Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states

Alexander Bernal, J. Alberto Casas, Jesus M. Moreno

TL;DR

The paper derives a closed-form expression for the maximal CHSH violation of any qubit-qudit state, reducing the optimization to a 3-parameter search over SO(3) rotations. The central result, $\mathcal{B}=2 \max_{R\in SO(3)} \sqrt{\| (R\beta)_1 \|_1^2 + \| (R\beta)_2 \|_1^2}$, generalizes Horodecki’s qubit-qubit formula and yields explicit observables achieving the maximum. It also provides simple, computable lower and upper bounds, connects nonlocality to purity, analyzes random states, and proves that embedding to higher dimensions does not improve CHSH violation, illustrated by a qubit-qutrit example. The findings enable efficient nonlocality assessments in higher-dimensional systems and offer a versatile framework for exploring entanglement versus nonlocality in diverse quantum platforms.

Abstract

We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.

Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states

TL;DR

The paper derives a closed-form expression for the maximal CHSH violation of any qubit-qudit state, reducing the optimization to a 3-parameter search over SO(3) rotations. The central result, , generalizes Horodecki’s qubit-qubit formula and yields explicit observables achieving the maximum. It also provides simple, computable lower and upper bounds, connects nonlocality to purity, analyzes random states, and proves that embedding to higher dimensions does not improve CHSH violation, illustrated by a qubit-qutrit example. The findings enable efficient nonlocality assessments in higher-dimensional systems and offer a versatile framework for exploring entanglement versus nonlocality in diverse quantum platforms.

Abstract

We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
Paper Structure (9 sections, 3 theorems, 39 equations, 2 figures)

This paper contains 9 sections, 3 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Lower and upper bounds on Bell violation
  5. Purity and non-locality
  6. Non-locality of $2\times d$ Random Matrices
  7. Embeddings
  8. A qubit-qutrit case study
  9. Summary and conclusions

Key Result

Lemma 2.1

Let $\vec{v} ,\vec{w}$ be two arbitrary vectors. Consider a simultaneous rotation of both vectors within the plane they span. Let $\vec{v} (\varphi)$ and $\vec{w} (\varphi)$ be the rotated vectors with $\varphi$ characterizing the rotation angle. Then the following identity holds: where the subscripts 1, 2 denote the components of the vectors.

Figures (2)

  • Figure 1: From right to left, probability distribution of $\mathcal{B}$ for $d= 2, 4, 10, 20, 40, 80$.
  • Figure 2: Values of the logarithmic negativity. $E=\log_2\left(\|\rho^{T_2}\|_1\right)$ (upper panel) and $\mathcal{B}$ (lower panel) for the qubit-qutrit model described by the density matrix of Eq.(\ref{['rho 2x3']}). The model is entangled ($E>0$) in the whole physical region, except for $x=y=1/3$, while it violates local realism for $\mathcal{B}>2$. The area enclosed by the white line corresponds to the region where CHSH violation is ruled out by the upper bound in Eq.(\ref{['bounds']}).

Theorems & Definitions (5)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 3.1