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{{see also| Frame of reference}}
A '''quantum reference frame''' is a reference frame which is treated quantum theoretically. It, like any [[Frame of reference|reference frame]], is an abstract coordinate system which defines physical quantities, such as [[time]], position, [[momentum]], [[
==Reference frame in classical mechanics and inertial frame==
{{see also| Frame of reference | Inertial frame}}
Consider a simple physics problem: a car is moving such that it covers a distance of 1 mile in every 2 minutes, what is its velocity in metres per second? With some conversion and calculation, one can come up with the answer "13.41m/s"; on the other hand, one can instead answer "0, relative to itself". The first answer is correct because it recognises a reference frame is implied
When speaking of a car moving towards east, one is referring to a particular point on the surface of the Earth; moreover, as the Earth is rotating, the car is actually moving towards a changing direction, with respect to the Sun. In fact, this is the best one can do: describing a system in relation to some reference frame. Describing a system with respect to an absolute space does not make much sense because an absolute space, if it exists, is unobservable. Hence, it is impossible to describe the path of the car in the above example with respect to some absolute space. This notion of absolute space troubled a lot of physicists over the centuries, including Newton. Indeed, Newton was fully aware of this stated that all inertial frames are [[Observational equivalence|observationally equivalent]] to each other. Simply put, relative motions of a system of bodies do not depend on the inertial motion of the whole system.<ref name = "Dickson">{{cite journal|doi = 10.1016/j.shpsb.2003.12.003|last = Dickson|first = Michael|title = A view from nowhere: quantum reference frames and uncertainty| journal = Studies in History and Philosophy of Modern Physics | volume = 35|issue = 2 |year = 2004 |pages = 195–220|bibcode = 2004SHPMP..35..195D}}</ref>
An [[inertial]] reference frame (or [[inertial frame]] in short) is a frame in which all the physical laws hold. For instance, in a [[rotating reference frame]], Newton's laws have to be modified because there is an extra Coriolis force (such frame is an example of non-inertial frame). Here, "rotating" means "rotating with respect to some inertial frame". Therefore, although it is true that a reference frame can always be chosen to be any physical system for convenience, any system has to be eventually described by an inertial frame, directly or indirectly. Finally, one may ask how an inertial frame can be found, and the answer lies in the [[Newton's laws]], at least in [[Newtonian mechanics]]: the first law guarantees the existence of an inertial frame while the second and third law are used to examine whether a given reference frame is an inertial one or not.
It may appear an inertial frame can now be easily found given the Newton's laws as empirical tests are accessible. Quite the contrary; an absolutely inertial frame is not and will most likely never be known. Instead, inertial frame is approximated. As long as the error of the approximation is undetectable by measurements, the approximately inertial frame (or simply "effective frame") is reasonably close to an absolutely inertial frame. With the effective frame and assuming the physical laws are valid in such frame, descriptions of systems will ends up as good as if the absolutely inertial frame was used. As a digression, the effective frame [[Astronomers]] use is a system called "[[International Celestial Reference Frame]]" (ICRF), defined by 212 radio sources and with an accuracy of about <math>10^{-5}</math> radians. However, it is likely that a better one will be needed when a more accurate approximation is required.
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==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
For example, if a [[spin-1/2]] particle is said to be in the state <math>\left|\uparrow z \right\rangle</math>, a reference frame is
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{{Citation needed|reason="Dequantisation" does not appear to be a common term in this field when applied to reference frames. Please find an example in the literature.|date=April 2014}}/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.<ref>{{cite journal|last=Barlett|first=Stephen D. |author2=Rudolph, Terry |author3=Spekkens, Robert W.|title=Dialogue concerning two views on quantum coherences: factist and fictionist|journal=
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.<ref>{{cite journal|doi=10.1103/PhysRevD.30.368|last=Aharonov |first=Y.|author2=T. Kaufherr |title=Quantum frames of reference| year=1984|journal=Phys. Rev. D|volume=30|issue=2|pages = 368–385|bibcode = 1984PhRvD..30..368A }}</ref> 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
==Further considerations of quantum reference frame==
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===An example of treatment of reference frames in quantum theory===
Consider
:::<math>V(r) = \frac{-Ze^2}{r}</math>
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where <math>l,</math> <math> m</math>, and <math>n</math> are the orbital angular momentum, magnetic, and energy quantum numbers, respectively.
Now consider the [[Schrödinger equation]] for the proton and the electron:
:::<math>\frac{\partial}{\partial t}\Psi(x_1,y_1,z_1,x_2,y_2,z_2,t)=-iH\Psi(x_1,y_1,z_1,x_2,y_2,z_2,t)</math>
A change of variables to relational and centre-of-mass coordinates yields
:::<math>\frac{\partial \Psi(x,y,z,X,Y,Z)}{\partial t} = -i[-\frac{1}{2M}\nabla_{c.o.m.}^2
where <math>M</math> is the total mass and <math>\mu</math> is the reduced mass. A final change to spherical coordinates followed by a separation of variables will yield the equation for <math>\Phi(r,\theta,\phi)</math> from above.
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:::<math>Z=\frac{m_1z_1+m_2z_2}{m_1+m_2}</math>
The importance of this result is that it shows the wavefunction for the compound system is [[Quantum entanglement|entangled]], contrary to what one would normally think in a classical
===Superselection rules===
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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 (though not always easy) 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
For example, for a spin-<math>j</math> system, the maximum number of measurements that can be made before the error tolerance, <math>\epsilon</math>, is
In this spin-<math>j</math> 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.<ref name=":0">{{cite journal| doi = 10.1088/1367-2630/9/5/156| last = Poulin| first = D. |author2=J. Yard | title = Dynamics of a quantum reference frame|year=2007|journal=New J. Phys.|volume = 9| issue = 5|pages=156|arxiv = quant-ph/0612126 |bibcode = 2007NJPh....9..156P | s2cid = 8337465}}</ref>▼
▲In this spin-<math>j</math> system, the degradation is due to the loss of purity of the reference frame state. On the other hand, degradation can also 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.<ref>{{cite journal| doi = 10.1088/1367-2630/9/5/156| last = Poulin| first = D. |author2=J. Yard | title = Dynamics of a quantum reference frame|year=2007|journal=New J. Phys|volume = 9| issue = 5|pages=156|arxiv = quant-ph/0612126 |bibcode = 2007NJPh....9..156P }}</ref>
==References==▼
{{reflist}}▼
==See also==
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*[[Information theory]]
*[[Quantum information]]
*[[Quantum spacetime]]
▲==References==
▲{{reflist}}
[[Category:Quantum mechanics]]
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