networking - Remotely turn on a device that is powered off?

Verbatim from the interview between Brian Williams and Edward Snowden last night:

Snowden: Any intelligence service in the world that has significant funding and a real technological research team can own that phone [refering to Williams' iPhone] as soon as it connects to a network; it can be theirs.

Williams: Can anyone turn it on remotely if it's off?

Snowden: They can absolutely turn it on with the power turned off on the device.

I concede Snowden's first point, fine. But the last? Does Snowden's last claim, that a powered off mobile device can be remotely turned, have any merit? It seems highly improbable and seemingly impossible.

There are lights-off management systems in high end servers from 9 years ago. Those systems "keep a flame burning" and when an event happened they woke up the whole system. Now, events that are software generated, such as a button being tapped or some such, could ping one of these lights-out devices.

Are those backdoors built in for use by wireless contact? I don't know. It is technically feasible, it just means a bit more power drain when totally powered off. Is it universally true? Not unless there is something that breaks the rules.

How do you know your phone is REALLY OFF, without removing the battery altogether ?

Everything on the phone is software controlled. And that includes the on/off button.

With low-level software control, which NSA and other agencies can obtain via hacking or backdoors as Snowden claims, it is completely feasible (even EASY) to make the phone appear to be OFF to the owner, while still keeping a connection with the agency.

The possibilities are endless: Just use the phones microphone to permanently "bug" the owner. Use the GPS to track where he is. Use the camera (in case the phone isn't in his pocket) to take photos of his surroundings.

This is very much feasible. Usually all the high end devices maintain a small "battery" other than the main one which you normally see. On paper, its purpose is to maintain things like time. That is why when you turn off your device for a day and turn it back on, it has the right time since the clock was very much "on" and was being powered by the smaller invisible battery (usually a bunch of capacitors but it might have evolved into something more sophisticated).

Most (all?) of the Apple and Google devices don't even allow removing the main battery so they have quite higher power at their disposal to power on/off components in a device.

These batteries can be used to power certain signalling components easily which an average human would be totally oblivious to.