How Many Pinching Antennas Are Enough?

Programmable wireless environments (PWEs) have emerged as a key paradigm for next-generation communication networks, aiming to transform wireless propagation from an uncontrollable phenomenon into a reconfigurable process that can adapt to diverse service requirements. In this framework, pinching-antenna systems (PASs) have recently been proposed as a promising enabling technology, as they allow the radiation location and effective propagation distance to be adjusted by selectively exciting radiating points along a dielectric waveguide. However, most existing studies on PASs rely on the idealized assumption that pinching-antenna (PA) positions can be continuously adjusted along the waveguide, while realistically only a finite set of pinching locations is available. Motivated by this, this paper analyzes the performance of two-state PASs, where the PA positions are fixed and only their activation state can be controlled. By explicitly accounting for the spatial discreteness of the available pinching points, closed-form analytical expressions for the outage probability and the ergodic achievable data rate are derived. In addition, we introduce the pinching discretization efficiency to quantify the performance gap between discrete and continuous pinching configurations, enabling a direct assessment of the number of PAs required to approximate the ideal continuous case. Finally, numerical results validate the analytical framework and show that near-continuous performance can be achieved with a limited number of PAs, offering useful insights for the design and deployment of PASs in PWEs.