Enhancing Non-classical Properties of Entangled Coherent States via Post-Selected von Neumann Measurements

TL;DR

This work investigates how post-selected weak measurements, modeled via the von Neumann interaction, modulate the non-classical properties of entangled coherent states (ECSs). By representing the ECS input, applying a phase shift, and performing post-selection, the authors connect weak-value amplification to observable enhancements in squeezing, entanglement, and phase estimation, controlled by the coupling strengths $s_1$ and $s_2$. The key findings show that increasing coupling drives the two-mode system from anti-squeezed toward highly squeezed regimes, expands negative regions in the joint Wigner function into multi-branch interference, strengthens Hillery-Zubairy entanglement, and lowers the quantum Cramér-Rao bound via higher Quantum Fisher Information. Overall, the results provide a tunable framework for measurement-based manipulation of continuous-variable entangled states with potential impact on quantum metrology and state engineering.

Abstract

The present study systematically investigates the modulation mechanism of post-selected weak measurement (WM) on the non-classical properties of entangled coherent states (ECSs). The primary goal is achieving controllable enhancement of quantum resources such as entanglement and squeezing, while minimising disturbance to the quantum state. We theoretically analyze the post-selected weak measurement process, and its effectiveness in amplifying the non-classical features of ECSs. The von Neumann measurement model is employed to analytically describe the weak-value amplification of the pointer state. It is demonstrated that a significant enhancement of squeezing can be achieved by tuning the measurement coupling strength. The joint Wigner function analysis in phase space further reveals that, as the coupling strength increases, the coherent structure of the state evolves from symmetric double peaks to multi-branch quantum interference fringes. The entanglement, quantified by the Hillery-Zubairy criterion, exhibits a pronounced increase with stronger coupling, while the quantum Fisher information indicates a systematic improvement in phase estimation precision. The results obtained establish a tunable weak measurement framework for precise manipulation of continuous-variable entangled states. This provides a feasible theoretical pathway for enhancing quantum metrology and measurement-based state engineering.

Figures (6)

• Figure 1: Schematic diagram of weak measurement theory and setup for preparing the state $|\Phi\rangle$ using a post-selected von Neumann measurement. Schematic diagram of the measurement on mode-a, showing three key steps: (1) Preparation: The measured system and pointer are initially prepared in states $|\psi_{i}\rangle$ and $|\phi\rangle$, respectively. (2) Weak Interaction: A weak interaction between the measured system and measuring device governs the evolution of the composite system. (3) Post-selection: Following evolution, the composite system is projected onto the measured system state $\vert\Phi\rangle$, deliberately and technically chosen to obtain the desired system observable values.
• Figure 2: Post-selection success probability of the final pointer state $\left|\Phi\right\rangle$ as a function of coupling strength ($s_{1}=s_{2}$) for different weak value parameters $(\theta_{1}=\theta_{2}$), with the parameters $\mu=\varphi=\delta_{1}=\delta_{2}=\pi/2$, and $r=0.1$.
• Figure 3: The variation of sum squeezing ($\mathrm{S_{2s}}$) in the state $|\Phi\rangle$ as a function of coupling coefficients $s_{1}$ and $s_{2}$, with the parameters $\Theta=\mu=\varphi=\delta_{1}=\delta_{2}=\pi/2$, $\theta_{1}=\theta_{2}=4\pi/5$, and $r=0.1$.
• Figure 4: Cross-section of the scaled joint Wigner function $P_{J}(\gamma,\beta)$ for state $|\Psi\rangle$ in the $Re(\gamma)$-$Re(\beta)$ plane, with $Im(\gamma)=Im(\beta)=0$. (a) $s_{1}=s_{2}=0$; (b) $s_{1}=s_{2}=1$; (c) $s_{1}=s_{2}=2$. Darker shades indicate negative values, a signature of nonclassicality. Other parameters: $r=0.1$, $\mu=\varphi=\delta_{1}=\delta_{2}=\pi/2$, and $\theta_{1}=\theta_{2}=4\pi/5$.
• Figure 5: The variation of Hillery-Zubairy correlation($\mathrm{E}$) in the state $\vert\Phi\rangle$ as a function of coupling coefficients $s_{1}$ and $s_{2}$, with the parameters $\mu=\varphi=\delta_{1}=\delta_{2}=\pi/2$, $\theta_{1}=\theta_{2}=4\pi/5$, and $r=0.1$.