It's not particularly theoretical; full-frame cameras often offer a crop mode (Nikons default to DX crop mode when a DX lens is attached). Per pixel, the image is the same whether shot cropped or cropped to the same dimensions in post processing. That said, the image quality for the "same" picture (cropped versus full-frame at the same output size) is not the same, since the same per-pixel noise values represent 2.25 times more of the final image's total data. There may be every bit as much noise per pixel in a full-frame image, but the total error (difference from the "ideal" values in the image or in any object represented in the image, assuming the same composition/framing) will come closer to approximating zero simply because there are more of them.
(A decent-enough, though still inaccurate, rough analogy would be inkjet or process printing. Almost none of the actual individual values in an inkjet print is correct, but the higher the printer DPI for the same image pixel dimensions and PPI the more the resulting print looks like a continuous-tone image without altering the accuracy of any of the individual dots.)
With a smaller sensor (assuming the same desired output size), you either need to use the same number of smaller pixels, each with a greater individual error, or fewer pixels, the error from each of which is going to be more visible in the final image.