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

Yuki Fujii, Toyohiro Tsurumaru

TL;DR

This work addresses the problem of estimating the spectral radius rho([A,B]) for two orthogonal projections by connecting it to the Bell-CHSH operator in quantum information theory. It develops a finite dimensional approximation framework for projections, extends Jordan's lemma via a one-shifted form, and derives upper and lower bounds for rho([A,B]) from eigenvalues of finite dimensional approximants, achieving exact results in constant-angle one-shifted cases. A key contribution is a new Tsirelson-type inequality proof obtained through reduction to finite dimensions, plus explicit constructions enabling practical computations. The results provide a principled method to bound Bell-CHSH operator spectra in infinite dimensional settings, with potential impact on device-independent quantum information tasks and related spectral problems in quantum mechanics.

Abstract

In quantum mechanics, the following problem is important: For two orthogonal projections P and Q on a separable infinite dimensional Hilbert space V, estimate the spectral radius of the commutator [A,B], where A=2P-I, B=2Q-I. This problem is known to be equivalent to determining the spectral radius of the Bell-CHSH operator. To approach this problem, we give a method to approximate orthogonal projections on V by those on finite dimensional subspaces. This method provides an upper bound and a lower bound for the spectral radius of [A,B], which become exact when the matrix representations of P and Q are "constant-angle one-shifted form".

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

TL;DR

This work addresses the problem of estimating the spectral radius rho([A,B]) for two orthogonal projections by connecting it to the Bell-CHSH operator in quantum information theory. It develops a finite dimensional approximation framework for projections, extends Jordan's lemma via a one-shifted form, and derives upper and lower bounds for rho([A,B]) from eigenvalues of finite dimensional approximants, achieving exact results in constant-angle one-shifted cases. A key contribution is a new Tsirelson-type inequality proof obtained through reduction to finite dimensions, plus explicit constructions enabling practical computations. The results provide a principled method to bound Bell-CHSH operator spectra in infinite dimensional settings, with potential impact on device-independent quantum information tasks and related spectral problems in quantum mechanics.

Abstract

In quantum mechanics, the following problem is important: For two orthogonal projections P and Q on a separable infinite dimensional Hilbert space V, estimate the spectral radius of the commutator [A,B], where A=2P-I, B=2Q-I. This problem is known to be equivalent to determining the spectral radius of the Bell-CHSH operator. To approach this problem, we give a method to approximate orthogonal projections on V by those on finite dimensional subspaces. This method provides an upper bound and a lower bound for the spectral radius of [A,B], which become exact when the matrix representations of P and Q are "constant-angle one-shifted form".

Paper Structure

This paper contains 21 sections, 18 theorems, 94 equations.

Table of Contents

  1. Preliminaries
  2. Notations
  3. Unitary representations of $\mathbb{Z}_2 * \cdots * \mathbb{Z}_2$
  4. Bell-CHSH operator
  5. Spectral radius of Bell-CHSH operator
  6. Tsirelson's inequality
  7. Tridiagonal matrices
  8. Sufficient conditions for distinct eigenvalues
  9. Explicit eigenvalues and eigenvectors of Toeplitz matrices
  10. Finite dimensional approximation of orthogonal projections
  11. Basic properties of finite dimensional approximation
  12. Reduction of inequalities to the finite dimensional case
  13. Application: New proof of Tsirelson's inequality
  14. One-shifted form of two orthogonal projections
  15. Basic properties of one-shifted form
  16. ...and 6 more sections

Key Result

Theorem 1

There exists an orthonormal basis $\mathcal{U} = \{u_i\}_{i=1}^\infty$ and a pair of sequences $\Theta = \{\theta_i\}_{i=1}^\infty, \Omega = \{\omega_i\}_{i=1}^\infty \subset (0,\pi)$ such that, for any vector $v \in V$, where $\pi_n$ is the orthogonal projection onto the finite dimensional subspace $V_n$ spanned by $\{u_1, \dots, u_{2n}\}$, and $P_n(\Theta)$, $Q_n(\Omega)$ are orthogonal project

Theorems & Definitions (56)

  • Theorem 1: Approximation of one-shifted form
  • Theorem 2
  • proof : Proof of (i)
  • proof : Proof of (ii)
  • proof : Proof of (iii)
  • Theorem 1.4: Tsirelson's inequality BSTsirelson1980
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 46 more