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In condensed matter systems we can have paired quantum states, for instance vortex-antivortex pairs and Cooper pairs, etc. How do the partners within each pair recognize each other and stay together (or do they?) as long as it is below the critical temperature?

To be precise, suppose we have two pairs $A$, consisting of $A_1,A_2$ and $B$, consisting of $B_1,B_2$. What is the thing that determines the pairing? It seems that nothing can stop us from identifying $A_1,B_2$ as a pair.

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If your question is: “how does one determine which particle has paired with which particle”, the answer is relatively simple: the pairs are in an entangled state, which means they influence each other; you change the wave function of the first particle and it will change the wave function of the second.

If your question is: “how does one know beforehand which particles are going to form a pair”, the answer is vague. I can only say: if two particles at some moment would have a lower energy if they formed a pair, then they probably will. Which of the available particles they will choose seems like a random process to me.

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