Enhancing the Size of Phase-Space States Containing Sub-Planck-Scale Structures via Non-Gaussian Operations

TL;DR

This work investigates enlarging phase-space states with sub-Planck-scale structures by applying non-Gaussian photon-addition to Gaussian-prepared superpositions (SSD, SS, Sq) and comparing them to parity-matched cat and compass states. Using quantum Fisher information $F_Q^{\psi}$ and fidelity, it identifies regimes where photon-added variants share metrological power with targets while achieving larger amplitudes $|\beta|$ and reduced Wigner-fringe sizes, thereby enabling efficient Gaussian preprocessing plus PA. The results reveal equal-$F_Q$ contours that move toward higher amplitudes as the photon-number $n$ increases, with high-fidelity instances (e.g., $n$-PASSD vs KS$^{l}(-)$ and $n$-PASq vs Cat$^{l}$) at large $|\beta|$, suggesting practical routes for metrology and cat-code error correction. The work highlights energy-resource considerations and proposes near-deterministic approaches (e.g., small $r$ and $|\alpha|$) to realize these non-Gaussian operations in platforms such as circuit QED, trapped ions, and optomechanics, with implications for enhancing phase-space sensitivity and robust quantum information encoding.

Abstract

We observe a metrological advantage in phase-space sensitivity for photon-added cat and kitten states over their original forms, due to phase-space broadening from increased amplitude via photon addition, albeit with higher energy cost. Using accessible non-classical resources, weak squeezing and displacement, we construct a squeezed state and two superposed states: the squeezed cat state and the symmetrically squeezed state. Their photon-added variants are compared with parity-matched cat and KSs using quantum Fisher information and fidelity. The QFI isocontours reveal regimes where KS exhibit high fidelity and large amplitude, enabling their preparation via Gaussian operations and photon addition. Similar regimes are identified for cat states enhanced by squeezing and photon addition, demonstrating improved metrological performance. Moreover, increased amplitude and thus larger phase-space area reduces the size of interferometric fringes, enhancing the effectiveness of quantum error correction in cat codes.

Figures (4)

• Figure 1: (Color online) Evident from phase space distribution (first two left column) and central fringe area (CFA), there is an indication of enhancement for phase-space area (support) and metrological power (in $3^{rd}$ column for $O_{\lambda_{o}}$) for both cat and compass state upon photon addition. The relation is clearly seen in the rightmost column for CFA between adding photons and increasing amplitude $(\alpha)$ for the cat state and KS.
• Figure 2: (Color online) QFI and fidelity ($\mathcal{F}$) comparison among $n-$PACatE, $n-$PAKS, and their corresponding parity-matched CatE and KS counterparts. The top panels (a–d) illustrate QFI constraints in space of coherent amplitudes $|\alpha|\in \hat{a}^{\dagger\,n}\psi$ and $|\beta|\in\psi$. Specifically: (a) takes pair $\{\hat{a}^{\dagger\,n}\text{CatE},\psi_{\text{Cat}}^{n}\}$; (b) compares $\hat{a}^{\dagger\,n}\psi_{KS}^{0}(-)$ with $\psi_{KS}^{l}(-)$ where $l = \sin(n\pi/2)$; (c) contrasts $\hat{a}^{\dagger n}\psi_{KS}^{1}(-)$ with $\psi_{KS}^{l}(-)$ where $l = |\cos(n\pi/2)|$; and (d) identifies a QFI equality condition between $\hat{a}^{\dagger n}\psi_{KS}^{0}(+)$ and $\psi_{KS}^{n}(+)$. The lower panels (e-h) present fidelity across the same parameter space, showing large $\mathcal{F}$ for amplitudes as a consequence of multiple $n$ photon addition, shifting the equal QFI contours to enhanced amplitudes i.e., $|\beta|>|\alpha|$ .
• Figure 3: (Color online) Equal QFI ($F_Q^\psi=F_Q^\phi$) regimes are shown for $n$-PASSD vs. $\psi_{\text{KS}}^l(-)$ in (a-b), $n$-PASS vs. $\psi_{ks}^l(+)$ in (c), and $n$-PACatE vs. $\psi_{\text{Cat}}^n$ in (d) across relevant parameter spaces. The parameters $\{r,\alpha\}\in n-$PASSD, while $\beta\in\psi_{\text{KS}}^l(-)$ in plots (a) and (b); $r\in n$-PASS, and $\beta\in\psi_{\text{KS}}^l(+)$ in (c); and $r\in n$-PACatE, and $\beta\in\psi_{\text{Cat}}^l$ in (d). The plots (a-b) are 2D projections of the 3D contours satisfying $F_Q^{n-\text{PASSD}} = F_Q^{\psi_{\text{KS}}^l(-)}$. The scattered data points form curves, indicating an increase in the target's amplitude ($\beta$) with each successive photon addition $n$.
• Figure 4: (Color online) The effect of photon additions on proposed SSD, SS and Sq, is clearly seen in high fidelity ($\mathcal{F}$), leading to an increase in $\beta$ of KS in the plots (a) and (b), and of CatE in (c). (a) Fidelity remains close to one for $1$-PASSD against $\psi^{l}_{\text{KS}}(-)$ for $\beta$ less than 1.8, while the addition of more than 1 photon leads to a decrease in the maximum of fidelity to 0.9 around $\beta=2$. (b) Each subsequent PA to SS, leads to an increase in $\beta$ for fidelity$\,\sim\,1$. In (a-b), the $\{r,\alpha\}$ parameters satisfy equal Fisher information for states: ($n-$PASSD, $\psi^{l}_{\text{KS}}(-)$) and ($n-$PASS, $\psi^{l}_{\text{KS}}(+)$). Specifically, in plot (a), $\psi^{l}_{\text{KS}}(-)$ is parametrized with $l=(1+e^{in\pi})/2$, while in plot (b), $\psi^{l}_{\text{KS}}(+)$ takes $l=n$ as same as in (c) b/w $n-$PASq and Catn.