bibkey: yuetal2021absolutelyentangled
authors: Baichu Yu; Pooja Jayachandran; Adam Burchardt; Yu Cai; Nicolas Brunner; Valerio Scarani
year: 2021
title: "Absolutely entangled sets of pure states for bipartitions and multipartitions"
doi: 10.1103/PhysRevA.104.032414
url: https://arxiv.org/abs/2106.16216v3
claim: "The paper studies absolutely entangled sets of pure states, sets for which every global unitary leaves at least one state entangled; it defines them for multipartitions of a non-prime dimension, constructs a set that is absolutely entangled with respect to every partition, and asks whether some absolutely entangled set has, for every unitary, one state entangled according to all partitions."
strata_touched:
  - D5/S3/Quantum/Entanglement/Witness/AbsolutelyEntangledAllPartitions
license: citation-only
triage: anchor

Yu, Jayachandran, Burchardt, Cai, Brunner and Scarani, absolutely entangled sets for multipartitions

B. Yu, P. Jayachandran, A. Burchardt, Y. Cai, N. Brunner and V. Scarani, Physical Review A 104, 032414 (2021), DOI 10.1103/PhysRevA.104.032414; arXiv:2106.16216v3 (4 October 2021; v1 30 June 2021). Quotations are from the arXiv v3 TeX source, with cross-references omitted.

The definition (partitions are the factorizations $d=d_1\cdots d_k$ of the dimension):

Consider a set of quantum states $S=\{{\rho }_1,\dots ,{\rho }_N\}$ in a Hilbert space ${\mathbb{C}}^d$ of non-prime dimension $d$. (i) The set $S$ is said to be absolutely entangled with respect to the $(d_1,d_2,\dots ,d_k)$-partition where ${\prod }_{i=1}^k{d}_i=d$, if for every unitary matrix $U\in SU(d)$, at least one state $U{\rho }_n{U}^{†}$ is entangled with respect to multipartition ${\mathbb{C}}^{d_1}\otimes \cdots \otimes {\mathbb{C}}^{d_k}$. (ii) Let P be the set of all possible partitions of ${\mathbb{C}}^d$. The set is said to be absolutely entangled with respect to any partition, if for every partition $p\in P$ and every unitary matrix $U\in SU(d)$, at least one state $U{\rho }_n{U}^{†}$ is entangled with respect to $p$.

After constructing a set that is absolutely entangled with respect to any partition (a Corollary with a fixed orthonormal basis $\{{\xi }_i\}$ of ${\mathbb{C}}^d$), the source asks:

This result guarantees that, given any $U$ and any partition $P$, there exists at least one entangled state. An interesting open question is whether one can find an AES, such that, for every $U$, there exists a state that is entangled according to all partitions. This would arguably be the strongest form of genuine multipartite entanglement.

Verified locator

DOI: https://doi.org/10.1103/PhysRevA.104.032414 (Physical Review A 104, 032414 (2021); the journal text was not compared line by line with v3).

Source: https://arxiv.org/abs/2106.16216v3

• URL: https://arxiv.org/abs/2106.16216v3 (the e-print is a gzipped tar archive; member main.tex, 750 lines, md5 e11d4dc540598e9ef271fc11bad389a0):
  • partitions as factorizations, l. 308;
  • Definition 3, Defi3, l. 310–317;
  • the Corollary for every partition, l. 499–508;
  • the question, l. 509.
• The author list, the journal reference and the DOI are from Crossref.