Gaussian entanglement of formation beyond bisymmetry

Problem

Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state? Let \(\rho_{AB}\) be Gaussian, with integers \(n_A,n_B\geq1\) and \(n_A+n_B\geq3\). Assume finite total mean photon number. For \(S(\tau):=-\operatorname{Tr}(\tau\log_2\tau)\), define

\begin{equation} E_F(\rho_{AB}):=\inf_{\mu}\int S(\operatorname{Tr}_B|\psi\rangle\langle\psi|)\,\mu(d\psi). \tag{1} \end{equation}

The probability measures in Eq. (1) range over normalized pure vectors with barycentre \(\rho_{AB}\). Non-Gaussian vectors are allowed.

The Gaussian restriction is equivalently

\begin{equation} E_F^{\mathrm G}(\rho_{AB}):= \inf_{\substack{V_p\preceq V\\V_p\ \mathrm{pure\ Gaussian}}}E(V_p), \tag{2} \end{equation}

where \(V\) is the covariance of \(\rho_{AB}\) and \(E(V_p)\) is the entropy of either reduced state of the pure Gaussian component. A covariance is bisymmetric when it is invariant under all mode permutations within Alice’s block and, independently, within Bob’s block. Is \(E_F(\rho_{AB})=E_F^{\mathrm G}(\rho_{AB})\) outside this family, with the quantities defined by Eqs. (1) and (2)?

Source

Adesso’s preprint proves Gaussian optimality for all two-mode and bisymmetric multimode states and leaves the generic nonsymmetric multimode case open [Ade26]. The equivalent universal Gaussian-discord formulation below follows from purification duality, rather than being a second independent problem [KW04], [AD10].

Progress

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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’s arXiv preprint establishes \(E_F=E_F^{\mathrm G}\) for all two-mode Gaussian states, then extends it to bisymmetric multimode states by localization to a two-mode core. Theorem 1 and Eqs. (18)–(22) state these results; The supplemental subsection Completion of the proof and infinite-dimensional limit, Eqs. (S54)–(S57) and the discussion after Eq. (S62), addresses limiting and continuous pure-state ensembles. Generic nonsymmetric multimode states remain open [Ade26].
For the discord formulation, Adesso–Datta Eqs. (3)–(4) solve the Gaussian measurement optimization for two modes [AD10]. Pirandola et al., Eqs. (12)–(14), prove unrestricted measurement optimality for states obtained by sending one half of a two-mode squeezed vacuum through a phase-insensitive Gaussian channel. Heterodyne detection on the other half attains the minimum. Their further Gaussian-unitary decompositions cover additional states, but do not settle the general two-mode discord problem, let alone the universal multimode problem. A generic two-mode discord instance has a two-mode purifying system and hence maps to a \(1\times2\)-mode entanglement-of-formation problem, outside the two-mode theorem; compare Wilde, Remark 3 [Wil18] [PSBCL14].

Comment

The generic multimode equality remains open. The resolved two-mode case is recorded separately as 01M26KH5NF45HBHXTPG8ZBD75H; its resolution is currently an arXiv preprint.

An equivalent universal question asks whether Gaussian measurements attain one-sided discord for every finite-energy bipartite Gaussian state \(\omega_{AX}\). For a POVM \(\{M_x\}\) on \(X\), set \(p_x=\operatorname{Tr}[(I_A\otimes M_x)\omega_{AX}]\) and \(\omega_{A|x}=\operatorname{Tr}_X[(I_A\otimes M_x)\omega_{AX}]/p_x\) for \(p_x>0\). Define

\begin{equation} D_X(\omega_{AX}):=S(\omega_X)-S(\omega_{AX}) +\inf_M\int p_x S(\omega_{A|x})\,dx . \tag{3} \end{equation}

All POVMs are allowed in Eq. (3); integrals include discrete sums. Define \(D_X^{\mathrm G}\) by restricting to general-dyne measurements, including joint multimode seeds and homodyne limits.

For a finite-energy Gaussian purification \(\Psi_{ABX}\), the ensemble-measurement correspondence gives

\begin{equation} D_X^{\mathrm G}(\Psi_{AX})-D_X(\Psi_{AX}) =E_F^{\mathrm G}(\Psi_{AB})-E_F(\Psi_{AB}). \tag{4} \end{equation}

The ordinary identity follows from Koashi–Winter Theorem 1, Eq. (2); Adesso–Datta discuss its Gaussian version after Eq. (7) [KW04], [AD10]. For the Gaussian restriction, every pure covariance \(V_p\preceq V\) in Eq. (2) can be prepared by a rank-one Gaussian measurement on a Gaussian purification. Singular boundary cases use homodyne limits; mixed measurement seeds cannot improve the minimum over their pure refinements. The unrestricted correspondence permits probability-measure ensembles.

Every finite-energy Gaussian state has a finite-mode, finite-energy Gaussian purification. Thus Eq. (4) identifies the two universal optimality questions. This equivalence, together with the resolved two-mode and bisymmetric cases, is why the discord formulation is consolidated here.

References

[WGKWC04]: M. M. Wolf, G. Giedke, O. Krüger, R. F. Werner, and J. I. Cirac, “Gaussian Entanglement of Formation,” Physical Review A 69, 052320 (2004).DOI arXiv
[Ade26]: G. Adesso, “Optimality of Gaussian Entanglement of Formation,” arXiv:2608.01909v2 (2026).DOI arXiv
[KW04]: M. Koashi and A. Winter, "Monogamy of Quantum Entanglement and Other Correlations," Physical Review A 69, 022309 (2004).DOI arXiv
[AD10]: G. Adesso and A. Datta, "Quantum versus Classical Correlations in Gaussian States," Physical Review Letters 105, 030501 (2010).DOI arXiv
[PSBCL14]: S. Pirandola, G. Spedalieri, S. L. Braunstein, N. J. Cerf, and S. Lloyd, "Optimality of Gaussian Discord," Physical Review Letters 113, 140405 (2014).DOI arXiv
[Wil18]: M. M. Wilde, "Entanglement Cost and Quantum Channel Simulation," Physical Review A 98, 042338 (2018).DOI arXiv

Gaussian entanglement of formation beyond bisymmetry

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