Comparative study of nonclassicality, entanglement, and dimensionality of multimode noisy twin beams

The nonclassicality, entanglement, and dimensionality of a noisy twin beam are determined using a characteristic function of the beam written in the Fock basis. One-to-one correspondence between the negativity quantifying entanglement and the nonclassicality depth is revealed. Twin beams, which are either entangled or nonclassical (independent of their entanglement), are observed only for the limited degrees of noise, which degrades their quantumness. The dimensionality of the twin beam quantified by the participation ratio is compared with the dimensionality of entanglement determined from the negativity. Partitioning of the degrees of freedom of the twin beam into those related to entanglement and to noise is suggested. Both single-mode and multimode twin beams are analyzed. Weak nonclassicality based on integrated-intensity quasidistributions of multimode twin beams is studied. The relation of the model to the experimental twin beams is discussed.

Figure 1

Negativity $N$ as a function of the mean photon-pair number $B_{\mathrm{p}}$ for noiseless twin beams (i.e., $B_{\mathrm{s}}=B_{\mathrm{i}}=0)$ according to Eq. (16).

Figure 2

Negativity $N$, given in Eq. (15), as a function of the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ in the signal and idler modes, respectively, assuming the mean photon-pair number $B_{\mathrm{p}}$ equal to 0.5 [bottom, light-grey (yellow) area], 1 [grey (green) area], 2 [dark-grey (blue) area], and 4 [top, black area]. The larger $B_{\mathrm{p}}$, the larger the negativity $N$.

Figure 3

Curves giving the boundaries between entangled and separable twin beams and determined according to Eq. (18) plotted in the plane spanned by the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ assuming the mean photon-pair number $B_{\mathrm{p}}$ equal to 0.01 [dotted (red) curve], 0.1 [dash-dotted (yellow) curve], 0.5 [dashed (green) curve], 2 [long-dashed (blue) curve], and 100 (solid black curve). Entangled states are localized in the lower-left corner of the plane. The larger $B_{\mathrm{p}}$, the larger the area containing entangled states.

Figure 4

Distribution $d_N$ of negativity $N$ given in Eq. (19) assuming $B_{\mathrm{p}}=2$ and (a) $B_{\mathrm{s}}=B_{\mathrm{i}}=0$ and (b) $B_{\mathrm{s}}=B_{\mathrm{i}}=0.1$. Note that $-d_N\left(M\right)$ corresponds to the sum of all the negative eigenvalues for the $(M+1)$–dimensional block of the partially transposed statistical operator ${\widehat{\rho }}^{\Gamma }$. Thus, $d_N\left(M\right)$ shows the internal structure of entanglement in the Liouville space.

Figure 5

Nonclassical depth $\tau$ given in Eq. (21) as it depends on the mean photon-pair number $B_{\mathrm{p}}$ for noiseless twin beams, i.e., $B_{\mathrm{s}}=B_{\mathrm{i}}=0$.

Figure 6

Nonclassical depth $\tau$ given in Eq. (20) as a function of the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ for the mean photon-pair number $B_{\mathrm{p}}$ equal to 0.1 [bottom, light-grey (yellow) area], $0.5$ [grey (green) area], 4 [top, dark-grey (blue) area]. The greater the value of $B_{\mathrm{p}}$, the greater the value of $\tau$.

Figure 7

Negativity $N$ as a function of the nonclassical depth $\tau$, according to Eq. (22).

Figure 8

Modified entanglement dimensionality ${\tilde{K}}_{\mathrm{ent}}$ given in Eq. (31) [lower, dark-grey (blue) area] and signal-field participation ratio $R_{\mathrm{s}}$ given in Eq. (27) [upper, grey (red) area] as they depend on the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ assuming the mean photon-pair number $B_{\mathrm{p}}=1$.

Figure 9

Coefficient $r_{\mathrm{ent}}$ given in Eq. (32) versus the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ for the mean photon-pair number $B_{\mathrm{p}}$ equal to 1 [upper, dark-grey (blue) area] and 10 [lower, grey (red) area].

Figure 10

von Neumann entropy $S_{\mathrm{s}}$ as a function of the participation ratio $R_{\mathrm{s}}$ according to Eq. (35).

Figure 11

Nonclassical intensity depth ${\tau }_W$ as a function of the mean noise photon numbers $B_{\mathrm{s}}$ and $B_{\mathrm{i}}$ for the mean photon-pair number $B_{\mathrm{p}}$ equal to: 2 [bottom, light-grey (yellow) area], 4 [grey (green) area], and 8 [top, dark-grey (blue) area], assuming $M_{\mathrm{p}}=M_{\mathrm{s}}=M_{\mathrm{i}}=1$. The greater the value of $B_{\mathrm{p}}$ the greater the value of ${\tau }_W$.