Optimality of Gaussian Entanglement of Formation

Abstract

We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.

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BibTeX RIS

BibTeX

@misc{adesso2026optimality,
  title = {{Optimality of Gaussian Entanglement of Formation}},
  author = {Gerardo Adesso},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.01909},
  eprint = {2608.01909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01909},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Optimality of Gaussian Entanglement of Formation
AU  - Gerardo Adesso
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2608.01909
UR  - https://arxiv.org/abs/2608.01909
AB  - We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.
ER  -

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