bibkey: tononilewenstein2025temporalbell
authors: Andrea Tononi; Maciej Lewenstein
year: 2025
title: "Temporal Bell inequalities in non-relativistic many-body physics"
doi: 10.1088/2058-9565/adcbce
url: https://arxiv.org/abs/2409.17290v4
claim: "The paper evaluates temporal Clauser-Horne correlators for the XX spin chain in a transverse field through the Jordan-Wigner fermions and their propagator, and conjectures a contraction identity Re[<0|f_N f_1 σ^x_N(t) f_1†|0> + <0|f_N f_1 σ^x_N(t) f_N†|0>] = Re[G_NN(t) - G_1N(t)], verified for 2 ≤ N ≤ 8 and assumed for every N."
strata_touched:
  - D5/S3/Quantum/SpinChains/TemporalBellContractionIdentity
license: citation-only
triage: anchor

Temporal Bell inequalities in non-relativistic many-body physics

A. Tononi and M. Lewenstein, arXiv:2409.17290v4 [quant-ph, cond-mat.quant-gas]; Quantum Sci. Technol. 10 (2025). Quotations are from the v4 TeX source.

The Jordan–Wigner Hamiltonian:

$H=-J{\sum }_{i=1}^{N-1}\left(f_i^{†}{f}_{i+1}+f_{i+1}^{†}{f}_i\right)-\mu {\sum }_{i=1}^N{f}_i^{†}{f}_i={\sum }_{\genfrac{}{}{0ex}{}{m=1,}{k=k_m}}^N{ϵ}_k{c}_k^{†}{c}_k,$

The propagator:

In particular, we find that $f_j^{†}(t)={\sum }_{i=1}^N{G}_{ij}(t)f_i^{†}$, where we define the propagator $G_{ij}(t)={\sum }_{\genfrac{}{}{0ex}{}{m=1,}{k=k_m}}^N{u}_{ik}{u}_{jk}{e}^{i{ϵ}_{k}t}$, with $u_{jk}=(-1{)}^{j-1}{U}_{j-1}({\lambda }_k)/[{\sum }_{l=1}^N{U}_{l-1}^2({\lambda }_k){]}^{1/2}$ normalized eigenfunctions expressed in terms of the Chebyshev polynomials of second kind $U_{j-1}({\lambda }_k)=\sin (jk)/\sin (k)$.

The Jordan–Wigner transformation (Methods):

The inverse transformation is given by ${\sigma }_i^{+}={\prod }_{j<i}(1-2f_j^{†}{f}_j^{})f_i^{†}$ and ${\sigma }_i^{-}={\prod }_{j<i}(1-2f_j^{†}{f}_j^{})f_i^{}$.

The conjecture:

However, we conjecture that $\text{Re}[⟨0|f_N{f}_1{\sigma }_N^x(t)f_1^{†}|0⟩+⟨0|f_N{f}_1{\sigma }_N^x(t)f_N^{†}|0⟩]=\text{Re}[G_{NN}(t)-G_{1N}(t)]$, and we verified \cite{sneg} that the exact analytical form of the left- and right-hand sides coincides for $2\le N\le 5${, and with exact numerics} up to $N=8$ \cite{mullerrigat2025}. In the absence of a general proof, we assume the relation to hold for any value of $N$.

Verified locator

Source: https://arxiv.org/abs/2409.17290v4

• URL: https://arxiv.org/abs/2409.17290v4 (source arxiv.tex, md5 225051e786fe4712ac07f5226a0a99ac):
  • the Hamiltonian, l. 144–150;
  • the propagator, l. 153–154;
  • the Jordan–Wigner transformation, l. 285–286;
  • the conjecture, l. 343–353.
• DOI: 10.1088/2058-9565/adcbce (https://doi.org/10.1088/2058-9565/adcbce; Crossref lists Quantum Science and Technology, volume 10, issued 17 April 2025).