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arXiv:2608.01909

Optimality of Gaussian Entanglement of Formation

Gerardo Adesso

quant-phcond-mat.stat-mechmath-phSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations• Revised Aug 13, 2026 (v2)
v1 · Aug 3, 2026v2 · Aug 13, 2026v3 · Oct 15, 2026
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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. Our result extends to bisymmetric multimode Gaussian states, and also provides a measurable lower bound on the entanglement of formation of arbitrary non-Gaussian states.

Comments
9 pages, 1 figure; references updated

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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. Our result extends to bisymmetric multimode Gaussian states, and also provides a measurable lower bound on the entanglement of formation of arbitrary non-Gaussian states.
Entanglement of formation quantifies the entanglement of a state in terms of the entropy of entanglement of the least entangled pure state needed to prepare it. An analytical expression for this measure exists only for special cases, and finding a closed formula for an arbitrary state remains an open problem. In this work we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for the measure that get tight for several special cases. Further, we show that the problem of calculating the actual value of the entanglement of formation for arbitrary two-mode Gaussian states reduces to a trivial single parameter optimization process, and we provide an efficient algorithm for the numerical calculation of the measure.
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Entangled two-mode Gaussian states are a key resource for quantum information technologies such as teleportation, quantum cryptography, and quantum computation, so quantification of Gaussian entanglement is an important problem. Entanglement of formation is unanimously considered a proper measure of quantum correlations, but for arbitrary two-mode Gaussian states no analytical form is currently known. In contrast, logarithmic negativity is a measure that is straightforward to calculate and so has been adopted by most researchers, even though it is a less faithful quantifier. In this work, we derive an analytical lower bound for entanglement of formation of generic two-mode Gaussian states, which becomes tight for symmetric states and for states with balanced correlations. We define simple expressions for entanglement of formation in physically relevant situations and use these to illustrate the problematic behavior of logarithmic negativity, which can lead to spurious conclusions.