I'm sorry, "non-invertible symmetry"? In what way is that still a symmetry?
Imagine a pair of denim pants, one could say it has the property of being made of denim, and when you bend the pants or move them around the room, they keep being made of denim. Anything you do to the pants that you can undo, like moving them around or bending them, those are invertible symmetries. Now imagine I rip the pants in half: they're still made of denim, but this action is not so easy to invert. So there are actions I can perform on these pants that are hard or impossible to undo, but still maintain the property of them being made of denim, these are non-invertible symmetries.
The example given in the article is a system being split into a superposition in a way that's hard or impossible to undo, sort of like the pants being torn in half, but just because the system is now in a superposition doesn't mean there weren't properties preserved along the way.
Is this something like the physical operation to invert it doesn't exist, even though it's logically possible? The "pants" example, and maybe the superposition example, makes it sound like they're uninvertible for entropy reasons, which seems interesting if it's not an illusion.
In seems like physicists are finding more general conserved quantities associated with more general spaces of physical operations such that not every action is reversible, but there are still conserved quantities associated to the underlying systems, AKA symmetries.
I don't have any solid physical intuition as to what makes an operation irreversible in this context, though, whether it's related to entropy or not.
I can understand that popularizations have to gloss over the math, but they usually at least identify the important points even if they don't get into the weeds of explaining the details. This seems like more of a sleight of hand.
https://physics.stackexchange.com/questions/561118/why-shoul...