Two-way distillable entanglement of mixed two-mode Gaussian states

Problem

What is the exact two-way distillable entanglement of a finite-energy mixed two-mode Gaussian state?

Let \(\rho_{AB}\) be a finite-energy mixed two-mode Gaussian density operator. Let \(\Lambda_k\) range over protocols using local operations and unlimited two-way classical communication. The complete protocols are trace preserving, and non-Gaussian operations are allowed. Finite energy means finite total mean photon number. Alice and Bob hold one mode each. Define the Bell-pair density operator and trace norm by

\begin{equation} \Phi_2:=\tfrac12(|00\rangle+|11\rangle)(\langle00|+\langle11|), \qquad \lVert X\rVert_1:=\operatorname{Tr}\sqrt{X^\dagger X}. \tag{1} \end{equation}

Using the Bell pair in Eq. (1), call \(r\geq0\) achievable if an allowed sequence \((\Lambda_k)_{k\geq1}\) satisfies

\begin{equation} \lim_{k\to\infty}\left\lVert\Lambda_k(\rho_{AB}^{\otimes k})-\Phi_2^{\otimes\lfloor rk\rfloor}\right\rVert_1=0. \tag{2} \end{equation}

Determine \(D_{\leftrightarrow}(\rho_{AB}):=\sup\{r\geq0:r\text{ is achievable}\}\), with achievability defined by Eq. (2).

Source

This exact-rate question is formulated from the gap between quantitative distillation bounds [DW05], [VW02], together with the Gaussian distillability criterion [GDCZ01].

Progress

Giedke et al., Theorem 1, prove that a bipartite Gaussian state is distillable exactly when its partial transpose is not positive. In particular, every entangled two-mode Gaussian state is distillable. This settles positivity of the rate, not its exact value. [GDCZ01]
Devetak–Winter, Theorem 10, proves the finite-dimensional hashing inequality. The hashing bound, applied in either communication direction and extended to finite-energy bosonic states by local truncation, gives

\begin{equation} D_{\leftrightarrow}(\rho_{AB})\geq\max\{0,S(\rho_A)-S(\rho_{AB}),S(\rho_B)-S(\rho_{AB})\}, \tag{3} \end{equation}

In Eq. (3), \(\rho_A:=\operatorname{Tr}_B\rho_{AB}\), \(\rho_B:=\operatorname{Tr}_A\rho_{AB}\), and \(S(\omega):=-\operatorname{Tr}(\omega\log_2\omega)\). [DW05]
Vidal–Werner, Proposition 7 and Section V.C, bounds the rate by logarithmic negativity:

\begin{equation} D_{\leftrightarrow}(\rho_{AB})\leq\log_2\lVert\rho_{AB}^{T_B}\rVert_1 =\sum_{j=1}^2\max\{0,-\log_2\widetilde\nu_j\}, \tag{4} \end{equation}

In Eq. (4), \(T_B\) is partial transposition and \(\widetilde\nu_j\) are the symplectic eigenvalues of the partially transposed covariance in vacuum-\(I\) units. [VW02]

Comment

Separable states have rate zero. For general entangled mixed two-mode Gaussian states, the known lower and upper bounds need not coincide. Gaussian-operation no-go theorems do not settle the unrestricted LOCC rate.

References

[DW05]: I. Devetak and A. Winter, "Distillation of Secret Key and Entanglement from Quantum States," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, 207–235 (2005).DOI arXiv
[VW02]: G. Vidal and R. F. Werner, "Computable Measure of Entanglement," Physical Review A 65, 032314 (2002).DOI arXiv
[GDCZ01]: G. Giedke, L.-M. Duan, J. I. Cirac, and P. Zoller, "Distillability Criterion for All Bipartite Gaussian States," Quantum Information and Computation 1(3), 79–86 (2001).arXiv

Two-way distillable entanglement of mixed two-mode Gaussian states

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