Quantum magnets, which involve strong quantum fluctuation stemming from noncommutative properties of spin operators, have attracted much attention because of those fascinating properties. In this paper, we report detailed behaviors of the ferroelectricity, driven by the field-induced Bose-Einstein condensation of magnon quasiparticles in the quantum spin gap system TlCuCl3. Superposition of the wave functions inherent in a quantum magnet in its ground state plays a key role in the appearance of this ferroelectricity. The field dependence of the spontaneous electric polarization clearly demonstrates that the ferroelectricity is caused by the emergence of the vector spin chirality in the magnon Bose-Einstein condensate. The ferroelectricity is suggested to be significantly enhanced by quantum entanglement in the spin dimer. Furthermore, reflecting the isotropic nature of TlCuCl3, the ferroelectricity is very soft with a low electric coercive field Er≃0.03 MV/m. Our analysis indicates that vector components of the electric polarization, which are not caused by the spin current mechanism, appear in TlCuCl3.