Conservation relation of nonclassicality and entanglement for Gaussian states in a beam splitter

We study the relation between single-mode nonclassicality and two-mode entanglement in a beam splitter. We show that single-mode nonclassicality (the entanglement potential) of incident light cannot be transformed into two-mode entanglement completely after a single beam splitter. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as an increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam splitters.

Figure 1

A nonclassical Gaussian state ${\rho }_{\text{NCS}}$ is mixed with a vacuum state at a BS, generating an output state ${\rho }_{12}$. Each output mode of the bipartite state after the BS is given by ${\rho }_1={\text{Tr}}_2\left({\rho }_{12}\right)$ and ${\rho }_2={\text{Tr}}_1\left({\rho }_{12}\right)$, respectively.

Figure 2

Nonclassicality of the input single-mode state transforms into two-mode entanglement in a beam splitter. When the amount of extracted two-mode entanglement $\left(E_N\right)$ increases, the nonclassicality in the output states $\left(\tau \right)$ decreases. Not all of the nonclassicality could be converted to two-mode entanglement, even for an optimum BS mixing angle.

Figure 3

The two-mode output state of the first BS is partially traced and put in to a second BS, ${\tilde{\rho }}_{12}^{\left({\mathrm{BS}}_2\right)}={\text{Tr}}_2\left({\rho }_{12}^{\left({\mathrm{BS}}_2\right)}\right)\otimes {\text{Tr}}_1\left({\rho }_{12}^{\left({\mathrm{BS}}_2\right)}\right)$, in order to extract the remaining nonclassicality as entanglement. We repeat the same procedure for four BSs. The nonclassical depth before a BS, ${\tau }_{i-1}$, displays parallel behavior with the two-mode entanglement extracted from this BS, $E_N^{{(\mathrm{BS}}_i)}$. Nonclassicality is depleted at the fourth BS.

Figure 4

Output entanglement from mixing a nonclassical state with a vacuum at a BS quantified by two measures of the degree of entanglement. (a) For constant nonclassicality, the relation of the degree of entanglement vs $\theta$ for different purities of the input nonclassical state. (b) At an optimal BS angle, the monotonic relation of the degree of entanglement vs the inverse of the minimum eigenvalue.

Figure 5

$N_{\text{noncl}}^{\text{in}},N_{\text{noncl}}^{\text{out}}$ and $S_{\mathcal{N}}$ vs the BS angle $\theta$ for a nonclassical state mixing with a vacuum state.

Figure 6

$N_{\text{noncl}}^{\text{in}},N_{\text{noncl}}^{\text{out}}$ and $S_{\mathcal{N}}$ vs the BS angle $\theta$ for a pure nonclassical state mixing with a thermal state.

Figure 7

$N_{\text{noncl}}^{\text{in}},N_{\text{noncl}}^{\text{out}}$ and $S_{\mathcal{N}}$ vs the BS angle $\theta$ for a pure nonclassical state mixing with another nonclassical state.

Figure 8

A nonclassical state ${\rho }_{\text{NCS}}$ is mixed with a vacuum state at a BS, generating an output state ${\rho }_{12}$. Each output mode ${\rho }_1$ or ${\rho }_2$ is mixed with a vacuum state at another BS generating two sets of output bipartite Gaussian states ${\rho }_{12}^{11}$ and ${\rho }_{12}^{12}$. This process can be extended further with more BSs.