bibkey: ortheyaugusiak2025steering
authors: Alexandre C. Orthey; Remigiusz Augusiak
year: 2025
title: "Certifying classes of d-outcome measurements with quantum steering"
doi: 10.1088/1367-2630/adda73
url: https://arxiv.org/abs/2410.20477v2
claim: "The paper constructs steering inequalities that self-test classes of d-outcome measurements built from Heisenberg-Weyl operators with admissible phases, proves the self-testing statement, and for the robustness analysis with d > 3 assumes Conjecture 1: the transformed operators satisfy B̃_k^(n) = B̃_k^n for every prime d and every n in {1, …, d-1}, noted as true for d = 3."
strata_touched:
  - D5/S3/Quantum/Measurements/SteeringPowerPreservationRefutation
license: citation-only
triage: anchor

Certifying classes of d-outcome measurements with quantum steering

A. C. Orthey Jr and R. Augusiak, arXiv:2410.20477v2 [quant-ph]; New J. Phys. 27 (2025). Quotations are from the v2 TeX source.

The reference measurements and their phases:

where $\{{\varphi }_{j,y}{\}}_{j=0}^{d-1}$ is a set of any real numbers satisfying ${\sum }_{j=0}^{d-1}{\varphi }_{j,y}=0$ and $\oplus$ stands for addition modulo $d$.

The coefficients:

${\gamma }_{yk}^{(n)}:=\frac{1}{d}{\omega }^{kn(n-1)/2}{\sum }_{l=0}^{d-1}{e}^{-\text{bbm}i{\sum }_{m=1}^n{\varphi }_{l\oplus m,y}}{\omega }^{knl}$

The admissibility condition:

${\sum }_{k=0}^{d-1}{\left({\gamma }_{y^{\prime }k}^{(n)}\right)}^{\ast }{\gamma }_{yk}^{(n)}={\delta }_{y{y}^{\prime }},\forall y,y^{\prime }.$

The substitution rule:

${\left(X{Z}^k\right)}^n⟶{\tilde{B}}_k^{(n)}:={\sum }_{y=0}^{d-1}{\gamma }_{yk}^{(n)}{B}_{n|y}$, where $B_y=\{B_{n|y}\}$, for $y=0,\dots ,d-1$ and $n=1,\dots ,d-1$, represent the $d$-outcome arbitrary measurements performed by Bob

Projective measurements:

(i) Bob’s measurements are projective, that is, all operators $B_{n|y}$ are unitary such that $B_{n|y}^d=\text{I}$,

The robustness setting and Conjecture 1 (Appendix “Robust self-testing”):

For simplicity, we will assume that the state is pure and the measurements performed by Bob are projective.

To prove the robustness of our self-testing statement for the qutrit case, we will make use of the fact that ${\tilde{B}}_k^{(n)}={\tilde{B}}_k^n,\forall n\in \{1,2\}\text{and}d=3$, which is true from the fact that ${\tilde{B}}_k^{(2)}={\left({\tilde{B}}_k^{(1)}\right)}^{†}$ and ${\left({\tilde{B}}_k^{(2)}\right)}^3=\text{I}$. We can conjecture that the above relation is also valid for any prime $d>3$:

Conjecture 1. For any prime $d$, we have that ${\tilde{B}}_k^{(n)}={\tilde{B}}_k^n$, for every $n\in \{1,\dots ,d-1\}$. Note that this statement is true for $d=3$.

Verified locator

Source: https://arxiv.org/abs/2410.20477v2

• URL: https://arxiv.org/abs/2410.20477v2 (source main2.tex, md5 b3c5b40a34f721352981dcf58659087e):
  • the phases, l. 190–194;
  • the coefficients, l. 204–206;
  • the admissibility condition, l. 234–236;
  • the substitution rule, l. 246–249;
  • projective measurements, l. 355;
  • the robustness setting, l. 1263;
  • Conjecture 1 and its context, l. 1265–1273.
• DOI: 10.1088/1367-2630/adda73 (https://doi.org/10.1088/1367-2630/adda73; Crossref lists New Journal of Physics, volume 27, issued 29 May 2025; arXiv lists v1 of 27 October 2024 and v2 of 18 May 2026).