Operational Quantification of Continuous-Variable Correlations

We quantify correlations (quantum and/or classical) between two continuous-variable modes as the maximal number of correlated bits extracted via local quadrature measurements. On Gaussian states, such “bit quadrature correlations” majorize entanglement, reducing to an entanglement monotone for pure states. For non-Gaussian states, such as photonic Bell states, photon-subtracted states, and mixtures of Gaussian states, the bit correlations are shown to be a monotonic function of the negativity. This quantification yields a feasible, operational way to measure non-Gaussian entanglement in current experiments by means of direct homodyne detection, without a complete state tomography.

Figure 1

\Bit quadrature correlations vs normalized negativity for 18 000 random two-mode Gaussian states. The lowermost dotted curve accommodates pure states. The leftmost solid vertical line denotes separable states parametrized by $c_p=0$, $c_x=ϵ({\lambda }^2-1)/\lambda$, with $0\le ϵ\le 1$, and ${\lambda }_{a,b}=\lambda \to \infty$. The uppermost dashed horizontal line denotes perfectly correlated states parametrized by $c_x=({\lambda }^2-1)/\lambda$, $c_p=ϵc_x$ (and $ϵ$, $\lambda$ as before). Product states (totally uncorrelated) lie at the origin.

Figure 2

(a) Bit quadrature correlations $Q$ vs squeezing $r$ and beam splitter transmittivity $T$ for photon-subtracted states (shaded surface) and two-mode squeezed states (wireframe surface). (b) $Q$ vs normalized negativity.

Figure 3

Bit quadrature correlations $Q$ vs squeezing $r$ (in decibels) for the de-Gaussified states demonstrated in 7 with beam splitter reflectivities $R$ equal to (from top to bottom): 3%, 5%, 10%, 20%. The dashed curve depicts $Q$ for a two-mode squeezed state. Cf. Fig. 6 of 7.