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I often encounter the argument that quantum mechanics reduces to classical mechanics at sufficiently big scales, as soon as h becomes sufficiently small respect to the actions involved. I clearly understand this for the uncertainty relations.

But when it comes to superposition, what is preventing it to happen at any scale? Is there actually something that prevents superposition for system bigger (in spatial or any other dimension) than some threshold?

As a practical example, is there an insuperable limit in how many q-bits we can have in entanglement?

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  • $\begingroup$ Related: Transition from quantum to classical mechanics. You may not find a direct answer there, but I'm not sure that there's really an agreed-upon answer to this question; it remains a subject of debate & research. $\endgroup$ – Michael Seifert Feb 12 at 12:56
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    $\begingroup$ Quantum mechanics does not reduce to classical mechanics even when h is sufficiently small. The classical limit is a result of decoherence: physics.stackexchange.com/questions/456575/… $\endgroup$ – alanf Feb 12 at 15:48
  • $\begingroup$ I confess I'd like it much more to be like that $\endgroup$ – J.Ask Feb 12 at 17:04
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Nothing prevents superposition at macroscopic scales. Schrodinger's cat states exist according to the Schrodinger equation. If they didn't, it would violate unitarity.

What we don't get is interference at macroscopic scales. One reason for this is that macroscopic objects simply have very short wavelengths. So, e.g., you can't observe double-slit diffraction with a baseball, because the diffraction angles would be too small. Also, the baseball's phase gets randomized too rapidly by interactions with its environment. This is called decoherence.

As a practical example, is there an insuperable limit in how many q-bits we can have in entanglement?

Not in principle, but in larger and larger systems it gets more and more difficult to prevent decoherence.

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  • $\begingroup$ But I guess not all interferences, only those whose magnitude is related to $h$. Because if it's true that it is possible to run, for example, a Grover algorithm on a register of billions of q-bits, then there would be a large-scale interference at work there $\endgroup$ – J.Ask Feb 12 at 17:13
  • $\begingroup$ @J.Ask: The time scale for superposition is related to $h$, but because of units, clearly $h$ can't be the only factor. It also depends on things like the temperature of the environment and the cross-section of interaction between the environment and the system. $\endgroup$ – Ben Crowell Feb 12 at 20:02
  • $\begingroup$ Time is surely related with decoherence, in more than one manner. But I think in the ideal setting of our big quantum register being isolated it wouldn't be a specific problem. Because quantum gates can be decomposed down to be 2 or 3 local (I don't remember exactly). I mind here only about theorical limitations, not technological ones $\endgroup$ – J.Ask Feb 12 at 20:19
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As you are already aware, classical mechanics (which includes the principle of superposition) breaks down at a quantum level, yet holds at more macroscopic levels, due to the effect of h meaning that quantum effects become greatly reduced as momentum of an object increases.

This trend only keeps going, even as we tend to colossally large objects/systems. In fact, something like a galaxy should follow classical mechanical laws even closer than we humans do, albeit to a negligible degree.

So while there is a limit on the validity of the principle of superposition as we decrease our observation size to nanoscopic levels and beyond, there is absolutely no upper bound we know of.

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