Lower-bound conjecture for the effective squeezing parameter

Let $\rho$ be a physical state on an $n$-mode quantum system with average photon number

$$\overline{N}=\operatorname{Tr}(\rho\hat{N}).$$

For a symplectic basis $M=(\bs{\xi}_1,\hdots,\bs{\xi}_{2n})^T$, define the effective squeezing parameter by

$$\tilde{\Delta}^2(\rho)\coloneqq \min_{M\in\operatorname{Sp}_{2n}(\mathbb{R})}\min_i\Delta^2_{\bs{\xi}_i}(\rho).$$

Effective-squeezing lower-bound conjecture. The effective squeezing satisfies

$$\tilde{\Delta}^2(\rho)\geq \frac{4}{\overline{N}}.$$

The conjecture would give a lower bound depending only on the state's average photon number, supporting the physicality assumptions used to control local GKP shadow sample complexity. The supplied text motivates the bound from approximate GKP states but does not establish it for general physical states.