Metrological advantage at finite temperature for Gaussian phase estimation

In the context of phase estimation with Gaussian states, we introduce a quantifiable definition of metrological advantage that takes into account thermal noise in the preparation procedure. For a broad set of states—isotropic nonpure Gaussian states—we show that squeezing is not only necessary but sufficient to achieve metrological advantage. We interpret our results in the framework of resource theory and discuss possible sources of advantage other than squeezing. Our work is a step towards using phase estimation with pure and mixed states to define and quantify nonclassicality. This work is complementary with studies that defines nonclassicality using quadrature displacement estimation.

Figure 1

(a) Schematic depiction of the metrological setup. The input state $\widehat{\rho }$ is modified by a unitary $\widehat{U}$, sent inside the interferometer, then measured. We also consider a perfect phase reference (red dotted line) which allows us to make homodyne measurements. (b) Different possible configurations for $\widehat{U}$. ${\widehat{J}}_z$ rotations are only phase differences. A rotation $e^{i\varphi {\widehat{J}}_x}$ corresponds to a beam splitter with transmission coefficient $cos\frac{\varphi }{2}$ (which can also be emulated with phase shifters and balanced beam splitters). The Mach-Zehnder in (a) is described by a rotation $e^{i\varphi {\widehat{J}}_y}$; such an operation can also be directly implemented with an unbalanced beam splitter as displayed in (b). In all figures, we have omitted additional $\frac{\pi }{2}$ phase shifts for readability.

Figure 2

Left column: ${\varphi }_1+{\varphi }_d=0$, right: ${\varphi }_1+{\varphi }_d=\frac{\pi }{2}$. The top figures are schematic depictions of the Wigner function of the states in both modes. The curves below show the evolution of $\frac{I_F^{\text{opt}}-I_F^{\text{ref}}}{I_F^{\text{ref}}}$ as a function of the other parameters. From top to bottom: evolution with $\left|\gamma \right|$, for $\nu =10$ and various squeezing; evolution with $r$ (in dB), for $\left|\gamma \right|=100$ and $\nu =10$; evolution with $\nu$, for $\left|\gamma \right|=100$ and $r=10dB$ of squeezing. In the case ${\varphi }_1+{\varphi }_d=0$, one can distinguish two distinct regimes; the optimal strategy changes when one goes from one regime to the other.

Figure 3

Summary of the links between squeezing, particle entanglement, and metrological advantage, with and without SSR, for Gaussian or arbitrary states. Full arrows are established results, dotted arrows are conjectures. Our results and conjecture correspond to the green arrows on the first cartoon. The third cartoon corresponds to the results of [9].

Figure 4

Addition of ancilla. A displaced thermal state ${\widehat{\rho }}_{\text{anc}}$ is sent in a third mode; the three modes interacts via a controlled free unitary, and then the ancilla is discarded. If the ancilla is a coherent state with a lot of photons and $\widehat{U}$ is just a mode-mixing operation between the ancilla and the first mode, this operation amounts to a displacement of the first mode.