Gaussian states and geometrically uniform symmetry

Quantum Gaussian states can be considered as the majority of the practical quantum states used in quantum communications and more generally in quantum information. Here we consider their properties in relation to the geometrically uniform symmetry, a property of quantum states that greatly simplifies the derivation of the optimal decision by means of the square root measurements. In a general framework of the $N$-mode Gaussian states we show the general properties of this symmetry and the application of the optimal quantum measurements. An application example is presented to quantum communication systems employing pulse position modulation. We prove that the geometrically uniform symmetry can be applied to the general class of multimode Gaussian states.

Figure 1

A pure Gaussian state $\left|\psi \right(p)\rangle$ is sent through a Gaussian channel specified by the triplet $(E,\ell ,F)$. The state at the output is still Gaussian, but in general mixed and described by the density operator $\rho (\psi \left(p\right))$.

Figure 2

Squared inner product $\Gamma =|\langle r{e}^{i\theta },\alpha |0,0\rangle {|}^2$ versus the phase $\theta$ for $r=1.0$ and two values of the displacement.

Figure 3

Representation of the states $|z,\alpha \rangle$ and $|0,0\rangle$ in the phase space for different values of $z=r{e}^{i\theta }$. Pictorially the noise variances are represented by a tilted “error ellipse.”

Figure 4

Error probability of quantum 8-PPM for coherent states and three values of squeezing. For $z=-0.1$ we must have $N_R\ge 3.3\times {10}^{-3}$, for $z=-0.5$, $N_R\ge 0.09$, and for $z=-1$, $N_R\ge 0.46$.