Entanglement of formation for Gaussian states

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Entanglement of formation is defined as a minimum over all convex decompositions of a bipartite state into pure states (see problem 7). It has been shown, that for certain two-mode Gaussian states this minimum can be taken over decompositions of the given state into pure states, all of which are translates of the same squeezed Gaussian state with Gaussian weights. Show (or disprove) that this is true for all Gaussian states.


If the optimization over convex decompositions of a bipartite state is restricted to decompositions into Gaussian states, entanglement of formation becomes a new entanglement measure, the Gaussian entanglement of formation introduced in [1]. Whith this, the above question reads: Does entanglement of formation equal all entanglement of formation for all Gaussian states?

Partial Results

  • It has been shown in [2] that Gaussian entanglement of formation equals entanglement of formation for two-mode Gaussian states which are symmetric with respect to interchange of the modes.


  1. M. Wolf, G. Giedke, O. Krüger, R. F. Werner, and J. I. Cirac, Phys. Rev. A 69, 052320 (2004) and quant-ph/0306177 (2003)
  2. G. Giedke, M. M. Wolf, O. Krüger, R. F. Werner, and J. I. Cirac, Phys. Rev. Lett. 91, 107901 (2003) and quant-ph/0304042 (2003)