Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 67 / 4 / a / Solution

Here is a finite-dimensional form of the Kitaev geometrical lemma. Let $A,B$ be positive operators, with nullspaces $S_A,S_B$ and all positive eigenvalues at least $\gamma >0$. First assume $S_A\cap S_B=\{0\}$. Let $P,Q$ be the orthogonal projections onto those nullspaces, and define their smallest angle between two subspaces byIf either nullspace is zero, set $\cos \vartheta =0$. Then

To prove it, spectral decomposition gives $A\ge \gamma (I-P)$ and $B\ge \gamma (I-Q)$. Write $C=P+Q$ and $c=\cos \vartheta$. For every vector $z$,Thus $C^2\le (1+c)C$. Since $C$ is positive semidefinite, each of its eigenvalues lies in $[0,1+c]$, so $2I-C\ge (1-c)I$. Consequentlyas claimed.

If there is a common nullspace $S=S_A\cap S_B$, restrict to $S^{\perp }$ and define the angle between $S_A\cap S^{\perp }$ and $S_B\cap S^{\perp }$ there. The same proof bounds the smallest positive eigenvalue of $A+B$ by $2\gamma {\sin }^2(\vartheta /2)$, while $S$ is its zero-energy subspace. This version gives a spectral gap above a possibly degenerate ground state space.