It is known that a semiclassical analysis is not always adequate for atomtronics devices, but that a fully quantum analysis is often necessary to make reliable predictions. While small numbers of atoms at a small number of sites are tractable using the density matrix, a fully quantum analysis is often not straightforward as the system becomes larger. We show that the fully quantum positive-P representation is then a viable calculational tool. We postulate an atomtronic phase gate consisting of four wells in a Bose-Hubbard configuration, for which the semiclassical dynamics are controllable using the phase of the atomic mode in one of the wells. We show that the quantum predictions of the positive-P representation for the performance of this device have little relation to those found semiclassically, and that the performance depends markedly on the actual quantum states of the initially occupied modes. We find that initial coherent states lead to closest to classical dynamics, but that initial Fock states give results that are quite different. A fully quantum analysis also opens the door for deeply quantum atomtronics, in which properties such as entanglement and Einstein-Podolsky-Rosen steering become valuable technical attributes of a device.