bibkey: morelliyamasakihubertavakoli2022imprecise
authors: Simon Morelli; Hayata Yamasaki; Marcus Huber; Armin Tavakoli
year: 2022
title: "Entanglement detection with imprecise measurements"
doi: 10.1103/PhysRevLett.128.250501
url: https://arxiv.org/abs/2202.13131v2
claim: "The paper bounds entanglement witnesses when lab measurements deviate from the targets by a fidelity-based inaccuracy epsilon; for the CHSH witness with targets sigma_X and sigma_Z on two qubits it gives an explicit separable model attaining 4(1-2eps)sqrt(eps(1-eps)) + sqrt(2-16eps(1-eps)(1-2eps)^2) for eps up to 1/2-1/(2 sqrt 2) and 2 beyond, and conjectures that this model is optimal."
strata_touched:
  - D5/S3/QuantumBounds/ImpreciseCHSHSeparableOptimum
license: citation-only
triage: anchor

Morelli, Yamasaki, Huber and Tavakoli, entanglement detection with imprecise measurements

S. Morelli, H. Yamasaki, M. Huber and A. Tavakoli, arXiv:2202.13131v2 [quant-ph] (15 May 2022; v1 26 February 2022); Phys. Rev. Lett. 128, 250501 (2022). Quotations are from the v2 TeX source, with the macros \Tr, \expect, \norm written out and cross-references omitted.

The two-outcome inaccuracy constraint, for lab observables $A_x=A_{1|x}-A_{2|x}$ and targets ${\tilde{A}}_x$:

$\mathrm{Tr}\left(A_x{\stackrel{x)\ge d(1-2\epsilon xA),\; Tr(ByB}{A}}_y\right)\ge d\left(1-2{\epsilon }_y^{\text{B}}\right)$ … The observables can be arbitrary Hermitian operators whose extremal eigenvalue is bounded by unity, i.e. $\Vert A_x{\Vert }_{\infty }\le 1$ and $\Vert B_y{\Vert }_{\infty }\le 1$.

The CHSH witness and the conjecture (main text):

consider the CHSH quantity for entanglement detection, namely $\mathcal{W}=⟨{\sigma }_X\otimes \left({\sigma }_X+{\sigma }_Z\right)⟩+⟨{\sigma }_Z\otimes \left({\sigma }_X-{\sigma }_Z\right)⟩$. … However, as we show in Supplementary Material through an explicit separable model that we conjecture to be optimal, this fact does not qualitatively improve the robustness of idealised ($\epsilon =0$) entanglement detection to small inaccuracies. We obtain ${\mathcal{W}}_{\text{sep}}=4\left(1-2\epsilon \right)\sqrt{\epsilon (1-\epsilon )}+\sqrt{2-16\epsilon \left(1-\epsilon \right){\left(1-2\epsilon \right)}^2}$, when $\epsilon \le \frac{1}{2}-\frac{1}{2\sqrt{2}}$ and ${\mathcal{W}}_{\text{sep}}=2$ otherwise.

The supplementary model (Section AppCHSH) takes $A_1=B_1$, $A_2=B_2$ with Bloch vectors in the $XZ$-plane at angles $\alpha =\arccos (1-2\epsilon )$, $\beta =\arcsin (1-2\epsilon )$, and a product state $|\varphi ⟩\otimes |\varphi ⟩$.

Verified locator

Source: https://arxiv.org/abs/2202.13131v2

• DOI: https://doi.org/10.1103/PhysRevLett.128.250501 (Phys. Rev. Lett. 128, 250501 (2022); author list, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2202.13131v2 (the e-print is a gzipped tar archive; member Manuscript_epsilonEW.tex, 405 lines, md5 cd8e9557361d4082b54a066c0a876cce):
  • the framework, the fidelity constraint and the two-outcome observables, l. 77–100;
  • the CHSH witness, the conjecture and the separable value, l. 111–118;
  • the explicit separable model, Section AppCHSH, l. 232–257.
• The author list and the arXiv dates are from the arXiv metadata.