Entanglement in the classical limit: Quantum correlations from classical probabilities

We investigate entanglement for a composite closed system endowed with a scaling property which allows the dynamics to be kept invariant while the effective Planck constant ${\hslash }_{\mathit{eff}}$ of the system is varied. Entanglement increases as ${\hslash }_{\mathit{eff}}\to 0$. Moreover, for sufficiently low ${\hslash }_{\mathit{eff}}$ the evolution of the quantum correlations, encapsulated, for example, in the quantum discord, can be obtained from the mutual information of the corresponding classical system. We show this behavior is due to the local suppression of path interferences in the interaction that generates the entanglement.

Figure 1

(a) The entanglement rate as a function of the number of kicks is given for quantum systems characterized by $k=0.25$ and ${\hslash }_{\mathit{eff}}=0.5,0.25,0.1,0.01$ (from bottom to top). The dashed lines result from applying the scaling relation (11) to the three upper curves. (b) Top: The black ring encircling the $\mathbf{N}$ axis is the initial classical distribution of $\mathbf{J}$ with $J=10$ (corresponding to the quantum case with ${\hslash }_{\mathit{eff}}=0.1$), centered on $N_0=T=50$. Bottom: The distribution at $t=10$ kicks (black dots), showing a spread. The surface of section for $k=0.25$ is shown in yellow (light gray).

Figure 2

For chosen values of the coupling $k$ and initial state, each row shows on the left-hand side entanglement evolution as computed from the entropy $H\left(t\right)$, and on the right the mutual information $M\left(t\right)$ of the corresponding classical counterpart. The time $t$ is computed at the discrete kick times $t=q{\tau }_{\epsilon }$. The parameters are $J=100,T=500$ (corresponding to ${\hslash }_{\mathit{eff}}=0.01$ for the quantum system), and (a) $k=1,N_0=402$; (b) $k=1,N_0=430$; (c) $k=0.01,N_0=498$; and (d) $k=10,N_0=460$.

Figure 3

Classical evolution and surfaces of section for the parameters corresponding to the plots of Fig. 2. (a)–(c) For $k=1$ (mixed phase-space), (a) gives the initial distributions for $N_0=402$ [blue (gray) ring] and $N_0=430$ (black ring); (b) displays the distributions after ten kicks; (c) at $t=25$. (d) Position of the initial distribution defined by $N_0=498$ for $k=0.01$ (regular dynamics). (e), (f) Distributions for $k=10$ (mostly chaotic phase space) and $N_0=460$ at $t=0$ [(e)] and at $t=5$ kicks [(f)].