Entanglement of quantum circular states of light

We present a general approach to calculating the entanglement of formation for superpositions of two-mode coherent states, placed equidistantly on a circle in phase space. We show that in the particular case of rotationally invariant circular states the Schmidt decomposition of two modes, and therefore the value of their entanglement, are given by analytical expressions. We analyze the dependence of the entanglement on the radius of the circle and number of components in the superposition. We also show that the set of rotationally invariant circular states creates an orthonormal basis in the state space of the harmonic oscillator, and this basis is advantageous for representation of other circular states of light.

Figure 1

A symmetric beam splitter. The in-state of mode A is transformed into an (in general) entangled out-state of modes A and B.

Figure 2

Coherent states on a circle of radius $|{\alpha }_0|$. Each coherent state is represented by a circle of radius $\frac{1}{2}$, corresponding to the $\sigma$ area of its Wigner function, being a two-dimensional Gaussian distribution.

Figure 3

Entanglement of formation for a two-mode RICS as a function of the coherent amplitude for various numbers of components and $q=1$. The asymptotic value in all cases is ${log}_{2}N$.

Figure 4

Entanglement of formation for a two-mode RICS as a function of the number of components (a) for various values of $|{\alpha }_0|$ and $q$ and (b) for the case where $|{\alpha }_0|=3$ and $q=4$, represented as the sum of entropies.

Figure 5

Entanglement of formation for a two-mode RICS, $|{\alpha }_0|=4$, and for a Kerr state with the same amplitude. The dependence has three regions: (i) logarithmic growth up to $N_1=\pi \left|{\alpha }_0\right|$ during which the components are almost orthogonal; (ii) oscillations when the components start to overlap and interfere; and (iii) an asymptotically constant value after $N_2=2e{\left|{\alpha }_0\right|}^2$. $q$ max corresponds to the maximal entanglement for a given $N$.