $\chi$ has a significant effect on distinguishability for finite $\Delta$.