bibkey: seongberahiesmayrchruscinskibae2026mirrored
authors: Jiheon Seong; Anindita Bera; Beatrix C. Hiesmayr; Dariusz Chruściński; Joonwoo Bae
year: 2026
title: "Mirrored entanglement witnesses for multipartite and high-dimensional quantum systems"
doi: 10.1088/2058-9565/ae30a3
url: https://arxiv.org/abs/2510.06863v2
claim: "The paper develops mirrored entanglement witnesses for multipartite and high-dimensional systems, generalizes mirroring to pairs W, M with W+M=K block-positive, and leaves open whether a pair of optimal witnesses can be mirrored through another optimal witness."
strata_touched:
  - D5/S3/Quantum/Entanglement/Witness/MirroredOptimalWitnessSum
license: citation-only
triage: anchor

Mirrored entanglement witnesses for multipartite and high-dimensional quantum systems

J. Seong, A. Bera, B. C. Hiesmayr, D. Chruściński and J. Bae, arXiv:2510.06863v2 [quant-ph]; Quantum Sci. Technol. 11 (2026). Quotations are from the v2 TeX source.

Entanglement witnesses:

a state $\rho$ is entangled if and only if there exists an observable $W=W^{†}$ such that $\mathrm{tr}[W\rho ]<0\mathrm{whereas}\mathrm{tr}[W{\sigma }_{sep}]\ge 0\forall {\sigma }_{sep}\in S_{sep}$,

Optimality:

Suppose that for some $ϵ>0$ and $P\ge 0$, we have $W^{{}^{\prime }}=W-ϵP$. Here $W^{{}^{\prime }}$ is also an EW. Let $D_W$ denote the set of states detected by an EW $W$, $D_W=\{\rho \in S(\stackrel{˝}{\otimes }\stackrel{˝}{)}:\mathrm{tr}[W\rho ]<0\}$. It holds that $D_W\subset D_{W^{{}^{\prime }}}$. In this case, two EWs can be compared and we call $W^{{}^{\prime }}$ is {\it finer} than $W$. EWs which cannot be improved anymore, that is, {\it finer} than any other, are called optimal EWs.

The spanning property:

We write by product states $|{\psi }_k\otimes {\varphi }_k⟩$ that vanish on a witness $W$: $⟨{\psi }_k\otimes {\varphi }_k|W|{\psi }_k\otimes {\varphi }_k⟩=0$. Then, we call a witness $W$ has the spanning property if \mathrm{spanning~property:~}\mathrm{span}\{|\psi_k\otimes \phi_k\rangle \} = \H \otimes \H, and $W$ is optimal.

Generalized mirroring (Section V, “Bipartite systems: generalization”):

We here generalize mirrored EWs in Eq. (\ref{eq:mr}) such that they are connected by some other observable, not an identity, $W+M=\text{mathbbm}K$, where it follows that $\text{mathbbm}K$ block-positive, i.e., it is nonnegative for all separable states since $W$ and $M$ are EWs.

Open problem 2 (Section VI):

Optimal EWs can be mirrored via another EW, as shown in Eq. (\ref{eq:gmeew_form}). Since optimal EWs are generally not extremal, yet, it remains unanswered whether a pair of optimal EWs can be mirrored through another optimal EWs.

Verified locator

Source: https://arxiv.org/abs/2510.06863v2

• URL: https://arxiv.org/abs/2510.06863v2 (source mn_revision_clean.tex, md5 08daa608d307e41e61bcd9c0d9f06c31):
  • entanglement witnesses, l. 276–280;
  • optimality and detection sets, l. 388–396;
  • the spanning property, l. 485–490;
  • generalized mirroring, l. 1688–1692;
  • open problem 2, l. 1749.
• DOI: 10.1088/2058-9565/ae30a3 (https://doi.org/10.1088/2058-9565/ae30a3; Crossref lists Quantum Science and Technology, volume 11, issued 12 January 2026; arXiv lists v1 of 8 October 2025 and v2 of 16 January 2026).