A Sensitive Quantumness Measure for One-Dimensional Continuous-Variable Systems

Abstract

For one-dimensional continuous-variable quantum systems such as single-mode quantum optical systems, we give a quantification of the quantumness of such a system's state, ρ, by introducing the measure of quantumness, Ξ, which works for all states, pure or mixed. Ξ is a measure which is universal, sensitive, monotonic, and unbounded. Ξ[ρ] yields a single positive value to quantify how nonclassical ρ is. Ξ employs phase space distributions to represent ρ and is a fixed function, Ξ[.], independent of the system, its environment or the type of state.

Figures (18)

• Figure 1: Left panel: $\xi(x,p)$, around the origin, for an odd 'cat' state $W_{x_0=6}(\theta = \pi)$. Middle panel: local quantumness ${\hbox{\boldmath${\Delta}$}}\xi(x,p)|_{\xi<0}$ for $W_{6}(\pi)$. Right panel: trend line for minimum value $\xi_-[W_{x_0}(\pi)]$ for odd 'cat' states with 'cat' size $2\, x_0$.
• Figure 2: Left to right: trend lines of $\Xi$ for positive 'cat', $W_{x_0}$ (compare Fig. \ref{['fig:Xi_I_pure_cats']}), Fock states, $W_{|n\rangle}$, and pure squeezed states, $W_{{s}}$. The plots confirm the monotonic growth with increasing quantumness of the states. The thick green curve for $\Xi[W_s]$ is the analytical expression discussed in \ref{['sec:PureSqueezedState']}.
• Figure 3: From left to right: Values of $\xi(x,p)$ for fourth excited Fock state $W_{|4\rangle}$. Values of local quantumness ${\hbox{\boldmath${\Delta}$}}\xi(x,p)|_{\xi<0}$ for $W_{|4\rangle}$. Values of $\xi(r)=\xi(\sqrt{x^2+p^2})$, up to quantum number $n=34$, for odd Fock states (red graphs) and even Fock states (green graphs), displaying respective minimum values oscillating between $\xi_-^{\rm odd}=-1/\pi$ and $\xi_-^{\rm even}\approx -0.131$.
• Figure 4: Normalized radial density $2\pi\, r\, W_{|n\rangle} (r)$ for Fock states $|7\rangle$, $|27\rangle$ and $|47\rangle$. Because of the rotational phase space symmetry of Fock states $\iint_{-\infty}^{\infty} dx dp \, W_{|n\rangle} = \int_{0}^{\infty} dr \, 2\pi r\, W_{|n\rangle} = 1$.
• Figure 5: Left panel: values of $\xi(x,p)|_{\xi<0}$ for a pure squeezed state $W_{{s}=4}^{[\mu=1]}$ (\ref{['eq:W_impure_squeezed']}), the zero contour is highlighted. Middle panel: values of local quantumness ${\hbox{\boldmath${\Delta}$}}\xi(x,p)|_{\xi<0}$ for the same state. Right panel: trend line for minimum value $\xi_-[W_{{s}}^{[\mu=1]}]$ for pure squeezed states (\ref{['eq:W_impure_squeezed']}).