Extending Hudson’s theorem to mixed quantum states

According to Hudson’s theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions.

Figure 1

(Color online) (a) Upper bound on non-Gaussianity $\delta {\left[\rho ,{\rho }_G\right]}^{ex}$ derived with the Lagrange multiplier method. The double thin orange line marks the boundary between the two branches of solutions. The dotted blue line indicates the intersection with the plane $\mu \left[\rho \right]=1$ (also noted ${\delta }^{u.ult}$ in Fig. 2) while the double thick blue line shows the intersection with the plane $\mu \left[{\rho }_G\right]=1$. The red thick straight line denotes the “left” extremity of the surface. (b) Lower bound on non-Gaussianity $\delta {\left[\rho ,{\rho }_G\right]}_{\text{CS}}$ implied by the Cauchy-Schwarz inequality. We plot this lower bound only up to the intersecting line of $\delta {\left[\rho ,{\rho }_G\right]}_{\text{CS}}$ and $\delta {\left[\rho ,{\rho }_G\right]}^{ex}$ ($\mu \left[{\rho }_G\right]=\mu \left[\rho \right]$ and $\delta \left[\rho ,{\rho }_G\right]=0$).

Figure 2

(Color online) Ultimate upper bound ${\delta }^{u.ult}$ (blue solid line) on the non-Gaussianity of states with positive Wigner function as a function of the purity of corresponding Gaussian state $\mu \left[{\rho }_G\right]$. A lower estimate ${\delta }^{l.ult}$ (black dashed line) on the ultimate upper bound that was obtained for a mixture of coherent states. The two curves limit the region where the tight ultimate upper bound on non-Gaussianity must be located.