bibkey: ducuarasusaskrzypczyk2024hidden
authors: Andrés F. Ducuara, Cristian E. Susa, Paul Skrzypczyk
year: 2024
title: "Locally inaccessible hidden quantum correlations"
doi: 10.1103/PhysRevA.110.022435
url: https://arxiv.org/abs/2406.03945v1
claim: "The paper proves, modulo a conjecture on quantum steering ellipsoids, the existence of two-qubit states whose hidden CHSH and F3 violations cannot be revealed by local filters on one side only; the conjecture states that the maximal CHSH and F3 violations are bounded by functions of the steering-ellipsoid centre length alone, with numerically estimated impossibility thresholds 0.5 and 0.66."
strata_touched:
  - D5/S3/Quantum/Contextuality/SteeringEllipsoidFThreeThreshold
license: citation-only
triage: anchor

Ducuara–Susa–Skrzypczyk, locally inaccessible hidden quantum correlations

Verified locator

DOI: 10.1103/PhysRevA.110.022435

Source: https://arxiv.org/abs/2406.03945v1

Phys. Rev. A 110, 022435 (2024), published 26 August 2024.

A. F. Ducuara, C. E. Susa and P. Skrzypczyk, arXiv:2406.03945v1 [quant-ph] (6 June 2024, the only version), source file QC_LI.tex.

Statement

For a two-qubit state $\rho$ with Bloch vectors $\mathbf{a}$, $\mathbf{b}$ and correlation matrix $T_{ij}=\mathrm{tr}[\rho {\sigma }_i\otimes {\sigma }_j]$, the maximal violation of the ${\mathrm{F}}_3$ steering inequality is ${\mathrm{F}}_3(\rho )=\sqrt{s_1^2+s_2^2+s_3^2}$, with $s_i$ the singular values of $T$ (Eq. (CAC), l. 211), and Bob’s steering-ellipsoid centre is ${\mathbf{c}}_B={\gamma }_a^2(\mathbf{b}-T^{\mathsf{T}}\mathbf{a})$ with ${\gamma }_a=(1-a^2{)}^{-1/2}$ (Eq. (QSEcentre), l. 261). The paper states (l. 282–288):

Conjecture (QSE conjecture). For any two-qubit state $\rho$, its maximal violation of the CHSH-inequality and the ${\mathrm{F}}_3$-inequality are upper bounded by functions that depend on the magnitude of the QSE’s centre as $\mathrm{B}(\rho )\le f_{\mathrm{CHSH}}(c_B)$ and ${\mathrm{F}}_3(\rho )\le f_{{\mathrm{F}}_3}(c_B)$, respectively.

Exploring the form of function $f_{{\mathrm{F}}_3}(c_B)$ is an interesting task which we leave for future research. … there seem to exist values $c_{\mathrm{CHSH}}=0.5$ and $c_{{\mathrm{F}}_3}=0.66$, such that states with QSE centres’ magnitudes greater than these values, can no longer lead to a violation of the CHSH-inequality and the ${\mathrm{F}}_3$-inequality, respectively.

The CHSH half restates Conjecture 1 of Milne, Jennings, Jevtic and Rudolph (2014).

Encoding and boundary

The repository defines ${\mathrm{F}}_3(\rho )$ as the supremum of $\frac{1}{\sqrt{3}}|{\sum }_i\mathrm{tr}[\rho ({\mathbf{\alpha }}^i\cdot \mathbf{\sigma })\otimes ({\mathbf{\beta }}^i\cdot \mathbf{\sigma })]|$ over unit vectors ${\mathbf{\alpha }}^i$ and orthonormal frames ${\mathbf{\beta }}^i$, and uses Bob’s centre with the convention ${\mathbf{c}}_B=\mathbf{b}$ when $|\mathbf{a}|=1$. It proves ${\mathrm{F}}_3(\rho )\le \max \{\sqrt{3(1-c_B)},1\}$ for every two-qubit density matrix and exhibits, for every $0\le t<\frac{2}{3}$, a state with $c_B=t$ and ${\mathrm{F}}_3>1$; the impossibility threshold is therefore exactly $\frac{2}{3}$. The optimal envelope of ${\mathrm{F}}_3$ at fixed centre length, the CHSH half and the hidden and one-sided-filter quantities are outside this conclusion.