Convexity properties of superpositions of degenerate bipartite eigenstates

The entanglement content of superpositions of pairs of degenerate eigenstates of a bipartite system is considered in the case that both eigenstates are also eigenstates of the $z$ component of the total angular momentum. It is shown that the von Neumann entropy of the state that is obtained tracing out one of the parts of the system has a definite convexity (concavity) as a function of the superposition parameter and that its convexity (concavity) can be predicted using a quantity of information that measures the entropy shared by the states at the extremes of the superposition. Several examples of two-particle systems, whose eigenfunctions and density matrices can be obtained exactly, are analyzed thoroughly.

Figure 1

The von Neumann entropy as a function of the superposition parameter $\alpha$, for states $\left|{\psi }_{\alpha }\right.〉=\sqrt{\alpha }\left|n,m,l,p\right.〉+\sqrt{1-\alpha }\left|n^{\prime },m^{\prime },l^{\prime },p^{\prime }\right.〉$ with $\lambda =0$ (noninteracting oscillators). Note that a negative quantum number, $-n$, is denoted as $\underline{n}$ in the legends. (a) For $|m+p|=1$, the superpositions of states that are shown are $\left|1,1,2,0\right.〉$ with $\left|0,0,3,-1\right.〉$ (red full), $\left|0,0,3,-1\right.〉$ with $\left|0,0,3,1\right.〉$ (blue full), and $\left|0,0,3,-2\right.〉$ with $\left|3,1,0,0\right.〉$ (green dots). (b) For $|m+p|=2$, the superpositions of states that are shown correspond to $\left|0,0,2,-2\right.〉$ with $\left|0,0,2,2\right.〉$ (black full), $\left|0,0,2,-2\right.〉$ with $\left|1,1,1,1\right.〉$ (blue full), $\left|0,0,2,2\right.〉$ with $\left|1,-1,1,-1\right.〉$ (green dots), and $\left|1,1,1,1\right.〉$ with $\left|1,-1,1,-1\right.〉$ (full red). See the corresponding values of $Q_c$ in Table 1.

Figure 2

The von Neumann entropy as a function of the combination parameter $\alpha$ for states $\left|{\psi }_{\alpha }\right.〉=\sqrt{\alpha }\left|n,m,l,p\right.〉+\sqrt{1-\alpha }\left|n^{\prime },m^{\prime },l^{\prime },p^{\prime }\right.〉$ of two interacting oscillators with $\lambda =0.7$ and $|m+p|=1\text{and}2$. The combinations of states that are shown are $\left|0,-2,0,0\right.〉$ with $\left|0,2,0,0\right.〉$ (red), $\left|1,-2,0,0\right.〉$ with $\left|1,2,0,0\right.〉$ (green), $\left|1,-1,0,0\right.〉$ with $\left|1,1,0,0\right.〉$ (blue), and $\left|2,-1,0,0\right.〉$ with $\left|2,1,0,0\right.〉$ (black).

Figure 3

The von Neumann entropy as a function of $\alpha$ for $L=2$ states of two electrons on a surface of a sphere. The states are a linear combination of the form shown in Eq. (51). The entropy with $\left|M\right|=1$ is depicted with the black line and that with $\left|M\right|=2$ is depicted with a red dashed line. In Table 3 the values of the entropies, the criterion prediction, and convexities of the curves are shown.

Figure 4

The von Neumann entropy as a function of the combination parameter $\alpha$, for states $\left|{\psi }_a\right.〉$ from Eq. (54). The symmetric combinations ($l=l^{\prime }$ and $m=-m^{\prime }$) $(l_{\alpha =0},m_{\alpha =0},l_{\alpha =1},m_{\alpha =1})=(1,-1,1,1),(3,-3,3,3)$, and $(2,-1,2,1)$ are shown with black, red pointed, and green dashed curves, respectively. The asymmetric combinations $(l_0,m_0,l_1,m_1)=(2,2,2,1),(2,-2,1,1)$ are shown with blue dashed and brown dash-dotted curves, respectively.

Figure 5

(a) The von Neumann entropy as a function of the combination parameter $\alpha$, for the states of Eq. (61) with $\ell =3$ and $L=\left|M\right|$. $\left|M\right|$ and $L$ values for each curve are given in the legend. (b) The von Neumann, remaining, and not-shared entropies calculated for the cases shown in (a) are shown using black, red, and blue solid circular dots, respectively. The square dots correspond to the not-shareable entropy calculated using the canonical angular momentum basis and the triangular ones correspond to the entropy calculated using another basis (see the text for details).