"Quantum teleportation" refers to any method or phenomena (such as short-range teleportation or long-range FTL travel) in Mobile Suit Gundam 00 used to circumvent traditional distances in normal space.
Description & Characteristics
Note: Most of what is written below is expanded writing based on observations/conjecture and limited information that is currently available.
Quantum teleportation is a phenomenon that allows an object or vehicle to circumvent traditional distances through the usage of GN Particles. The nature of this phenomena was totally based on time-scape control done by supersaturating a plane with further purified GN Particles.
Originally, quantization was a phenomenon created through extremely high purified GN Particle concentrations caused by the GN-0000+GNR-010 00 Raiser's Trans-Am System. This phenomenon allowed 00 Raiser to teleport away from its current position to another location nearby (often in dodge of danger). Much is still not known about how 00 Raiser uses this ability, as it was not originally designed with the 00 Raiser machine, and it is related to 00 Raiser"s Twin Drive System, which produces a squared output of the two synchronized drives. The way the pilot activates and uses the phenomenon with 00 Raiser is also not clear at this moment, although it seems like Setsuna has the ability to force-control 00 Raiser purely by thoughts alone for a short period of time. 00 Raiser's successor, the GNT-0000 00 Qan[T] possesses the ability to quantum teleport, but unlike 00 Raiser which uses it to move medium range distances in battles, the 00 Qan[T] can use it to travel extremely great distances in deep space, thereby functioning as a real FTL transportation mechanism (without the use of the Trans-Am system). The 00 Qan[T] quantum teleports by having its GN Sword Bits form a circle, which take part in GN Particle channelization and thus supersaturating an equator plane with purified GN Paticles, to create a quantum gateway in which it enters. 
However, the presence of the sword bits are not needed for instant teleportation when 00 Quanta is in its trans-am mode as the unit is already supersaturated with GN Particles.
The ability to quantum teleport is also available to Extraterrestrial Living-metal Shape-shifter or ELS, who uses it during the ELS Conflict. After the "large ELS" received information about Quantum Brainwaves on Earth from the scouting party, it quantum teleported into Jupiter and exited from its red spot. In spite of this being one of the forms of quantum teleportation is currently unknown how the physics/method is similar or different from the humans' method of quantum teleportation; if any.
After the ELS Conflict, the Earth Sphere Federation eventually came into possession of true GN Drive technology, and continued to refine the technology. They eventually developed new methods of controlling the flow of GN Particles to generate quantum gateways without the need for remote bits. In conjunction with refined GN Drive technology, new generation mobile suits like the GNW-100A Sakibure can now generate quantum gateways without the GN Particle requirements supplied by a Twin Drive System. However, it can be equally suggested that a single drive's output has been refined to generate near Twin Drive levels. The technology has since been diversified past Mobile Suit development. As the technology expanded beyond combat applications, the newest generation of starships like the Sumeragi have their own FTL-Drives for deep space exploration and travel.
- ↑ Mobile Suit Gundam 00N Chapter 16