Postselection

In probability theory, to postselect is to condition a probability space upon the occurrence of a given event. In symbols, once we postselect for an event $E$ , the probability of some other event $F$ changes from ${\textstyle \operatorname {Pr} [F]}$ to the conditional probability $\operatorname {Pr} [F\,|\,E]$ .

For a discrete probability space, ${\textstyle \operatorname {Pr} [F\,|\,E]={\frac {\operatorname {Pr} [F\,\cap \,E]}{\operatorname {Pr} [E]}}}$ , and thus we require that ${\textstyle \operatorname {Pr} [E]}$ be strictly positive in order for the postselection to be well-defined.

See also PostBQP, a complexity class defined with postselection. Using postselection it seems quantum Turing machines are much more powerful: Scott Aaronson proved^[1]^[2] PostBQP is equal to PP.

Some quantum experiments^[3] use post-selection after the experiment as a replacement for communication during the experiment, by post-selecting the communicated value into a constant.

References

^ Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time". Proceedings of the Royal Society A. 461 (2063): 3473–3482. arXiv:quant-ph/0412187. Bibcode:2005RSPSA.461.3473A. doi:10.1098/rspa.2005.1546.
^ Aaronson, Scott (2004-01-11). "Complexity Class of the Week: PP". Computational Complexity Weblog. Retrieved 2008-05-02.
^ Hensen; et al. (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres". Nature. 526 (7575): 682–686. arXiv:1508.05949. Bibcode:2015Natur.526..682H. doi:10.1038/nature15759. PMID 26503041.

P ≟ NP

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[PostBQP_=_PP-1] Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time". Proceedings of the Royal Society A. 461 (2063): 3473–3482. arXiv:quant-ph/0412187. Bibcode:2005RSPSA.461.3473A. doi:10.1098/rspa.2005.1546.

[2] Aaronson, Scott (2004-01-11). "Complexity Class of the Week: PP". Computational Complexity Weblog. Retrieved 2008-05-02.

[Hensen_et_al.-3] Hensen; et al. (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres". Nature. 526 (7575): 682–686. arXiv:1508.05949. Bibcode:2015Natur.526..682H. doi:10.1038/nature15759. PMID 26503041.

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