Distributed quantum metrology with a single squeezed-vacuum source

We propose an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number $M>1$ of unknown phase delays, distributed across an $M$-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number-resolving detectors, or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relies upon a single squeezed-state source, an antisqueezing operation at the interferometer output, and on-off photodetectors.

Figure 1

Interferometric setup with only a single squeezed-vacuum source for the Heisenberg-limited estimation of the linear combination of unknown phases ${\phi }_i$, with $i=1,...,M$, as in Eq. (1). The linear optical networks represented by $\widehat{U},{\widehat{U}}^{†}$ are set up in such a way as to satisfy Eq. (5). The operators $\widehat{S}\left(z\right),{\widehat{S}}^{†}\left(z\right)$ represent squeezing and antisqueezing operations, respectively.

Figure 2

Estimation of the linear combination two unknown phases ${\phi }_1,{\phi }_2$ with weights $w_1,w_2$. The interferometric scheme in Fig. 1 reduces to a Mach-Zehnder interferometer, with unbalanced beam splitters with $R/T=w_1/w_2$, where $R$ is the reflectivity and $T$ is the transmittivity.