Quantum Reference Frame
A reference frame can be treated in the formalism of quantum theory, and, in this case, such is referred as a quantum reference frame. Despite different name and treatment, a quantum reference frame still share much of the notions with a reference frame in classical mechanics. It is still defined with the same definition. It is still always associated to some physical system. And it is still always relational.
For example, if a spin-1/2 particle is said to be in the state, reference frame is implicitly implied, and it can be understood to be some reference frame with respect to an apparatus in a lab. It is obvious that the description of the particle does not place it in an absolute space, and doing so would make no sense at all because, as mentioned above, absolute space is empirical unobservable. On the other hand, if a magnetic field along y-axis is said to be given, behaviour of the particle in such field can then be described. In this sense, y and z are just relative directions. They do not and need not have absolute meaning.
One can observe that a z direction used in a laboratory in Berlin is totally different from a z direction used in a laboratory in Melbourn. Issues like aligning and sharing reference frames arise when a communication between two laboratories are tried to be set up. It is absolutely correct, and the study of communications like this has been the main agenda in quantum information theory.
Just as in this spin-1/2 particle example, a quantum reference frames are almost always treated implicitly in the definition of quantum states, and the process of including the reference frame in a quantum state is called quantisation/internalisation of reference frame while the process of excluding the reference frame from a quantum state is called dequantisation/externalisation of reference frame. Unlike the classical case, in which treating a reference internally or externally is purely an aesthetic choice, internalising and externalising a reference frame does make a difference in quantum theory.
One final remark may be made on the existence of a quantum reference frame. After all, a reference frame, by definition, has a well-defined position and momentum, while quantum theory, namely uncertainty principle, states that one cannot describe any quantum system with well-defined position and momentum simultaneously, so it seems there is some contradiction between the two. It turns out, an effective frame, in this case a classical one, is used as a reference frame, just as in Newtonian mechanics a nearly inertial frame is used, and physical laws are assumed to be valid in this effective frame. In other words, whether motion in the chosen reference frame is inertial or not is irrelevant.
The following treatment of a hydrogen atom motivated by Aharanov and Kaufherr can shed light on the matter. Supposing a hydrogen atom is given in a well-defined state of motion, how can one describe the position of the electron? The answer is not to describe the electron's position relative to the same coordinates in which the atom is in motion, because doing so would violate uncertain principle, but to describe its position relative to the nucleus. As a result, more can be said about the general case from this: in general, it is permissible, even in quantum theory, to have a system with well-defined position in one reference frame and well-defined motion in some other reference frame.
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