Polarization and entanglement of photon-added coherent states

Polarization of light has been used extensively in quantum information processing, and quantum entanglement is essential to many areas of research, including quantum computing. Here we investigate the degree of polarization and the entanglement of a family of quantum states known as photon-added entangled coherent states. Such states could serve as means of entanglement distribution and quantum key distribution. Using the quantum Stokes parameters and the $Q$ function, we demonstrate that, in general, the degree of polarization of two two-mode photon-added coherent states increases significantly with the number of added photons. And using the concurrence, we show that the amount of entanglement in this kind of superposition presents a behavior that is dependent on whether or not the number of added photons in each mode is the same.

Figure 1

Quantum degree of polarization (P) versus the expected energy in the underlying coherent states ${\left|\alpha \right|}^2+{\left|\beta \right|}^2$, for four superpositions with different numbers of added photons. As seen in the figure, in general, the quantum degree of polarization increases with the number of added photons. When the number of added photons is different from zero and $\alpha =\beta =0$, the results obtained agree with those in the literature (the quantum degree of polarization of number states) [20].

Figure 2

Percentage increase of P for each of the states $|{\psi }_{01}\rangle$, $|{\psi }_{11}\rangle$, and $|{\psi }_{12}\rangle$ in relation to $|{\psi }_{00}\rangle$, in the interval $2\le {\left|\alpha \right|}^2+{\left|\beta \right|}^2\le 8$.

Figure 3

The degree of polarization is analyzed in two cases: (1) a superposition with one added and (2) a superposition with zero added photons. For both cases, the degree of polarization increases when we increase the separation of the coherent-state amplitudes ($1/4$, $1/2$, and 1). By comparing any pair of curves of the same design (dotted, dashed, or solid) and color, we can see that one single photon added to one of the coherent states can substantially increase the degree of polarization.

Figure 4

Amount of entanglement, given by the concurrence $C$, versus the expected energy in the underlying coherent states ${\left|\alpha \right|}^2+{\left|\beta \right|}^2$, for four superpositions with different numbers of added photons. For superpositions where the number of added photons in each mode is equal, the concurrence starts at zero and increases with ${\left|\alpha \right|}^2+{\left|\beta \right|}^2$, and for states where the number of added photons in each mode is different, the amount of entanglement starts at maximum and after decreasing slightly, it starts to increase again.

Figure 5

The amount of entanglement, given by the concurrence $C$, is analyzed in two cases: (1) a superposition with one added and (2) a superposition with zero added photons. For both cases, the amount of entanglement increases when we increase the separation of the amplitudes.

Figure 6

Smallest phase shift allowing distinguishability when we consider (a) entangled NOON states with $N=3$, (b) entangled photon-added coherent states with three added photons and $\alpha =2$, and (c) entangled photon-added coherent states with three added photons and $\alpha =4$. Note that in case (b), the two states are not fully distinguishable.