bibkey: mallickgangulymajumdar2026partialannihilating
authors: Bivas Mallick; Nirman Ganguly; A. S. Majumdar
year: 2026
title: "On the characterization of partially entanglement breaking and annihilating channels"
doi: 10.1063/5.0274921
url: https://arxiv.org/abs/2411.19315v2
claim: "The paper characterizes r-partially entanglement breaking channels, shows that they are in general not closed under tensor products for r>1, introduces local and non-local r-partially entanglement annihilating channels acting on a bipartite system C^d ⊗ C^d through the Schmidt number of their outputs, and leaves open whether the non-local r-partially entanglement annihilating channels are closed under tensor products."
strata_touched:
  - D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation
license: citation-only
triage: anchor

On the characterization of partially entanglement breaking and annihilating channels

B. Mallick, N. Ganguly and A. S. Majumdar, arXiv:2411.19315v2 (v1 28 November 2024); J. Math. Phys. 67(8) (2026), 082201. Quotations are from the v2 TeX source.

Schmidt number (Section II):

A bipartite density matrix $\rho$ has Schmidt number $r$, if in every possible decomposition of $\rho$ into pure states, i.e. $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$, with $p_k\ge 0$, at least one of the pure states $|{\psi }_k⟩$ has Schmidt rank $r$.

The sentence continues with the existence of a decomposition whose pure states all have Schmidt rank at most $r$.

The remark on partially entanglement breaking channels (Section III):

the set of (r)-partially entanglement breaking channels is, in general, not closed under tensor products for (r>1).

Local and non-local channels (Section IV):

A channel $\mathcal{E}$ is said to be local if it can be expressed as a tensor product of individual channels, i.e. $\mathcal{E}={\mathcal{E}}_1\otimes {\mathcal{E}}_2$. If a quantum channel can not be written in the above form, then it is said to be a non-local channel.

Let, $\mathcal{E}$ be a quantum channel acting on subsystem B. We call $\mathcal{E}$ to be a non-local $r$-partially entanglement annihilating channel if $\mathcal{E}$ can not be written as ${\mathcal{E}}_1\otimes {\mathcal{E}}_2$ and SN$(\mathcal{E}({\rho }_B))\le r$, for all ${\rho }_B\in \mathcal{D}({\mathcal{H}}_{\mathcal{B}})$, where, $r<d$ and $B$ is a bipartite system $\in \mathcal{D}({\mathbf{C}}^d\otimes {\mathbf{C}}^d)$.

The open question (Conclusion):

We have also shown that $r$-partially entanglement breaking channels are not closed under tensor products, motivating the investigation of whether a similar property holds for $r$-partially entanglement annihilating channels.

The source also contains, commented out, a draft proposal that the tensor product “may not be non-local r-PEAT”, followed by an informal proof sketch through the Schmidt number of a tensor product of pure states and the remark “[doubt in writing]”. Neither is part of the published article.

Verified locator

Source: https://arxiv.org/abs/2411.19315v2

• URL: https://arxiv.org/abs/2411.19315v2 (the e-print is a tar archive; member main.tex, md5 02f5d47ee4475c5ba30a52b1e5931b9b, 1322 lines):
  • Schmidt number of a mixed state, l. 261;
  • the remark on partially entanglement breaking channels, l. 545;
  • local and non-local channels, l. 630;
  • non-local $r$-partially entanglement annihilating channels, l. 632;
  • the commented-out draft and proof sketch, l. 679–706;
  • the open question, l. 1167.
• DOI: 10.1063/5.0274921 (https://doi.org/10.1063/5.0274921; Crossref lists Journal of Mathematical Physics, volume 67, issue 8, article 082201, published 1 August 2026).