Witnessing the degree of nonclassicality of light

We introduce an experimentally accessible method to measure a unique degree of nonclassicality, based on the quantum superposition principle, for arbitrary quantum states. We formulate witnesses and test a given state for any particular value of this measure. The construction of optimal tests is presented as well as the general numerical implementation. We apply this approach on examples such as squeezed states, and we show how to formulate conditions to certify a particular degree of nonclassicality for single- and multimode radiation fields.

Figure 1

Schematic representation of the application of the Hahn-Banach separation theorem. The closed, convex, nested sets ${\mathcal{M}}_r$ are depicted for several $r$. The degree of nonclassicality is determined by tangent hyperplanes. Here, all states on the right-hand side of this tangent cannot be elements of ${\mathcal{M}}_r$. By the determination of the parameter $b_r$, it is possible to decide whether a state lies in ${\mathcal{M}}_r$ or not.

Figure 2

The maximal projection, $b_1=b_1\left(\right|\psi \rangle \langle \psi \left|\right)$ (solid line), of the even coherent state with the set of states with a minimal degree of nonclassicality ${\mathcal{S}}_1$ is shown in dependence of the coherent amplitude $\left|\beta \right|$ of the coherent components of $|\psi \rangle$. The dashed lines depict the limiting values 1 and $0.5$ for $\left|\beta \right|\to 0$ and $\left|\beta \right|\to \infty$, respectively.

Figure 3

The bounds $b_r=b_r\left(\right|\psi \rangle \langle \psi \left|\right)$ (solid line, $r=1$; dashed line, $r=2$; dotted line, $r=3$) are shown in dependence of the squeezing parameter $\xi$, which defines the squeezed vacuum state $|\psi \rangle$. We find that a stronger squeezing yields a smaller bound to be violated to verify a degree of nonclassicality above $r$.