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I was wondering how can we go from an analyzed system of particles, where we have considered them to be distinguishable, to results about the same particles where we consider them indistinguishable. For example, if I have a unitary that certain (distinguishable) two-particle states into some resultant two-particle states. How can I apply indistinguishability to those output states directly by doing appropriate amplitude additions?

Mathematically, Given two particles A and B with two possible states (1,0) & (0,1) :

distinguishable case, (call x_i) :  (1,0,0,0) , (0,1,0,0) ,(0,0,1,0) ,(0,0,0,1)

indistinguishable case : |1,1>, |2,0>,|0,2>

Given U|x_i> = |y_i>                (*where y_i are the output states*)

My questions are : can we represent the indistinguishable states in vector notation like in the distinguishable case? And, is there a way to find the resultant of the action of operator U on the indistinguishable states using the known action on the distinguishable ones ie the amplitudes in |y_i>?

Sorry for my English and the formatting, I don't know how to code in latex, etc on this website. Hope this was clear. Thanks for any responses!

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  • $\begingroup$ I'm not sure I understand fully. Yes, you can describe many-particle states in vector notation: though the space is generally infinite in this case, you'll get finite vectors assuming you can limit yourself to considering the states with some finite number of particles. Given a single-particle unitary, you can always compute the corresponding evolution on $n$ particles (bosonic or fermionic). This involves computing permanents/determinants of submatrices of the original matrix. I discuss some aspects of this in quantumcomputing.stackexchange.com/a/2246/55. Is this what you're asking? $\endgroup$
    – glS
    Commented Oct 6, 2022 at 9:58
  • $\begingroup$ also, see math.meta.stackexchange.com/q/5020/173147 to see how to format math $\endgroup$
    – glS
    Commented Oct 6, 2022 at 9:59

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