Entanglement of mixed macroscopic superpositions: An entangling-power study

We investigate entanglement properties of a recently introduced class of macroscopic quantum superpositions in two-mode mixed states. One of the tools we use in order to infer the entanglement in this non-Gaussian class of states is the power to entangle a qubit system. Our study reveals features which are hidden in a standard approach to entanglement investigation based on the uncertainty principle of the quadrature variables. We briefly describe the experimental setup corresponding to our theoretical scenario and a suitable modification of the protocol which makes our proposal realizable within the current experimental capabilities.

Figure 1

(Color online) Scheme of principle of the protocol to produce the generalized version of the catlike state (microscopic-macroscopic entangled state) and the second class of thermally weighted entangled coherent states (performed by the additional conditioned step depicted in the dashed box). The conditional-$\phi$ gate is the effective evolution induced by the cross-Kerr interaction between the microscopic part $m$ (in the pure state $\mid \psi {⟩}_m$) and the macroscopic one in the thermal displaced state ${\rho }_M^{\mathrm{th}}$. BS is a 50:50 beam splitter and the symbol for a detector is also shown.

Figure 2

Entanglement in the generalized catlike state ${\rho }_{mM}^{\mathrm{ent}}$ against the interaction phase $\phi$, for different values of the variance $V$. In panel (a) we have considered the displacement $d=1$. From bottom to top, the curves show the behavior of the entanglement as $V$ is increased. We have considered $V=2(◆)$, $V=5(\star )$, and $V=\left(10\right)(∎)$. Panel (b) shows the results for $d=7$ showing that, for larger $d$, the curves relative to different $V$’s get so close to become indistinguishable. The saturation to one ebit of entanglement is nearly independent of the temperature of the thermal distribution.

Figure 3

(Color online) Entangling power of the state $\int d^2{P}_M^{\mathrm{th}}(V,d){\rho }_{\psi }\left(\tau \right)$ for $V=10$ and $d=10$ ($◆$, blue line) and for $V=10$ with $d=7$ ($\star$, red line). In both the simulation, $\phi =\pi$ has been assumed. The horizontal axis shows the rescaled interaction time $\tau =\Omega t$.

Figure 4

(Color online) Mixedness of the state $\int d^2{P}_M^{\mathrm{th}}(V,d){\rho }_{\psi }\left(\tau \right)$ against the rescaled interaction time $\tau$ for the same parameters as Fig. 3. Also shown the mixedness threshold for a two-qubit state for quantum teleportation (thick straight line).

Figure 5

(Color online) (a) Entangling power of the state $\int d^2{P}_M^{\mathrm{th}}(V,d){\rho }_{\psi }\left(\tau \right)$ for $V=100$ and $d=20$. The inset shows the entanglement, against $\phi$, in the corresponding ${\rho }_{mM}^{\mathrm{ent}}$. (b) Mixedness of the same state as a function of the rescaled interaction time. The mixedness threshold for teleportation is also shown (straight line).

Figure 6

Scheme of principle of the experimental to be realized in order to test the entangling power as described in Sec. 3. Modes $m$ and $M$ are assumed to be addressing two cavities through which two independent qubits pass. The cavity-qubit interaction changes the refraction index of the cavity, allowing for the feeding by the external fields.

Figure 7

Scheme for the experimental verification of the analysis presented in this paper. The scheme reproduces the single-qubit transfer protocol we address in the body of the paper. In this case, the generalized catlike state is given by the state of qubit $a$ and the cavity mode field. Qubit $b$ is ancillary and is used just in order to test the entangling power in a less demanding way.

Figure 8

(Color online) (a) Entangling power of the state $\int d^2{P}_M^{\mathrm{th}}(V,d){\rho }_{\psi }\left(\tau \right)$ against $\tau$ for $V=10$ and $d=10$ $(◆)$ and $V=10$ with $d=7(\star )$. (b) Mixedness of the same state as a function of the rescaled interaction time. The mixedness threshold for teleportation is also shown (straight line).