Probing light polarization with the quantum Chernoff bound

We recall the framework of a consistent quantum description of polarization of light. Accordingly, the degree of polarization of a two-mode state $\mathrm{\rho ̂}$ of the quantum radiation field can be defined as a distance of a related state ${\mathrm{\rho ̂}}_b$ to the convex set of all SU(2)-invariant two-mode states. We explore a distance-type polarization measure in terms of the quantum Chernoff bound and derive its explicit expression. A comparison between the Chernoff and Bures degrees of polarization leads to interesting conclusions for some particular states chosen as illustrative examples.

Figure 1

Displaying the saddle-point evaluation of the Chernoff degree of polarization ${\mathbb{P}}_{\mathrm{C}}(|\Psi \rangle \langle \Psi |)$ for a state (4.11) with $N_1=1$, $N_2=2$, and $p=0.1$. The Rényi overlap, Eq. (4.13), is plotted vs $s$ and ${\pi }_1$. The saddle point has the coordinates $\mathrm{s̃}=0.124$ and ${\mathrm{\pi ̃}}_1=0.634$. The Chernoff overlap, Eq. (4.17), is $\mathrm{Q̃}=0.431$, so the degree of polarization is ${\mathbb{P}}_{\mathrm{C}}=0.569$.

Figure 2

Degree of polarization of the pure states (4.11) characterized by $N_1=1$, $N_2=2$ as a function of the probability $p$: the Chernoff measure (black full line) and the Bures measure (red dashed line).

Figure 3

Saddle-point evaluation of the Chernoff degree of polarization ${\mathbb{P}}_{\mathrm{C}}(\mathrm{\tau ̂})$ for a state (4.23) with $N_1=1$, $N_2=2$, $p=0.1$, $\alpha =0.1,\beta =0.01$, and $\gamma =0.04$. The Rényi overlap $Q_s$, Eq. (4.25), is plotted vs $s$ and ${\pi }_1$. The saddle point is reached at $\mathrm{s̃}=0.434$ and ${\mathrm{\pi ̃}}_1=0.209$. The optimal value $\mathrm{Q̃}$ is 0.544, so that the degree of polarization is ${\mathbb{P}}_{\mathrm{C}}(\mathrm{\tau ̂})=0.251$. For the same state, the Bures degree of polarization, Eq. (4.27), is ${\mathbb{P}}_{\mathrm{B}}(\mathrm{\tau ̂})=0.247$.

Figure 4

Degree of polarization of the mixed states (4.23) characterized by the same parameters $N_1,N_2,\alpha ,\beta ,\gamma$ as in Fig. 3 vs the mixing coefficient $p$: the Chernoff measure (black full line) and the Bures measure (red dashed line).