Gaussian entanglement of formation beyond bisymmetry
- Field
- Topics
Problem
Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state? Let \(\rho_{AB}\) be a Gaussian state of \(n_A+n_B\geq3\) bosonic modes, with covariance matrix \(V\). Its entanglement of formation is
where the infimum is over all pure-state ensembles. The Gaussian restriction of Eq. (1) is equivalently
where \(E(V_p)\) is the entropy of either reduced state of the pure Gaussian state with covariance matrix \(V_p\). A covariance matrix is bisymmetric when it is invariant under arbitrary permutations of Alice’s modes and, independently, of Bob’s modes. The question is whether \(E_F(\rho_{AB})=E_F^G(\rho_{AB})\) outside this bisymmetric family; the two quantities are fixed by Eqs. (1) and (2).
Source
Adesso proves Gaussian optimality for all two-mode and bisymmetric multimode states and explicitly leaves the general nonsymmetric multimode case open [Ade26].
Progress
Wolf, Giedke, Krüger, Werner, and Cirac introduced the Gaussian convex-roof restriction and derived the covariance-matrix optimization in Eq. (2). Their analysis does not establish equality with the unrestricted roof in Eq. (1) [WGKWC04].
Adesso proved \(E_F=E_F^G\) for every two-mode Gaussian state and for every bisymmetric multimode Gaussian state. The same work explicitly leaves the generic nonsymmetric multimode case open [Ade26].
Comment
As of August 31, 2026, equality of the two convex roofs is proved for all two-mode states and for bisymmetric multimode states, but neither a proof nor a counterexample is known for a general non-bisymmetric state with at least three modes.