Statistical mixtures of states can be more quantum than their superpositions: Comparison of nonclassicality measures for single-qubit states

A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan $P$ function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\tau$, related to the minimal amount of Gaussian noise which changes a nonpositive $P$ function into a positive one; (2) the nonclassical distance $D$, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance $D$ for a fixed value of both the depth $\tau$ and NP in the whole range $[0,1]$ of their values, as well as the NP for a given value of $\tau$ such that $\tau >0.3154$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\tau$ for a given $D$, NP versus $D$, and $\tau$ versus NP. We also show the “relativity” of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.

Figure 1

Allowed values of the nonclassicality measures for single-qubit states: (a) nonclassical distance $D$ versus nonclassical depth $\tau$, (b) negativity potential NP versus $\tau$, and (c) $D$ versus NP. The points correspond to a Monte Carlo simulation of ${10}^5$ states $\rho$. Each point is plotted for $\left[D\right(\rho ),\tau (\rho \left)\right]$ in (a) and analogously for panels (b) and (c). The vertical broken line in panel (b) is plotted at ${\tau }_0\approx 0.3154$. The boundaries are given by pure states ${\rho }_{\mathrm{P}}$ [vertical red lines in the far right of (a) and (b), and the red diagonal line in (c)], completely mixed states ${\rho }_{\mathrm{M}}$ (solid red upper curves), as well as partially mixed states ${\rho }_{+}$ (bottom broken lines) and ${\rho }_{\mathrm{opt}}$ [corresponding to blue points right above the curve for ${\rho }_{\mathrm{M}}$ in (b)]. In (b), it is barely visible that ${\rho }_{\mathrm{M}}$ is not the upper bound for $\tau <{\tau }_0$. Thus, this region is magnified in Fig. 2. Note that, in a mathematical sense, there are no states corresponding exactly to the broken lines at the bottom of (a) and (b) for $0<\tau \le 1$ and $D=\mathrm{NP}=0$. However, one can find states being arbitrarily close to these lines.

Figure 2

(a) The inset of Fig. 1 showing, in greater detail, the boundaries for the NP versus nonclassical depth $\tau$. These boundaries are reached by the partially mixed optimal states ${\rho }_{\mathrm{opt}}$ for $\tau <{\tau }_0$ and completely mixed states ${\rho }_{\mathrm{M}}$ for $\tau \ge {\tau }_0$. For clarity, we do not plot here points corresponding to our Monte Carlo simulation shown in Fig. 1. (b) Optimal parameters $p_{\mathrm{opt}}=\langle 1|{\rho }_{\mathrm{opt}}|1\rangle$ and $x_{\mathrm{opt}}=|\langle 0|{\rho }_{\mathrm{opt}}|1\rangle |$ as a function of $\tau$. These parameters are discontinuous at $\tau ={\tau }_0$, but, for clarity, we have plotted the red and green vertical connecting lines at this point.

Figure 3

Particular single-qubit states ${\rho }_n$ defined explicitly in Table 2 and plotted in analogy to Fig. 1. As in Fig. 1, here the boundaries are given by pure states ${\rho }_{\mathrm{P}}$, completely mixed states ${\rho }_{\mathrm{M}}$, together with the partially mixed states ${\rho }_{+}$ and ${\rho }_{\mathrm{opt}}$. There are no states corresponding exactly to the broken lines and empty circles. Any inequality listed in Table 3 can be satisfied by properly choosing pairs of these states. This analysis demonstrates the relativity of nonclassicality measures.