Additivity of entanglement of formation for two-mode Gaussian states

Problem

Is unrestricted entanglement of formation additive on tensor powers of every finite-energy two-mode Gaussian state?

Let \(\rho_{AB}\) be any finite-energy two-mode Gaussian density operator, shared between one mode at each party. Finite energy means finite total mean photon number. Define the unrestricted pure-state convex roof

\begin{equation} \begin{gathered} E_F(\omega):=\inf_\mu\int S(\operatorname{Tr}_B|\psi\rangle\langle\psi|)\,\mu(d\psi), \\ S(\tau):=-\operatorname{Tr}(\tau\log_2\tau), \end{gathered} \tag{1} \end{equation}

In Eq. (1), \(\mu\) ranges over pure-state probability measures with barycentre \(\omega\).

The target is \(E_F(\rho_{AB}^{\otimes k})=kE_F(\rho_{AB})\) for every integer \(k\geq1\). The bipartition for the tensor power is \(A^k:B^k\). Non-Gaussian vectors are allowed in all decompositions.

Source

Adesso explicitly leaves equality of entanglement cost and single-copy entanglement of formation open in the Conclusion and Outlook [Ade26]. Under the finite-entropy cost theorem this is equivalent to tensor-power additivity [YKHL25].

Progress

Theorem 7 of Yamasaki et al. identifies entanglement cost with the regularized convex roof:

\begin{equation} E_C(\rho_{AB})=\lim_{k\to\infty}\frac1kE_F(\rho_{AB}^{\otimes k}), \tag{2} \end{equation}

Equation (2) includes the continuous pure-state roof. Finite local entropy suffices; finite-energy states on finitely many modes satisfy this hypothesis. Here \(E_C\) is the asymptotic Bell-pair consumption per target copy under LOCC with vanishing trace-norm error. [YKHL25]
Tensor products of single-copy ensembles give only

\begin{equation} E_C(\rho_{AB})\leq\frac1kE_F(\rho_{AB}^{\otimes k})\leq E_F(\rho_{AB}), \tag{3} \end{equation}

The reverse inequality needed to saturate Eq. (3) remains open, as stated in Adesso’s Conclusion and Outlook. [Ade26]
Wilde, Proposition 6 and Eqs. (141)–(143), proves Eq. (4) for \(\rho_{AB}=(\operatorname{id}\otimes\mathcal N)(|\Phi_{N_s}\rangle\langle\Phi_{N_s}|)\). Here \(|\Phi_{N_s}\rangle:=\sum_{j\geq0}\sqrt{N_s^j/(N_s+1)^{j+1}}|j,j\rangle\) is a two-mode squeezed vacuum with \(N_s\geq0\). The channel \(\mathcal N\) is either pure loss with transmissivity \(0<\eta<1\), or a quantum-limited amplifier with gain \(G>1\).

\begin{equation} \begin{gathered} E_F(\rho_{AB}^{\otimes k})=kE_F(\rho_{AB}),\\ E_C(\rho_{AB})=E_F(\rho_{AB})=E_F^{\mathrm G}(\rho_{AB}). \end{gathered} \tag{4} \end{equation}

Here \(E_F^{\mathrm G}\) is the restriction to Gaussian pure-state decompositions. [Wil18]
Wilde, Remark 2 and Eq. (157), extends the additivity of optimized conditional entropy to the full two-mode Gaussian family considered by Pirandola et al. By purification duality this gives entanglement-of-formation additivity for the corresponding complementary states, whose mode counts must be checked separately. Remark 3 explains why Proposition 6 does not directly extend to generic noisy channel-output states: a faithful two-mode state needs a purification with at least four modes [Wil18], [PSBCL14].
Marian and Marian published a claim of additivity for arbitrary two-mode Gaussian states [MM08]. Adesso’s Conclusion and Outlook explains the unresolved step: tensoring the proposed single-copy ensembles gives only \(E_F(\rho^{\otimes k})\leq kE_F(\rho)\), not the required lower bound against collective non-Gaussian ensembles [Ade26].
Wolf et al., Section VII, Proposition 3, prove additivity of the Gaussian entanglement of formation for symmetric two-mode Gaussian states. This restricted-roof result does not exclude collective non-Gaussian decompositions in the unrestricted many-copy roof [WGKWC04].

Comment

The missing inequality must exclude collective non-Gaussian decompositions of many copies. A single-copy covariance optimization does not provide this exclusion. The cost is measured in asymptotic Bell pairs per copy with vanishing trace-norm error under unrestricted LOCC.

References

[YKHL25]: H. Yamasaki, K. Kuroiwa, P. Hayden, and L. Lami, "Entanglement Cost for Infinite-Dimensional Physical Systems," Communications in Mathematical Physics 406, 277 (2025).DOI arXiv
[Ade26]: G. Adesso, "Optimality of Gaussian Entanglement of Formation," preprint (2026), version 2, 13 August 2026.DOI arXiv
[Wil18]: M. M. Wilde, "Entanglement Cost and Quantum Channel Simulation," Physical Review A 98, 042338 (2018).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
[MM08]: P. Marian and T. A. Marian, “Entanglement of Formation for an Arbitrary Two-Mode Gaussian State,” Physical Review Letters 101, 220403 (2008).DOI arXiv
[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

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