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An idea that popped into my mind last night

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Could heat be transfered by quantum entanglement? It would be an interesting new mode of heat transfer occuring at unlimited distance.

There is no way I could do any mathematics of this, but I know there are a lot of smarter people here.

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Short answer: no.

Simplifying it massively, for heat to be transferred between two particles, you require energy to be transferred between the two (i.e one particle loses energy, the other gains it). Just because two particles have their states entangled doesn't mean that energy is transferred between the two, so it's not a mechanism that can be used to transfer heat.

Also on a side note, the concept of "heat" at the level of quantum particles doesn't really mean very much, heat is very much a macroscopic phenomenon.

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