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I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm finding myself a bit confused as to the notations used.

In page 435, above (10.15), I'd like to ask what exactly is meant by a "Projection Operator". What does it project, and from where? Does it project a $2^n \times 2^n$ Hilbert space onto a $2^m \times 2^m$ Hilbert space (where, in the book example, $n=3$ and $m=1$)?

Thank you very much :)

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    $\begingroup$ Hi and welcome on Quantum Computing SE. Please post each question separately. A rule of this site is that each question should be laser-focussed on one topic. $\endgroup$ Nov 17 '21 at 8:40
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    $\begingroup$ My bad, thank you :) $\endgroup$
    – XP_Gate
    Nov 17 '21 at 8:42
  • $\begingroup$ en.wikipedia.org/wiki/Projection_(linear_algebra) In quantum mechanics, we usually mean "orthogonal projection" if we say "projection" or "projector". $\endgroup$ Nov 18 '21 at 5:33
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In this context, the projection operator projects from a $2^n$ dimensional Hilbert space onto an $m$-dimensional subspace of the same Hilbert space. So, if you were viewing it as a matrix, it would be a $2^n\times 2^n$ matrix of rank $m$ (with the special properties that $P^2=P$, $P=P^\dagger$ and $\text{Tr}(P)=m$, and others that are related).

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