Entanglement of Formation for an Arbitrary Two-Mode Gaussian State

Abstract

We write the optimal pure-state decomposition of any two-mode Gaussian state and show that its entanglement of formation coincides with the Gaussian one. This enables us to develop an insightful approach of evaluating the exact entanglement of formation. Its additivity is finally proven.

Quantifying entanglement in two-mode Gaussian states

“…The optimum decomposition, and consequently r o , cannot in general be found analytically [5,12,24,26] Another way to decompose a state is as…”

“…One of them is entanglement of formation, defined as the convex-roof extension of the von Neumann entropy, E F (ρ) := inf{ i p i S(tr B |ψ i ψ i |)}, where the infimum is taken over all ensembles {p i ,ψ i } of ρ := i p i |ψ i ψ i | [4]. Specifically, for two-mode Gaussian states, where E F has been proven to be additive [5] (and thus strongly superadditive as well [6]), it coincides with the entanglement cost, E C (ρ) := lim n→∞ E F (ρ ⊗n )/n [7]. For a given state ρ, entanglement cost has a clear operational meaning, since it quantifies the minimum entanglement needed (cost of quantum resources) to produce ρ [7], which is of great importance in quantum technologies.…”

“…The reason this lower bound is tight for balanced states is that for these states the local squeezing parameters of the optimum decomposition are found to be r 1 = r 2 = 0 [5,12,26], and thus the two decompositions, i.e., Eq. (A5) and Eq.…”

“…(B1), coincide. For symmetric states, where the local squeezing parameters of the optimal decomposition are r 1 = r 2 = a+c 2 a−c 1 [5,12,24,26], the bound is tight since the operation L(r 1 ,r 2 )S 2 (r)S T 2 (r)L T (r 1 ,r 2 ) is identical to S 2 (r)L(r 1 ,r 2 )L T (r 1 ,r 2 )S T 2 (r) for r = r 1 = r 2 , so again, the two decompositions [Eq. (A5) and Eq.…”

Mixedness and entanglement for two-mode Gaussian states

“…It is well known today, both theoretically and experimentally, that in general such effects tend to destroy quantum properties; in the case of systems with one degree of freedom this happens only asymptotically [5][6][7]. The studies concerning the degradation of quantum effects are vast in the literature, specially those related to continuous variables systems subjected to noisy channels, see [8,9]. Some recent works devote their attention to the robustness of Gaussian and non-Gaussian states under dissipative channels [10].…”