In arnold you can turn off "camera ray visibility" for the backdrop. It will still contribute to all calculations but it wont be visible to the camera. Edit: You might need to add the "arnold properties" to the object to get this setting. I don´t know how its handled in maya, i only used arnold in softimage and houdini.
Thats exactly what I did .Wiki! Just turned off primary visibility of the object in the arnold tab (even though I was using vray) and kept all the shadows and diffuse/spec reflections and hey presto I had the alpha!