Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections
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| Format: | Preprint |
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2025
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| _version_ | 1866914499900997632 |
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| author | Fujii, Yuki Tsurumaru, Toyohiro |
| author_facet | Fujii, Yuki Tsurumaru, Toyohiro |
| contents | 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". |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections Fujii, Yuki Tsurumaru, Toyohiro Representation Theory Functional Analysis 20C07, 47A25 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". |
| title | Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections |
| topic | Representation Theory Functional Analysis 20C07, 47A25 |
| url | https://arxiv.org/abs/2511.10939 |