Separability rigidity of Gaussian states under positive domination
Abstract
We prove a separability rigidity theorem for finite-mode Gaussian states. Let ρG be Gaussian and let ℐ be a fixed partition of the modes. If ρG dominates a nonzero positive operator T that is separable across ℐ, 0≠ Tpreccurlyeq ρG, then ρG itself is separable across ℐ. The dominated operator is arbitrary and need not be Gaussian. The theorem has three applications. First, any Gaussian channel admitting a nonzero entanglement-breaking completely positive subchannel is itself entanglement breaking, even when that subchannel is non-Gaussian. Second, if a resource state dominates a positive multiple of a faithful separable state, no finite-copy, postselected separability-preserving protocol can prepare an entangled Gaussian state exactly with nonzero success probability. Third, a connected graph of entangled Gaussian two-mode marginals certifies genuine multipartite entanglement of every compatible global state, even if the global state is non-Gaussian; the certification is robust to sufficiently small trace-norm errors in the marginals. As a structural corollary, mixing states separable across different partitions cannot produce new Gaussian states, so full inseparability and genuine multipartite entanglement coincide for finite-mode Gaussian states and identical copies of a biseparable Gaussian state cannot activate genuine multipartite entanglement.
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BibTeX
@misc{shchukin2026separability,
title = {{Separability rigidity of Gaussian states under positive domination}},
author = {Evgeny Shchukin and Peter van Loock},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36336},
eprint = {2609.36336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36336},
}
RIS
TY - UNPB
TI - Separability rigidity of Gaussian states under positive domination
AU - Evgeny Shchukin
AU - Peter van Loock
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.36336
UR - https://arxiv.org/abs/2609.36336
AB - We prove a separability rigidity theorem for finite-mode Gaussian states. Let ρ_G be Gaussian and let ℐ be a fixed partition of the modes. If ρ_G dominates a nonzero positive operator T that is separable across ℐ, 0≠ Tpreccurlyeq ρ_G, then ρ_G itself is separable across ℐ. The dominated operator is arbitrary and need not be Gaussian. The theorem has three applications. First, any Gaussian channel admitting a nonzero entanglement-breaking completely positive subchannel is itself entanglement breaking, even when that subchannel is non-Gaussian. Second, if a resource state dominates a positive multiple of a faithful separable state, no finite-copy, postselected separability-preserving protocol can prepare an entangled Gaussian state exactly with nonzero success probability. Third, a connected graph of entangled Gaussian two-mode marginals certifies genuine multipartite entanglement of every compatible global state, even if the global state is non-Gaussian; the certification is robust to sufficiently small trace-norm errors in the marginals. As a structural corollary, mixing states separable across different partitions cannot produce new Gaussian states, so full inseparability and genuine multipartite entanglement coincide for finite-mode Gaussian states and identical copies of a biseparable Gaussian state cannot activate genuine multipartite entanglement.
ER -