LOCC entanglement cost of PPT-entangled Gaussian states

Problem

What is the exact LOCC entanglement cost of a finite-energy PPT-entangled Gaussian state?

Let \(\rho_V\) be a zero-mean Gaussian state of \(m\) modes held by Alice and \(n\) modes held by Bob, with integers \(m,n\geq2\). Assume that \(\rho_V\) is entangled and its partial transpose \(\rho_V^{T_B}\) is positive. Its finite covariance matrix and canonical quadratures satisfy

\begin{equation} \begin{gathered} V_{jk}:=\operatorname{Tr}\rho_V\{R_j,R_k\},\qquad [R_j,R_k]=i(\Omega_{m+n})_{jk},\\ \Omega_k:=\bigoplus_{j=1}^k\begin{pmatrix}0&1\\-1&0\end{pmatrix}. \end{gathered} \tag{1} \end{equation}

Equation (1) uses vacuum covariance \(I\). The allowed channels \(\Lambda_N\) use unrestricted local operations and classical communication (LOCC). Define the Bell-pair density operator by

\begin{equation} \Phi_2:=|\phi_2\rangle\langle\phi_2|,\qquad |\phi_2\rangle:=(|00\rangle+|11\rangle)/\sqrt2. \tag{2} \end{equation}

Using Eq. (2), the entanglement cost \(E_C(\rho_V)\) is the infimum of rates \(r\geq0\) for which LOCC channels \(\Lambda_N\) satisfy

\begin{equation} \lim_{N\to\infty}\|\Lambda_N(\Phi_2^{\otimes\lceil rN\rceil})-\rho_V^{\otimes N}\|_1=0. \tag{3} \end{equation}

Determine the cost specified by Eq. (3) as a function of \(V\). Separability means the trace-norm closed convex hull of product states. Here \(\|X\|_1:=\operatorname{Tr}\sqrt{X^\dagger X}\).

Source

This exact-evaluation question combines the PPT-entangled Gaussian family of Werner–Wolf, Section IV, with the infinite-dimensional operational cost theorem [WW01], [YKHL25].

Progress

Positive partial transpose does not imply Gaussian separability once both parties have two modes, and the logarithmic negativity vanishes on every state in this question:

\begin{equation} \rho_V^{T_B}\geq0\quad\Longrightarrow\quad \log_2\|\rho_V^{T_B}\|_1=0. \tag{4} \end{equation}

Despite Eq. (4), explicit four-mode counterexamples to the converse separability implication exist. [WW01]
Theorem 7 of Yamasaki et al. applies because the local entropies are finite. Finite-energy Gaussian states satisfy the infinite-dimensional entanglement-cost formula

\begin{equation} \begin{gathered} E_C(\rho_V)=\lim_{N\to\infty}\frac{E_F(\rho_V^{\otimes N})}{N},\\ E_F(\omega):=\inf_{\mu:\,\int|\psi\rangle\langle\psi|\,d\mu(\psi)=\omega} \int S(\operatorname{Tr}_B|\psi\rangle\langle\psi|)\,d\mu(\psi), \end{gathered} \tag{5} \end{equation}

In Eq. (5), \(\mu\) ranges over probability measures on normalized pure states and \(S(\tau):=-\operatorname{Tr}(\tau\log_2\tau)\); this variational formula permits non-Gaussian decompositions. [YKHL25]
Lami, Serafini, and Adesso further restrict where PPT-entangled Gaussian examples can occur. Theorem 9 locally reduces mono-symmetric states to a \(1\times n\)-mode core plus uncorrelated local modes, so PPT implies separability in that family. Theorem 11 proves PPT equivalence to separability for isotropic Gaussian states as well. Neither family therefore supplies a PPT-entangled input for the present cost question [LSA18].

Comment

The regularized convex-roof characterization is established. Its exact evaluation for arbitrary PPT-entangled Gaussian covariances remains open. Replacing the unrestricted roof by a Gaussian roof or changing LOCC to PPT-preserving operations changes the question.

References

[WW01]: R. F. Werner and M. M. Wolf, "Bound Entangled Gaussian States," Physical Review Letters 86, 3658–3661 (2001).DOI arXiv
[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
[LSA18]: L. Lami, A. Serafini, and G. Adesso, "Gaussian Entanglement Revisited," New Journal of Physics 20, 023030 (2018).DOI arXiv

LOCC entanglement cost of PPT-entangled Gaussian states

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