Question

Exercise 5.1.1: Let H = C², M1 = C|0) and M2 = C(0) + 1)) Let [2) = a|0) + B|1) with (al2 + 1B12 = 1. Show that Pr(span{M1, M2}) # Pr(M1)+Pr(M2) - Pr(MinM2). O

Answer 1

It follows that M1 is invariant subspace of operator ProjM2. Indeed, if v∈M1 then

ProjM1ProjM2v=ProjM2ProjM1v=ProjM2v,

so

ProjM1(ProjM2v)=ProjM2v,

but this can happen only if ProjM2v∈M1.

Similarly, it can be proved that M⊥1 is invariant for ProjM2, and also M2,M⊥2 are invariant for ProjM1 by symmetry.

Since M1 is invariant for ProjM2 and ProjM2 has two eigenspaces M2,M⊥2, then M1 can be split into

M1=(M1∩M2)⊕(M1∩M⊥2)

Similarly,

M2=(M2∩M1)⊕(M2∩M⊥1)

Now, clearly, (M1∩M⊥2)⊥(M2∩M⊥1), so

M1+M2=(M1∩M2)⊕(M1∩M⊥2)⊕(M2∩M⊥1),

hence

Pr(M1+M2)=Pr(M1∩M2)+Pr(M1∩M⊥2)+Pr(M2∩M⊥1)=

=Pr(M1)+Pr(M2)−Pr(M1∩M2)