Nonlinear squeezing of superpositions of quadrature eigenstates

TL;DR

This work introduces a nonlinear squeezing framework for evaluating how well approximations to ideal superpositions of quadrature eigenstates (SQE) perform as non-Gaussian resources. By constructing the operator family $\hat{O}(u,\varphi,c)$ with $\hat{O}_x(u)=(\hat{x}^2-u^2)^2$ and $\hat{O}_p(u,\varphi)$, the authors define a nonlinear squeezing measure $\xi_{\mathrm{SQE}}$ that acts as a non-Gaussianity witness and links to state fidelity in virtual interactions such as $\hat{D}_x(u)+\hat{D}_x(-u)$ and to GKP state breeding. They develop a numerical Pareto-frontier approach in truncated Fock spaces to bound the minimal interaction fidelity $F_{\min}$ for a given $\xi_{\mathrm{SQE}}$, and to bound (and even optimize) GKP squeezing in breeding protocols. The paper further shows that optimal SQE approximations correspond to ground states of $\hat{O}^{[N]}(u,\varphi,c)$ in dimension $N$ and connects these to stellar rank, offering practical target states for experimental realization within resource-constrained settings. Overall, SQE squeezing provides a concrete, computable bridge between non-Gaussianity witnesses and protocol performance in continuous-variable quantum information.

Abstract

We introduce a family of operators exploiting the symmetry of superpositions of quadrature eigenstates (SQE) and demonstrate how the associated nonlinear squeezing, quantified by the expectation value of such operators, serves both as a witness of non-Gaussianity and as an indicator of the quality of SQE approximations. To establish the usefulness of this measure, we connect it to quantum state fidelity and evaluate its implications in breeding protocols. Finally, we construct optimal approximations of SQE states in truncated Fock spaces.

Figures (5)

• Figure 1: Schematic visualization of the virtual interaction-based method for evaluating SQE approximations. Modes containing the evaluated state $\ket{\mathrm{R}}$ and an ideal superposition of quadrature eigenstates $\ket{C_{u,\infty,\varphi}}$ are split on identical beam splitters. Output states $\ket{\Omega}$ are obtained by homodyne measurement of $p=0$ in the remaining modes and subsequently utilized to calculate the virtual interaction based fidelity $F_\mathrm{BS}$.
• Figure 2: Numerically optimized lower bounds for virtual interaction fidelity $F_\mathrm{min}$ given SQE squeezing $\xi_\mathrm{SQE}$. Subplot (a) represents how the boundary changes with changes in relative weight $c$ between the constituents of $\hat{O}\left(u,\varphi,c\right)$, see \ref{['eq:o']}. Subplot (b) represents how the boundaries with specific $c$ differ with changes in dimensionality of the space $N$. The squeezing was evaluated in truncated Fock spaces using $\hat{O}^{[N]}(3,0,c)$. The bounds represent worst-case scenarios, no states underneath the curves exist in their respective spaces. Therefore, we can infer minimum values of virtual interaction fidelity given SQE squeezing and vice versa.
• Figure 3: Numerically optimized upper bounds for GKP squeezing after two rounds of breeding given SQE squeezing $\xi_\mathrm{SQE}$. Subplot (a) represents how the boundary changes with changes in relative weight $c$ between the constituents of $\hat{O}\left(u,\varphi,c\right)$, see \ref{['eq:o']}. Subplot (b) represents how the boundaries with specific $c$ differ with changes in dimensionality of the space $N$. The squeezing was evaluated in truncated Fock spaces using $\hat{O}^{[N]}(3,0,c)$. The area with gray filling represents the values of $\xi_\mathrm{GKP_0}$ for which fault tolerance is possible marek2024. Therefore, the intersections of the boundaries with this area provide a necessary condition for $\xi_\mathrm{SQE}$ for the resource states in order to breed GKP state approximations sufficient for fault tolerant quantum computation under ideal conditions.
• Figure 4: Optimal approximations for superpositions of quadrature eigenstates in truncated Fock spaces. Each subplot visualizes the Wigner function of the state best approximating an even eigenket superposition $\ket{x=3}+\ket{x=-3}$ in an $N-$dimensional truncated Fock space. The plotted states were obtained as ground states of $N-$dimensional representations of $\hat{O}\left(u,\varphi,c\right)$ with $u=3, \varphi = 0, c=10$\ref{['eq:o']}. As such, these states minimize SQE squeezing \ref{['eq:sqesq']} in their respective truncated Fock spaces and are the optimal approximations of $\ket{x=3}+\ket{x=-3}$ with stellar rank at most $r^\star$\ref{['eq:stellar']}.
• Figure 5: Accuracy of $\tilde{\hat{O}}_p(u,\varphi)$ in infinite dimensional spaces. Each plot represents the accuracy of the infinite dimensional approximation \ref{['ap:aprox']} for a)$u=1$, b)$u=2$, c)$u=3$, d)$u=4$, $\varphi = 0$ in all cases. The approximation was calculated for individual Fock states and its accuracy is expressed as the normalized density of points, for which $\left\vert1-\expval{\tilde{\hat{O}}_p(u,0)}{n}/\expval{\hat{O}_p(u,0)}{n}\right\vert\leq 1\%$, i.e. if a single point is picked in the 0.9 contour, the probability of the approximation being off by more than 1 % is smaller than 0.1.