bibkey: korzekwaetal2019distinguishing
authors: Korzekwa, Kamil; Czachórski, Stanisław; Puchała, Zbigniew; Życzkowski, Karol
year: 2019
title: "Distinguishing classically indistinguishable states and channels"
doi: 10.1088/1751-8121/ab30f7
url: "https://arxiv.org/abs/1812.09083v3"
claim: "The paper defines M-distinguishability regions of the probability simplex (classical versions admitting M perfectly distinguishable quantum states), bounds them by permutohedra, analyses the case d=4, M=3 (edges of the permutohedron belong to the region, faces do not), and conjectures from numerical evidence that the points of the region have a product structure over the four face centres (Eq. (30)); it also studies classically indistinguishable channels."
strata_touched:
  - D5/S3/Quantum/Measurements/ThreeDistinguishabilityProductRefutation
license: citation-only
triage: anchor

Korzekwa, Czachórski, Puchała and Życzkowski, Distinguishing classically indistinguishable states and channels

K. Korzekwa, S. Czachórski, Z. Puchała and K. Życzkowski, J. Phys. A: Math. Theor. 52, 475303 (2019), DOI 10.1088/1751-8121/ab30f7; arXiv:1812.09083v3. Quotations are from the arXiv v3 TeX source Distinguishability.tex; macros are expanded.

The classical version of a state $\rho$ is $\mathrm{diag}\rho$ in the preferred basis (l. 236–241). “A $d$-dimensional probability vector $p$ belongs to $M$-distinguishability region ${\mathcal{A}}_d^M$ of the probability simplex ${\Delta }_d$ if and only if there exist $M$ perfectly distinguishable quantum states with the same classical version $p$” (l. 148). For $d=4$, $M=3$ the bounding permutohedron is the tetrahedron of the face centres $f^i$, with $f^1=\frac{1}{3}(1,1,1,0)$ and the other $f^i$ its permutations, and the paper states (l. 462–467):

Moreover, based on numerical evidence, we conjecture that points belonging to ${\mathcal{A}}_4^3$ have the following product structure: $$ab{f}^1+a(1-b)f^2+(1-a)b{f}^3+(1-a)(1-b)f^4,$$ with $a,b\in [0,1]$.

Verified locator

• DOI: https://doi.org/10.1088/1751-8121/ab30f7
• Source: https://arxiv.org/abs/1812.09083v3