# Projection operators and their subspaces (of Hilbert space)

I've been watching Susskind's lectures on Quantum Entanglement, and something he said regarding (non-)commuting projection operators confused me.

Consider two subspaces {$$|a\rangle$$} and {$$|b\rangle$$} of Hilbert space, with operators $$K$$ and $$L$$ for which:

• $$K |a\rangle = \lambda |a\rangle (1)$$
• $$L |b\rangle = \mu |b\rangle (2)$$

Now considers operators $$P_K$$ and $$P_L$$ that project any vector in Hilbert space onto their respective subspaces, that is:

• $$K (P_K |\psi\rangle) = \lambda (P_K |\psi\rangle)$$
• $$L (P_L |\psi\rangle) = \mu (P_L |\psi\rangle)$$

We want to find simultaneous eigenstates of both $$K$$ and $$L$$. If $$P_K$$ and $$P_L$$ commute: $$P_K (P_L |\psi\rangle) = P_L (P_K |\psi\rangle)$$. Now the left-hand satiesfies $$(1)$$, and the right-hand side satisfies $$(2)$$, so these are the required states.

In fact, if $$P_K$$ and $$P_L$$ operators commute, they share a complete set of eigenstates. The eigenstates of projection operators are those that span the subspace they project onto, so apparently $$P_K$$ and $$P_L$$ project onto the same subspace, which means they're the same operator? Then, is the statement: "projection operators commute $$\rightarrow$$ they're the same" correct, or do they somehow project states onto the same subspace in a different way?

Furthermore, we can imagine the subspaces geometrically as 'planes', and where these planes intersect we can find states that satisfy both $$(1)$$ and $$(2)$$. Now, according to Susskind, if $$P_K$$ and $$P_L$$ do not commute, finding such states is impossible. If the previous paragraph holds (does it?), then them commuting implies the intersection of their subspaces is the entire subspace. I don't know what non-commuting means geometrically, but shouldn't there be a case where the intersection of their subspaces isn't the entire subspace (for example, imagine two 2D perpendicular planes intersecting each other on a 1D line)? Susskind's comment seems to contradict that, and can't see exactly where I'm going wrongly.

• Think of projection operators onto spin eigenstates; $\frac{1}{2} ( 1 + \sigma_i)$ Apr 12, 2014 at 23:16