Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections

We establish a new decomposition formula for two orthogonal projections P and Q on a separable Hilbert space V. This formula yields an orthogonal direct sum decomposition of V into invariant subspaces under P and Q, each of which is either at most two dimensional or infinite dimensional. On every infinite dimensional component, the pair (P,Q) admits a matrix representation that we call the "one-shifted form". This representation diagonalizes both P and Q into blocks of size at most two, and moreover, both projections can be explicitly approximated by orthogonal projections on finite dimensional subspaces. This approximation scheme offers a way to derive infinite dimensional results from their finite dimensional counterparts and is also useful in numerical computations. This decomposition provides a useful framework for analyzing a wide range of problems involving two orthogonal projections in infinite dimensions. In particular, several spectral problems for operators generated by P and Q (including polynomials in P and Q) can be reduced to the case where the pair admits a one-shifted form. More concretely, we can estimate the spectral radius of [P,Q], which is equivalent to estimating the spectral radius of the Bell-CHSH operator, a quantity of fundamental importance in quantum mechanics. We provide an upper bound and a lower bound for the spectral radius of [P,Q], which become exact when the matrix representations of P and Q are in "constant-angle one-shifted form".