bibkey: dinghu2020masking
authors: Ding, Feng; Hu, Xueyuan
year: 2020
title: "Masking quantum information on hyperdisks"
doi: 10.1103/PhysRevA.102.042404
url: "https://arxiv.org/abs/1909.11256v3"
claim: "The paper defines hyperdisks of pure states and their regular subsets, shows that the target states of a nondegenerate masking machine form a regular subset of a Schmidt hyperdisk, characterizes maskable sets in small dimensions, and asks whether a regular subset of a hyperdisk can be covered by finitely many hyperdisks."
strata_touched:
  - D5/S3/Quantum/Masking/HyperdiskRegularSubsetCover
license: citation-only
triage: anchor

Ding and Hu, Masking quantum information on hyperdisks

F. Ding and X. Hu, Phys. Rev. A 102, 042404 (2020), DOI 10.1103/PhysRevA.102.042404; arXiv:1909.11256v3. Quotations are from the arXiv v3 TeX source masking.tex.

For an orthonormal basis $\mathcal{B}=\{{\varphi }_j{\}}_{j<m}$ of a subspace, ${\mathbf{r}}_{\mathcal{B}}(\psi )=(|⟨{\varphi }_0|\psi ⟩|,\dots ,|⟨{\varphi }_{m-1}|\psi ⟩|)$. Definition 1 calls a set $\mathcal{S}$ of pure states a hyperdisk if some orthonormal basis $\mathcal{B}$ of $\mathrm{span}\mathcal{S}$ and a constant vector $\mathbf{r}$ with strictly positive entries satisfy ${\mathbf{r}}_{\mathcal{B}}(\psi )=\mathbf{r}$ for all $\psi \in \mathcal{S}$ and ${\mathbf{r}}_{\mathcal{B}}(\xi )\ne \mathbf{r}$ for all other pure states $\xi$. Definition 4 calls $\mathcal{C}$ a regular subset of the hyperdisk $\mathcal{S}$ if $\mathrm{span}\mathcal{C}\cap \mathcal{S}=\mathcal{C}$. A subset of a hyperdisk is the union of the hyperdisks it contains, and its optimal cover number, when finite, is the least number of hyperdisks needed to cover it.

The conclusion states:

Nevertheless, it is an open question for a nondegenerate masking machine whether the number of hyperdisks in the set of maskable states is always finite. A related open question is whether a regular subset of a hyperdisk can be covered with finite number of hyperdisks.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.102.042404
• Source: https://arxiv.org/abs/1909.11256v3