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”.

💡 Research Summary

The paper presents a comprehensive framework for analyzing two orthogonal projections (P) and (Q) acting on a separable Hilbert space (V). The authors first prove a new decomposition theorem that extends Jordan’s lemma from finite‑dimensional spaces to the infinite‑dimensional setting. They show that (V) can be written as an orthogonal direct sum of subspaces that are invariant under both (P) and (Q); each subspace is either at most two‑dimensional or infinite‑dimensional. On every infinite‑dimensional component the pair ((P,Q)) admits a canonical matrix representation called the “one‑shifted form”. In this form both projections are block‑diagonal with blocks of size at most two, and each block is a symmetric tridiagonal matrix determined by a single angle parameter.

A central technical contribution is Theorem 1.2, which constructs explicit finite‑dimensional approximations (P_n(\Theta)) and (Q_n(\Omega)) of the pair ((P,Q)). By choosing an orthonormal basis ({u_i}) and sequences of angles (\Theta={\theta_i}), (\Omega={\omega_i}\subset(0,\pi)), the authors define orthogonal projections onto the finite‑dimensional subspaces (V_n=\operatorname{span}{u_1,\dots,u_{2n}}). They prove that for any vector (v\in V), \