No-activation theorem for Gaussian nonclassical correlations by Gaussian operations

We study general quantum correlations of continuous variable Gaussian states and their interplay with entanglement. Specifically, we investigate the existence of a quantum protocol activating all nonclassical correlations between the subsystems of an input bipartite continuous variable system, into output entanglement between the system and a set of ancillae. For input Gaussian states, we prove that such an activation protocol cannot be accomplished with Gaussian operations, as the latter are unable to create any output entanglement from an initial separable yet nonclassical state in a worst-case scenario. We then construct a faithful non-Gaussian activation protocol, encompassing infinite-dimensional generalizations of controlled-not gates to generate entanglement between system and ancillae, in direct analogy with the finite-dimensional case. We finally calculate the negativity of quantumness, an operational measure of nonclassical correlations defined in terms of the performance of the activation protocol, for relevant classes of two-mode Gaussian states.

Figure 1

Pictorial representation of the no-activation theorem. ${\rho }_{AB}$ is a separable Gaussian state prepared from a Gaussian product state ${\rho }_A\otimes {\rho }_B$ by correlated displacements $D_A$ and $D_B$ distributed according to a Gaussian distribution with correlation matrix $P$, Eq. (1). $U_A\left(S_A\right)$ and $U_B\left(S_B\right)$ are local Gaussian unitaries which are adjusted such that without the displacements $D_A$ and $D_B$ the activation protocol produces from the product state ${\rho }_A\otimes {\rho }_B$ an output state which is separable across the $AB|A^{\prime }{B}^{\prime }$ splitting. As the unitaries $U_A\left(S_A\right)$, $U_B\left(S_B\right)$, and $U\left(S\right)$ induce a linear transformation of quadrature operators of the input modes, the displacements $D_A$ and $D_B$ can be relocated behind the global transformation $U\left(S\right)$ (dotted arrow). The new displacements $D_A,D_B,D_{A^{\prime }}$, and $D_{B^{\prime }}$, Eq. (2), cannot turn a separable state into an entangled state and therefore the protocol transforms the separable state ${\rho }_{AB}$ into a state which is separable across the $AB|A^{\prime }{B}^{\prime }$ cut (thick dashed line). See text for details.

Figure 2

Negativity of quantumness ${\mathcal{N}}_{\mathrm{p}}$ [Eq. (10)] (solid red line) and its lower bound ${\mathcal{L}}_{\mathrm{p}}$ [Eq. (28)] (dash-dotted brown line) for pure squeezed vacuum states, plotted as a function of the local mean number of thermal photons $\langle n\rangle ={sinh}^2\left(r\right)$. The upper bound on the negativity of quantumness ${\mathcal{N}}_{\mathrm{m}}$ [Eq. (13)] (dashed blue line) and its lower bound ${\mathcal{L}}_{\mathrm{m}}$ [Eq. (29)] (dotted black line) for separable mixed states which are obtained as unbiased mixtures of coherent states, plotted as a function of the local mean number of thermal photons $\langle n\rangle ={\sigma }^2$. Inset: Close-up for $\langle n\rangle \ll 1$, where the lower bounds become tight.