Distance-based degrees of polarization for a quantum field

It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.

Figure 1

Polarization degrees for two-dimensional states with $N_1=1$ and $N_2=2$. For any given $p$, the maximum and minimum fidelities are given by ${\mid q\mid }^2=p(1-p)$ and ${\mid q\mid }^2=0$, respectively. Region $A$ corresponds to states satisfying $p>4∕7$ and ${\mathbb{P}}_{\mathrm{B}}<1-\sqrt{3∕7}$. For any state ${\widehat{\rho }}_A$ in this region, we have ${\mathbb{P}}_{\mathrm{B}}\left({\widehat{\rho }}_A\right)<{\mathbb{P}}_{\mathrm{B}}\left({\widehat{\rho }}_B\right)$ while ${\mathbb{P}}_{\mathrm{HS}}\left({\widehat{\rho }}_A\right)>{\mathbb{P}}_{\mathrm{HS}}\left({\widehat{\rho }}_B\right)$, where the state ${\widehat{\rho }}_B$ is characterized by $p=4∕7$ and ${\mid q\mid }^2=0$. In region $C$, the states satisfy $p>2∕3$ and ${\mathbb{P}}_{\mathrm{HS}}>2∕3$. For any such state ${\widehat{\rho }}_C$, we have ${\mathbb{P}}_{\mathrm{HS}}\left({\widehat{\rho }}_C\right)>{\mathbb{P}}_{\mathrm{HS}}\left(\mid {\Psi }^{\left(2\right)}⟩\right)$ and ${\mathbb{P}}_{\mathrm{B}}\left({\widehat{\rho }}_C\right)<{\mathbb{P}}_{\mathrm{B}}\left(\mid {\Psi }^{\left(2\right)}⟩\right)$, where $\mid {\Psi }^{\left(2\right)}⟩$ is the (arbitrary) pure two-photon state corresponding to $p=0$.