# What is meant by a "projection operator" in the book "Quantum Computation and Quantum Information"?

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 :)

• 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. Commented Nov 17, 2021 at 8:40
• My bad, thank you :) Commented Nov 17, 2021 at 8:42
• en.wikipedia.org/wiki/Projection_(linear_algebra) In quantum mechanics, we usually mean "orthogonal projection" if we say "projection" or "projector". Commented Nov 18, 2021 at 5:33

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).