LOCC entanglement cost of PPT-entangled Gaussian states
- Field
- Topics
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
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
Using Eq. (2), the entanglement cost \(E_C(\rho_V)\) is the infimum of rates \(r\geq0\) for which LOCC channels \(\Lambda_N\) satisfy
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
Reports do not certify correctness or automatically change the problem's status. Progress policy.
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.