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I ask this question here knowing there are similar questions on this site, but not having found a satisfactory answer for myself below those. Or at least, one in which a comparison between different answers is properly considered.

The question is simple, yet one can provide a multitude of answers of increasing sophistication. These often take the following forms:

  1. It is a matter of size. Objects below a certain size threshold exhibit quantum behavior, while those above behave classically.

  2. It is a matter of degrees of freedom. Objects with too many degrees of freedom behave classically (a system of pariticles) while those with less degreees of freedom behave quantum mechanically.

  3. It is a matter of Action. If the action is small compared to $h$, then the system is quantum, while if it larger than $h$, it is classical.

  4. It is a matter of decoherence. System which become "too" entangled with their surroundings end up loosing their quantum nature, while only those which remain isolated remain quantum.

There are clear arguments against answers of type 1). For instance macroscopic quantum phenomena. Arguably answers of type 2,3,4 can be seen to be related to one another to different extents. What I would like to know is which of these criteria (or a different one) - and most importantly why - should count as the best criteria.

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If you can observe (aka measure) an object without substantially disturbing it, it's a classical object. If the object is so small that any type of observation does substantially disturb the object, it's a quantum mechanical object.

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    $\begingroup$ Hello. I am not sure to agree with this answer. Measuring a quantum state with an observable such that the quantum state is an eigenstate of this observable won't perturb it. $\endgroup$
    – StarBucK
    Commented May 25, 2022 at 16:05
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This is a complicated question in general.

One possible answer is to say that a system will be classical if you can conceptually access all its physical properties without changing them by measuring this system. What I mean by "all its physical properties" are all the quantities that allows to predict the future evolution (for any evolution) of this state (hence position and momentum).

To access the full state of a quantum system (in the sense that you will have all the information allowing you to predict its future evolution in general) will require quantum tomography. During this process, you have to perform measurements on the quantum system. If you want to fully characterize this quantum system, necessarily at some point, one measurement will perturb its state. This is the reason why quantum tomography requires several copy of a given quantum state in order to fully characterize it.

Note that in my last paragraph, it does not imply that you cannot measure a quantum state without perturbing it. It only means that if you want to access all the information it contains you will necessarily perturb it at some point.

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