Tunable Non-Gaussianity and Exact Higher-Order Coherences for Quantum Advantage

Abstract

Non-Gaussian states are essential for achieving a quantum advantage in continuous-variable (CV) information processing. Among these, coherent superpositions of squeezed states are a foundational resource. While exact higher-order statistics are available in the undisplaced case, a complete and analytically tractable treatment with a common displacement has been missing. We introduce and solve the displaced Janus state-a coherent superposition of two squeezed coherent states that share the same displacement-and develop an analytical framework, based on a family of Generalized Squeezing Polynomials, that yields closed-form expressions for arbitrary-order factorial moments and coherence functions \(g^{(k)}(0)\), the full Wigner function, and the quantum Fisher information. The analysis shows how interference at a fixed mean, driven by a mismatch of the component covariances rather than by mean separation, can be precisely engineered to transform the extreme photon bunching of the constituents into strong sub-Poissonian and even perfect multiphoton suppression. We further provide a rigorous quantum Fisher information analysis, proving that parameters encoded by linear generators (for example, the number operator) are bounded by the standard quantum limit, whereas parameters encoded by quadratic generators (for example, squeezing transformations) achieve Heisenberg-limited scaling. Together, these results furnish a complete analytical toolkit for a versatile class of non-Gaussian states, establishing the displaced Janus state as a key primitive for hybrid quantum protocols, quantum metrology, and fault-tolerant continuous-variable computation.

Figures (9)

• Figure 1: Generalized Squeezing Functions $F_{p,q}(z)$ for real $z\in[-1,1)$. Top: linear scale. Bottom: log magnitude highlighting the algebraic singularity at $z\to 1^{-}$ predicted by $F_{p,q}(z)=P_{p,q}(z)/(1-z)^{(p+q+1)/2}$. Curves shown: $F_{0,0}, F_{1,1}, F_{0,2}, F_{2,2}, F_{1,3}$.
• Figure 2: Phase–space maps of $g^{(2)}(0)$ for a squeezed coherent state with $r=1$. Each panel corresponds to a different squeezing angle $\theta$; since the dependence is through the relative phase $2\varphi-\theta$, varying $\theta$ rotates the pattern. The black contour indicates $g^{(2)}(0)=1$; blue (red) shades denote sub- (super-) Poissonian regions.
• Figure 3: Phase–space maps of $\log_{10} g^{(3)}(0)$ at $r=1$ for four $\theta$. The black contour marks $g^{(3)}(0)=1$ ($\log_{10}=0$).
• Figure 4: Phase-space maps of $\log_{10} g^{(4)}(0)$ at $r=1$ for four squeezing angles $\theta$. The black contour marks $g^{(4)}(0)=1$ ($\log_{10}=0$). A logarithmic scale reveals both the large central bunching (bright region) and the phase-rotated angular structure that would be compressed on a linear scale.
• Figure 5: Higher-order coherence of a squeezed coherent state ($r=1.0$). Zero-delay correlations $g^{(k)}(0)$ for $k=2,3,4,5$ versus displacement $|\alpha|$, shown for relative phases $2\varphi-\theta\in\{0,\pi/4,\pi/2,3\pi/4,\pi\}$. Dashed lines mark the coherent-state benchmark $g^{(k)}(0)=1$. Across all orders and phases, the curves approach unity with increasing $|\alpha|$; for small $|\alpha|$ the departure from unity amplifies with $k$ and is phase-ordered, with $2\varphi-\theta\approx\pi$ giving the strongest suppression.

Theorems & Definitions (3)

• Theorem 1: Symmetry of Generalized Squeezing Polynomials
• proof
• Remark 1: Covariance-driven negativity at $|\alpha|=0$