Quantum reference frame

A quantum reference frame is a reference frame which is treated quantum theoretically. It, like any reference frame, is an abstract coordinate system which defines physical quantities, such as time, position, momentum, spin, and so on. Because it is treated within the formalism of quantum theory, it has some interesting properties which do not exist in a normal classical reference frame.

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 shares much of the notions with a reference frame in classical mechanics. It is associated to some physical system, and it is relational.
For example, if a spin-1/2 particle is said to be in the state, a reference frame is 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 empirically unobservable. On the other hand, if a magnetic field along y-axis is said to be given, the 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 generally totally different from a z direction used in a laboratory in Melbourne. Two laboratories trying to establish a single shared reference frame will face important issues involving alignment. The study of this sort of communication and coordination is a major topic in quantum information theory.
Just as in this spin-1/2 particle example, 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 uncertainty 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.

Further considerations of quantum reference frame

An example of treatment of reference frames in quantum theory

Consider a hydrogen atom. Coulomb potential depends on the distance between the proton and electron only:
With this symmetry, the problem is reduced to that of a particle in a central potential:
Using separation of variables, the solutions of the equation can be written into radial and angular parts:
where, and are the orbital angular momentum, magnetic, and energy quantum numbers, respectively.
Now consider the Schrödinger equation for the proton and the electron:
A change of variables to relational and centre-of-mass coordinates yields
where is the total mass and is the reduced mass. A final change to spherical coordinates followed by a separation of variables will yield the equation for from above.
However, if the change of variables done early is now to be reversed, centre-of-mass needs to be put back into the equation for :
The importance of this result is that it shows the wavefunction for the compound system is entangled, contrary to what one would normally think in a classical standpoint. More importantly, it shows the energy of the hydrogen atom is not only associated with the electron but also with the proton, and the total state is not decomposable into a state for the electron and one for the proton separately.

Superselection rules

Superselection rules, in short, are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced to impose additional restriction to quantum theory beyond those of selection rules. As an example, superselection rules for electric charges disallow the preparation of a coherent superposition of different charge eigenstates.
As it turns out, the lack of a reference frame is mathematically equivalent to superselection rules. This is a powerful statement because superselection rules have long been thought to have axiomatic nature, and now its fundamental standing and even its necessity are questioned. Nevertheless, it has been shown that it is, in principle, always possible to lift all superselection rules on a quantum system.

Degradation of a quantum reference frame

During a measurement, whenever the relations between the system and the reference frame used is inquired, there is inevitably a disturbance to both of them, which is known as measurement back action. As this process is repeated, it decreases the accuracy of the measurement outcomes, and such reduction of the usability of a reference frame is referred to as the degradation of a quantum reference frame. A way to gauge the degradation of a reference frame is to quantify the longevity, namely, the number of measurements that can be made against the reference frame until certain error tolerance is exceeded.
For example, for a spin- system, the maximum number of measurements that can be made before the error tolerance,, is exceeded is given by. So the longevity and the size of the reference frame are of quadratic relation in this particular case.
In this spin- system, the degradation is due to the loss of purity of the reference frame state. On the other hand, degradation can also be caused by misalignment of background reference. It has been shown, in such case, the longevity has a linear relation with the size of the reference frame.