Towards a complete and continuous Wigner function for an ensemble of spins or qubits

We present a quasiprobability distribution function for ensembles of spin-half particles or qubits that has many properties in common with Wigner's original function for systems of continuous variables. We show that this function provides clear and intuitive graphical representation of a wide variety of states, including Fock states, spin-coherent states, squeezed states, superpositions, and statistical mixtures. The distribution is capable of simultaneously representing several angular momentum shells.

Figure 1

Isosurfaces of $\mathbb{W}$ for the state $\left|\uparrow \right.〉$. This Wigner-like function is positive within the blue regions and negative within the red regions.

Figure 2

Representation of the one-spin state $(\left|\uparrow \right.〉+\left|\downarrow \right.〉)/\sqrt{2}$ as a function in ${\mathbb{R}}^3$ (top), a function on the sphere (middle), and a point on the Bloch sphere (bottom). Blue indicates positive, red indicates negative, and green indicates 0.

Figure 3

Isosurfaces of Wigner functions for the triplet states $\left|\uparrow \uparrow \right.〉$, $\left(\left|\uparrow \downarrow \right.〉+\left|\downarrow \uparrow \right.〉\right)/\sqrt{2}$ and $\left|\downarrow \downarrow \right.〉$, and the singlet state $\left(\left|\uparrow \downarrow \right.〉-\left|\downarrow \uparrow \right.〉\right)/\sqrt{2}$. The Wigner functions are positive within the blue regions and negative within the red regions.

Figure 4

Wigner functions for Fock states with five spins represented on ${\mathbb{R}}^3$ (top row), the sphere (middle row), and the upside-down sphere (bottom row). The precise states plotted are (from left to right) $\left|\downarrow \downarrow \downarrow \downarrow \downarrow \right.〉$, $\sqrt{1/5}{S}_{+}\left|\downarrow \downarrow \downarrow \downarrow \downarrow \right.〉$, and $\sqrt{1/40}{\left(S_{+}\right)}^2\left|\downarrow \downarrow \downarrow \downarrow \downarrow \right.〉$.

Figure 5

Wigner functions for five spins. From left to right: the “up” state $\left|\uparrow \uparrow \uparrow \uparrow \uparrow \right.〉$, the “down” state $\left|\downarrow \downarrow \downarrow \downarrow \downarrow \right.〉$, the “cat” (or GHZ) state $\left|\uparrow \uparrow \uparrow \uparrow \uparrow \right.〉+\left|\downarrow \downarrow \downarrow \downarrow \downarrow \right.〉$, and the statistical sum of the up and down states. The label G indicates a hidden Gaussian-like peak.

Figure 6

Wigner-like functions for squeezed states $\exp (\beta {\left(S_{+}\right)}^2-\overline{\beta }{\left(S_{-}\right)}^2)\left|\uparrow \uparrow \uparrow \uparrow \uparrow \right.〉$. The left column shows $\beta =0.1$, and the right shows $\beta =0.2$. The spherical function ${\mathbb{W}}_{\mathbb{S}}$ has been plotted from two opposing viewpoints for greater clarity.