Gaussian optimality of two-mode entanglement of formation
- Field
- Topics
Problem
Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every finite-energy two-mode Gaussian state?
Let \(\rho_{AB}\) be any finite-energy two-mode Gaussian density operator, with one bosonic mode held by each party. Finite energy means finite total mean photon number. Define
The infimum in Eq. (1) is over probability measures on normalised pure vectors with barycentre \(\rho_{AB}\). The Gaussian entanglement of formation \(E_F^{\mathrm G}\) restricts the same infimum to pure Gaussian vectors.
The proposed equality is \(E_F(\rho_{AB})=E_F^{\mathrm G}(\rho_{AB})\), without an exchange-symmetry assumption.
Source
Wolf et al., Section IX, discuss equality of the Gaussian and unrestricted convex roofs [WGKWC04]. Adesso resolves the full two-mode instance in Eq. (19) [Ade26].
Progress
Gaussian decompositions give the covariance optimisation
\begin{equation} \begin{gathered} E_F(\rho_{AB})\leq E_F^{\mathrm G}(\rho_{AB}) =\inf_{\substack{\gamma\preceq V\\\gamma\text{ pure Gaussian covariance}}} g\!\left(\frac{\sqrt{\det\gamma_A}-1}{2}\right), \\ g(x):=(x+1)\log_2(x+1)-x\log_2x, \end{gathered} \tag{2} \end{equation}In Eq. (2), \(V\) is the covariance of \(\rho_{AB}\) in vacuum-\(I\) units and \(\gamma_A\) is the local covariance block of the pure Gaussian component. Use \(0\log_2 0:=0\). Proposition 1 of Wolf et al. establishes this restricted optimization. [WGKWC04]
Giedke et al., Proposition 2, establish \(E_F=E_F^{\mathrm G}\) for symmetric two-mode Gaussian states. [GWKWC03]
Adesso proves \(E_F=E_F^{\mathrm G}\) for every two-mode Gaussian state in Eq. (19). Theorem 1 applies an affine entanglement bound to every finite-energy pure two-mode state, including non-Gaussian states. Equations (18)–(19) average it over arbitrary decompositions. The supplemental subsection Completion of the proof and infinite-dimensional limit, Eqs. (S54)–(S57), treats the infinite-dimensional limit; the discussion following Eq. (S62) includes probability-measure decompositions. [Ade26]
The historical two-mode claims require distinguishing their assumptions. Marian and Marian construct a Gaussian candidate ensemble, but optimality against arbitrary non-Gaussian ensembles is not established by that construction. Ivan and Simon leave a generalized EPR extremality conjecture as an input. Akbari-Kourbolagh and Alijanzadeh-Boura prove a conditional rearrangement lemma with a geometric-ratio restriction on adjacent Schmidt coefficients. Adesso discusses these gaps in the Introduction and removes that restriction in the proof of Theorem 1 [Ade26].
Comment
The affirmative resolution is an arXiv preprint, version 2; no peer-reviewed publication was identified in the September 2026 audit. It determines the single-copy convex roof. Additivity across copies remains separate. The record 01M1HME78068MQY7E9KA81B7WX asks the distinct question for non-bisymmetric states with at least three modes. This is the two-mode instance of Open Quantum Problem 29; its historical attribution to the 2015 result must be read with the conditional Schmidt-coefficient hypothesis described above. The resolving preprint discloses assistance with its proof strategy; the catalog status records the stated resolution, not an independent refereeing of its technical proof.